A complex systems approach to the emergence of

Transcription

A complex systems approach to the
emergence of animal territoriality
Jonathan R. Potts
School of Biological Sciences
Bristol Centre for Complexity Sciences
University of Bristol
A dissertation submitted to the University of Bristol in
accordance with the requirements of the degree of
Doctor of Philosophy in the Faculty of Science.
July 2012
Word count: 38,000
Abstract
I present an agent-based model of animal movements and scent-mediated interactions whereby territories emerge as dynamic entities, though moving on a
time-scale much slower than that of the animals. Simulation analysis suggests
that the territory border movement depends upon the ratio of two quantities:
the so-called active scent time, measuring how long olfactory cues are recognised by conspecifics as fresh, and the time it takes for an animal to cover
its territory. By examining the interplay of adjacent territory boundaries, I
give analytic insights into this dependence. I also construct an approximate
analytic model of animal movement within dynamic territories that enables
quantification of the active scent time from the location distribution of animals. Fitting this to data on a red fox (Vulpes vulpes) population before and
after an outbreak of sarcoptic mange, I show how foxes change their behaviour
as a result of rapid declines in population. Finally, I examine an extension
of my model to the case where animals have fidelity towards a central place,
such as a den or nest site. In this case, stable home ranges emerge from the
territorial dynamics, enabling insights to be given into the mechanisms behind
allometric scaling laws of space use. This thesis represents the first example of
territorial emergence from a mechanistic model of individual-level movement
and interaction processes.
iii
Dedication and acknowledgements
It has been an immense privilege to be supervised by Luca Giuggioli and
Stephen Harris, who have helped me greatly during my PhD studies, often far
beyond the call of duty. My first words of thanks go to them. I also thank
my progress assessment monitors, Innes Cuthill and Mario di Bernardo, for
helpful conversations and fresh perpectives on my work, and my comrades
in the Bristol Centre for Complexity Science and the Mammal Group for fun
times and interesting conversations. For financial support, I thank the EPSRC
(grant number EP/E501214/1). On a more personal note, I thank my parents,
sister, in-laws, out-laws and friends for providing me with a life outside the
confines of science research. Finally, this thesis is dedicated to my wife, Anna,
for her ever-present love and support.
v
Author’s declaration
I declare that the work in this dissertation was carried out in accordance
with the requirements of the University’s Regulations and Code of Practice
for Research Degree Programmes and that it has not been submitted for any
other academic award. Except where indicated by specific reference in the
text, the work is the candidate’s own work. Work done in collaboration with,
or with the assistance of, others, is indicated as such. Any views expressed in
the dissertation are those of the author.
SIGNED: ............................................................. DATE:..........................
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Contents
1 Introduction
1
1.1
A complexity science approach . . . . . . . . . . . . . . . . . .
2
1.2
Mathematically analysing the complex system . . . . . . . . . .
4
1.3
The red fox: a paradigmatic territorial species . . . . . . . . . .
5
1.4
Comparisons with previous approaches . . . . . . . . . . . . . .
6
1.5
Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2 A simulation model of territorial emergence
9
2.1
Modelling and simulation methods . . . . . . . . . . . . . . . .
9
2.2
Results of the 1D system . . . . . . . . . . . . . . . . . . . . . .
12
2.2.1
Single file diffusion of the territorial border . . . . . . .
12
2.2.2
First passage times to cross the territory . . . . . . . . .
12
2.2.3
Territorial animal movement and home range overlap
.
15
Results of the 2D system . . . . . . . . . . . . . . . . . . . . . .
20
2.3.1
Varying the animal movement process . . . . . . . . . .
20
2.3.2
Subdiffusive territory border movement . . . . . . . . .
22
2.3.3
The effect of correlation on the animal’s MSD . . . . . .
25
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.3
2.4
3 An analytic model of territorial animal movement
27
3.1
A Fokker-Planck approach in 1D . . . . . . . . . . . . . . . . .
27
3.2
The mean square displacement of an animal . . . . . . . . . . .
29
3.3
Comparison with the agent-based model . . . . . . . . . . . . .
33
3.4
Extension to a 2D correlated walk . . . . . . . . . . . . . . . .
35
3.4.1
Circular territories . . . . . . . . . . . . . . . . . . . . .
37
3.4.2
Square territories . . . . . . . . . . . . . . . . . . . . . .
40
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
3.5
4 Application to data on urban red foxes
47
4.1
Data collection and analysis . . . . . . . . . . . . . . . . . . . .
48
4.2
Territorial dynamics pre- and post-mange . . . . . . . . . . . .
50
ix
4.3
Inferring active scent time from location data . . . . . . . . . .
54
4.4
Evolutionary invasion analysis . . . . . . . . . . . . . . . . . . .
55
4.5
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
5 Towards an analytic model of territory border movement
59
5.1
An anti-symmetric exclusion process . . . . . . . . . . . . . . .
60
5.2
Asymptotic analysis . . . . . . . . . . . . . . . . . . . . . . . .
63
5.3
The continuum limit . . . . . . . . . . . . . . . . . . . . . . . .
67
5.4
Anti-symmetric exclusion and territory border movement . . .
71
5.5
Analysing territory border movement in 1D: a way forward . .
73
5.6
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
6 Home range formation in territorial central place foragers
75
6.1
Agent-based simulations of territorial central place foragers . .
76
6.2
An analytic model of a central place forager within its territory
78
6.2.1
Movement in 1D . . . . . . . . . . . . . . . . . . . . . .
79
6.2.2
Movement in 2D . . . . . . . . . . . . . . . . . . . . . .
84
6.3
The marginal distribution of the animal . . . . . . . . . . . . .
86
6.4
Obtaining active scent time from animal position data . . . . .
88
6.5
Home range patterns and relations to allometry . . . . . . . . .
89
6.6
Comparison with previous approaches . . . . . . . . . . . . . .
91
6.6.1
The reaction-diffusion approach . . . . . . . . . . . . . .
91
6.6.2
A numerical comparison . . . . . . . . . . . . . . . . . .
92
6.6.3
An analytic comparison . . . . . . . . . . . . . . . . . .
94
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
6.7
7 Discussion and conclusions
99
7.1
Ecological implications . . . . . . . . . . . . . . . . . . . . . . . 100
7.2
Comparisons with previous work . . . . . . . . . . . . . . . . . 102
7.3
Possible future directions . . . . . . . . . . . . . . . . . . . . . 104
7.4
7.3.1
Building an analytic theory of border movement . . . . 104
7.3.2
Using approximate reaction-diffusion approaches . . . . 104
7.3.3
Extensions to the model . . . . . . . . . . . . . . . . . . 105
7.3.4
Epidemiological applications
. . . . . . . . . . . . . . . 106
Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Appendices
109
A Boundary return time calculation
109
B Solution of the Fredholm integral equation from section 5.1 111
C Solution of Fokker-Planck equation from section 5.3
113
D Asymptotic continuous-time expressions from section 5.3
117
E Matlab code for the distribution of a territorial animal
119
F Movie captions
123
F.1 Territory movement . . . . . . . . . . . . . . . . . . . . . . . . 123
F.2 Time dependence of the probability distribution . . . . . . . . . 123
F.3 Dynamics of territorial acquisition . . . . . . . . . . . . . . . . 124
List of Tables
2.1
Glossary of the key symbols used in Chapter 2 . . . . . . . . .
26
3.1
45
4.1
Details of fox radio-tracking data . . . . . . . . . . . . . . . . .
49
4.2
Glossary for Chapter 4 and best-fit values . . . . . . . . . . . .
57
5.1
74
6.1
97
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List of Figures
1.1
Representation of the reduced analytic model . . . . . . . . . .
4
2.1
Representation of the simulation model . . . . . . . . . . . . .
10
2.2
Gillespie versus synchronous simulations . . . . . . . . . . . . .
11
2.3
Universal curves for border movement in 1D . . . . . . . . . . .
13
2.4
Boundary return times . . . . . . . . . . . . . . . . . . . . . . .
16
2.5
Dynamics of animal and territorial border locations . . . . . . .
17
2.6
Standard deviation and home range size . . . . . . . . . . . . .
18
2.7
Exclusivity of space use . . . . . . . . . . . . . . . . . . . . . .
19
2.8
Contour plot of animal utilisation distributions . . . . . . . . .
21
2.9
Universal curves for border movement in 2D . . . . . . . . . . .
23
2.10 Oscillations in the animal MSD . . . . . . . . . . . . . . . . . .
24
3.1
Animal within subdiffusing territory borders . . . . . . . . . . .
28
3.2
Mean square displacement of 1D territorial animal . . . . . . .
32
3.3
Comparison of the simulation and analytic models . . . . . . .
36
3.4
Animal MSD in circular territory . . . . . . . . . . . . . . . . .
39
3.5
Animal MSD in square territory
. . . . . . . . . . . . . . . . .
43
4.1
Fox data with theoretical utilisation distribution . . . . . . . .
51
4.2
Home range overlap due to border movement . . . . . . . . . .
53
4.3
Time lag for territory acquisition . . . . . . . . . . . . . . . . .
55
5.1
Reduced model of territorial dynamics . . . . . . . . . . . . . .
59
5.2
Divergence timescales of three regimes . . . . . . . . . . . . . .
65
5.3
Comparison with simulations . . . . . . . . . . . . . . . . . . .
66
5.4
Comparison with asymptotic regimes . . . . . . . . . . . . . . .
67
5.5
Anti-symmetric exclusion and territory border movement . . .
72
6.1
Simulation output for territorial central place foragers . . . . .
78
6.2
The 1D model of territorial central place foragers . . . . . . . .
79
6.3
Simulation vs. analytic model for central place foragers . . . .
87
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6.4
Home ranges and allometry . . . . . . . . . . . . . . . . . . . .
90
6.5
Comparison with a previous model of territory formation . . .
93
6.6
Analytic comparison with the reaction diffusion model . . . . .
95
Chapter 1
Introduction
Understanding how animal territories form and change is of great importance
to many areas of ecology, from conservation biology (Beier, 1993) to wildlife
management (McCarthy & Destefano, 2011), from predator-prey dynamics
(Lewis & Murray, 1993) to epidemiology (Kenkre et al., 2007). Though the
broad nature of the conspecific avoidance processes underlying territory formation, such as scent marking in mammals (Johnson, 1973; Hurst, 2005;
Arnold, 2009) or displays and vocalisations in birds (Penteriani et al., 2007;
de Kort et al., 2009), are well documented in many species across a wide range
of taxa (Brown & Orians, 1970), quantifying how territories emerge from the
movements and interactions of animals has remained elusive (Börger et al.,
2008). Perhaps the greatest complicating factor is determining how avoidance mechanisms, taking place at relatively small temporal and spatial scales,
give rise to the population-level phenomena of extended and lasting territorial patterns. Understanding this transition from ‘microscopic’ processes to
‘macroscopic’ patterns is an issue common to many areas of ecology research
(Levin, 1992). Such analysis requires a combination of the traditional tools
of statistical physics (Okubo & Levin, 2002) and the more modern complex
systems simulations approaches (Grimm & Railsback, 2005).
Theoretical studies have tended to assume that territories, and the related
concept of home ranges (Burt, 1943), emerge as stationary entities, prompting questions as to how stability arises from dynamically moving animals
(Moorcroft & Lewis, 2006; Van Moorter et al., 2009). Despite this, flexible
territory borders have been observed across a wide variety of taxa, including in
birds such as wrens (Troglodytes aedon), when there are contests over resources
(Johnson & Kermott, 1990), reptiles like bronze anoles (Anolis aeneus) during
territorial establishment (Stamps & Krishnan, 1998) and mammals, including
red foxes (Vulpes vulpes) during an epizootic of a terminal disease (Baker et al.,
1
2000) and timber wolves (Canis lupus) when population density or resource
availability changes (Van Ballenbergh et al., 1975). These observations have
a history that spans almost 80 years since the seminal paper of Huxley (1934),
introducing the idea that territories are like ‘elastic discs’ that deform and
change according to fluctuations in neighbour pressure. In this thesis, I show
that the elastic disc phenomenon is a natural outcome of a dynamical system
of moving animals interacting via conspecific avoidance mechanisms, so that
territories do not form as stable entities.
1.1
A complexity science approach
Whilst the precise definition of ‘complexity science’ is still a subject of some
debate (Ladyman et al., 2011), it can roughly be characterised as the study
of systems of interacting agents, whose properties cannot be completely understood simply by examining the behaviour of each individual. These systems are usually termed ‘complex systems’. They tend to exhibit so-called
‘emergent’ properties that are far more complicated to describe than the simple underlying rules from which the system is built. For example, interactions between self-driven particles, used to model fish schools and bird flocks
(Couzin et al., 2002), can be straightforward to describe, but may exhibit rich
and varied resulting patterns (Vicsek et al., 1995). These complex systems
contrast with simpler ones, such as the two-body system, see e.g. Goldstein
(1980), whose macroscopic properties can be described very straightforwardly.
For example, the motion of two objects under long-range gravitational forces
can be explained just by using Kepler’s three laws of planetary motion (see
e.g. Hawkins (2002) for Kepler’s original writings, translated into English).
From a behavioural ecology perspective, the complexity approach differs
from the traditional one, where animals are studied individually, and focusses
on collective behaviour, whereby animal interactions cause phenomena such as
herding, swarming and flocking (Camazine et al., 2003). Territory formation
is a prime example of this. By definition, territories only occur when animals
use mechanisms to defend an area from conspecifics (Burt, 1943), so cannot
arise without some form of interaction. Furthermore, as will be made apparent
in this thesis, simple rules describing animal movements and interactions (e.g.
Figure 2.1) can cause complicated and varied territorial structures to emerge
(e.g. Figure 2.8).
The theoretical tools for studying systems of moving, interacting agents
come from the broad area of statistical physics. Advances over several decades
have allowed scientists to make the transition from individual-level descriptions
2
to properties of the system as a whole, see e.g. Liggett (1985); Privman et al.
(1997). This requires building a probabilistic model of the system that incorporates both the random and deterministic aspects of the agents’ movements
and interactions. If the system is simple enough, it can be solved exactly, using
analytic tools. More complex models require a mixture of computer simulations and approximate analytic techniques to uncover the emergent properties.
In this thesis, I construct a model of territory formation based on a random
walk paradigm for animal movement (Okubo & Levin, 2002), together with
an interaction mechanism of conspecific avoidance mediated by scent marks.
Each animal in the model is assigned the same movement process, which may
be a diffusive, correlated or Lévy random walk, or ballistic motion (see section
2.3.1). It may also include some other deterministic aspect, such as a tendency
to move towards a den or nest site (Chapter 6).
As an animal moves, it deposits scent to deter conspecifics. After a finite
amount of time, called the active scent time, TAS , the conspecifics stop recognising the scent marks as fresh territorial messages. Therefore the territory
of an animal at any point in time t is the area that contains ‘fresh’ scent,
i.e. scent deposited between time t − TAS and t. The animal is free to move
anywhere except into the territories of others (see Figure 2.1 for a graphical
illustration, where the movement process is that of a nearest neighbour lattice
random walker).
This causes the terrain to subdivide into territories that exhibit slow, random movement (see the movie territory movement.gif on the supplementary
CD ROM, explained in appendix F.1). Each territory has a boundary, and
if two boundaries touch, I say that a territorial border is formed. Due to
the random aspect of a model animal’s movement, certain parts of its border
may fail to be visited within a time TAS , causing that part of the border to
decay and some interstitial area to form. This may allow another animal to
acquire the newly-freed area for its territory. The subsequent movement of the
territories is an example of an exclusion process (Liggett, 1985), since territories cannot be in the same place at the same time. Unlike classical exclusion
processes, where the size and shape of each particle is fixed, territories continually change. On the other hand, like other exclusion processes, the territory
borders exhibit subdiffusive movement (Harris, 1965; Landim, 1992), i.e. the
variance of the probability distribution increases sublinearly with time.
As a consequence of this border movement, the utilisation distibution of
an animal, i.e. its ‘home range’ (Burt, 1943), will gradually increase over time.
This causes home ranges to overlap with neighbours, despite the fact that territory borders tend to be contiguous. In field studies, border fluctuations may
3
not be noticeable since, by the time sufficient independent location fixes have
been obtained to measure territory size (Harris et al., 1990), the borders could
have changed. This will result in an apparent overlap between neighbouring
territories. However, this overlap is simply an artefact of the time scale for
the data collection, arising from shifting territory borders, and not an implicit
biological phenomenon.
1.2
Mathematically analysing the complex system
To fit my model to data on territorial animal movement, an approximate
analytic expression of movement within subdiffusive territorial borders is constructed (Figure 1.1). Though it is possible in principle to measure the animal
probability distributions directly from the simulations, averaging over sufficiently many stochastic realisations to obtain an accurate empirical distribution takes many hours of computational time, to obtain results for just a single
set of parameters. The analytic theory obviates the need for running extensive
simulations and, though approximate, is very close to the simulation output
for the biologically realistic cases where the border movement is much slower
than that of the animal (see Figures 3.3 from Chapter 3 and 6.3 from Chapter
6).
Figure 1.1: Representation of the reduced analytic model. Territory
borders have intrinsic random movement but are connected to one another by
springs, modelling the fact that smaller than average territories will tend to
grow, whereas larger ones will tend to shrink. The animals are constrained to
move within their territory borders.
A so-called ‘adiabatic approximation’ is used, which is valid whenever
there are two interacting processes such that one occurs over a much shorter
4
timescale than the other. In the general situation, one assumes that
PJ (A, B, t) ≈ PF (A, t|B)PS (B, t)
(1.1)
where PF is the probability distribution of the fast process, PS that of the slow
process and PJ the joint probability function. In the present case, the animal
movement is fast compared to the slow territory borders. Exact solutions for
the probability distribution of the borders and that of an animal within fixed
borders were found. Via equation (1.1), this gave an approximate solution to
the joint probability of the borders and the animal, from which the marginal
distribution of the animal, over both space and time, could be derived. This
marginal distribution was then fitted, using maximum likelihood techniques,
to spatio-temporal animal location data in order to infer information about
the underlying movement and interaction processes employed by territorial
animals.
1.3
The red fox: a paradigmatic territorial species
I applied the theoretical findings to data on a red fox (Vulpes vulpes) population in Bristol gathered over more than 30 years (Baker et al., 2001). Foxes
live in small groups, consisting of a mating pair, cubs and occasionally some
subordinate adults (Iossa et al., 2008). Each group resides in a territory, which
is defended from other groups, predominantly by depositing urine scent marks
(Goszczyński, 2002). Though foxes of all ages and sex deposit scent throughout their territory for a variety of communication purposes (Henry, 1977),
the dominant male plays the main role as territorial defender (Harris, 1980;
Arnold et al., 2011). When applying the model to foxes, each individual is
a male dominant, one for each territory, with female, subordinate or juvenile
foxes ignored in the simulations.
Most fox activity takes place between 20:00 and 04:00 (Saunders et al.,
1993) so ‘one day’ is considered as an 8-hour period throughout this thesis.
However, their behaviour varies according to the season (White et al., 1996).
The female gives birth to cubs in the spring (March-May). During this time,
she and her cubs remain in or near the den and the male is responsible for
bringing food from within its territory. By the beginning of the summer (JuneAugust), the den is abandoned. However, the male dominant still plays a role
in feeding the cubs, therefore will remain within its territory, whereas the cubs
spend most of their time in a smaller area well inside the territory borders.
Throughout these two seasons, extra-territorial behaviour is rare.
By the autumn (September-November), the cubs are fully grown and
5
may disperse, moving away from their parents’ territory to establish their
own, often many kilometers away (Trewhella et al., 1988). During this time,
invading dispersers perturb the territorial structure causing an increase in
aggressive encounters (White & Harris, 1994).
Dispersal continues during
winter (December-February), when dominant males will also tend to exhibit
extra-territorial activity as they seek to mate with females from other groups
(White et al., 1996).
During 1994-6, a sarcoptic mange epizootic swept through the Bristol’s
foxes, decimating the population (Baker et al., 2000). As each fox group was
killed by the disease, neighbours increased their territories to take advantage
of the newly vacated areas. This shows a certain flexibility in the territorial
structure, echoing the elastic disc hypothesis proposed by Huxley (1934). I
took advantage of this natural experiment by applying the theory to data
before and after the outbreak of mange. This enabled elucidation of the various
changes in the foxes’ behaviour as the population was in decline.
1.4
Comparisons with previous approaches
Theoretical studies of territoriality have often been made in conjunction with
the related concept of home range. Whilst a territory is an area deliberately
defended against conspecifics, a home range is the area used by an animal for
‘everyday’ activities, such as foraging or communicating (Burt, 1943). Therefore the home range of a single animal can be measured without any reference
to its interactions with neighbours. In the field this is done by analysing
location data, using techniques such as minimum convex polygons or kernel
estimation (Harris et al., 1990), or analysing the mean square displacement of
animal movements (Giuggioli et al., 2006). Theoretical studies of home range
formation have tended to use random walk models with either an attraction towards a central location (Holgate, 1971; Okubo, 1980), or an internal memory
(Van Moorter et al., 2009; Briscoe et al., 2001).
However, whilst a home range can be defined for a solitary animal, interactions are implicit in the concept of territoriality. Studying the movement of a single animal on its own only has limited ecological applications
since interactions are key to understanding space use across a wide range
of taxa (Jetz et al., 2004). Until now, most models of territory formation
that include interactions have followed Lewis & Murray (1993), where conspecific avoidance mechanisms are modelled via a mean-field approach, using
a reaction-diffusion formalism (Moorcroft & Lewis, 2006; Hamelin & Lewis,
2010). Though movements are built mechanistically from individual-level pro6
cesses when using this type of approach (Moorcroft et al., 1999), the interactions are not.
Although deterministic reaction-diffusion equations are in general viable
approximations to represent spatio-temporal stochastic processes, they are not
well suited to model systems in which the individual components are present
in low concentrations, e.g. Kang & Redner (1985); Van Kampen (1981), and
may give results that are in disagreement with the stochastic description,
e.g. Levin & Durrett (1994); McKane & Newman (2004). In the present case,
since scent marks are completely ignored by conspecifics after a fixed amount
of time, the distribution of active scent of an animal drops to zero at certain
places. Therefore I have ensured that my model is built from individual-level
descriptions of both the movements and the interactions to account for these
low concentrations of scent. As well as providing a modelling framework that
is closer to reality, this has enabled the details of the interaction mechanisms to
be inferred from the data, which has previously not been possible (Potts et al.,
2012).
Another class of models in the literature are those that consider the territory (or home range) as an input into the model rather than an outcome of
animal movement and interaction mechanisms, e.g. Wang & Grimm (2007);
Barraquand & Murrell (2011). Whilst these studies are useful for linking territoriality to ecological phenomena such as population dynamics (Araujo et al.,
2010) or spread of diseases (Smith & Harris, 1991; Smith & Wilkinson, 2002;
Salkeld et al., 2010), understanding how the size and shape of the territories
relate to the details of the animal-level processes is vital for parametrising
these models. This is particularly relevant regarding disease spread, where
the transmission occurs by interactions between individuals, so needs to be
related in a non-speculative way to the conjoining of neighbouring territories
or overlapping of home ranges.
Optimal foraging approaches provide another class of territorial and home
range models. The assumption is that animals maintain home range sizes that
are optimal for their daily foraging tasks (Schoener, 1983; Cuthill & Houston,
1997; Mitchell & Powell, 2004, 2007). These have been successful for modelling animals such as small birds (Kacelnik et al., 1981; Houston, 1987), who
need to find resources in highly competitive foraging environments to survive
(Houston & McNamara, 1993). Bristol’s foxes, on the other hand, have on
average 150 times more food in their territories than they need for survival
(Ansell, 2004), so are maintaining territories far larger than is necessary for
resource requirements. Consequently, an optimal foraging approach is not
suitable for analysing such populations.
7
1.5
Thesis outline
In Chapter 2, I introduce the agent-based model and give results of simulation
output. Particularly, I show how the territory border movement is related to
the active scent time and animal population density. Much of this has been
published in Giuggioli et al. (2011a), with some results having appeared in
Giuggioli et al. (2011b) and Giuggioli et al. (2012).
Chapter 3 is devoted to mathematical analysis of animal movement within
subdiffusive territory borders. I use a Fokker-Planck formalism to describe
the territory border movement and an adiabatic approximation to derive the
animal movement within these borders. The 1D version of this model, with
the animal modelled as a Brownian particle, was published in Giuggioli et al.
(2011b), whereas Giuggioli et al. (2012) extended the model to a 2D version
where the animal performs a correlated random walk.
Data on Bristol’s red fox population are analysed in Chapter 4. I take
advantage of a natural experiment, caused by the 1994-6 mange outbreak,
to fit the model of territorial animal movement to data before and after the
disease outbreak, using maximum likelihood techniques. These results have
recently been submitted for publication (Potts et al., in review).
The calculations in Chapter 5 explain how territory border movement can
be derived from the interactions of neighbouring territories. I use this to
outline a programme for constructing an analytic theory of territory border
movement that will obviate the need for simulation analysis. The results were
first published in Potts et al. (2011).
In Chapter 6, I consider territorial animal movement with a deterministic
aspect, that of attraction towards a central place such as a den or nest site.
Unlike the cases where the movement process is purely stochastic, stable home
ranges form. I show how to quantify the emergent home ranges from the
movement and interaction processes. These results were originally published
in Potts et al. (2012). Although not relevant to Bristol foxes, there are a
number of animal populations that appear to maintain stable home ranges,
see e.g. Börger et al. (2008).
Each Chapter from 2 to 6 concludes with a glossary of the various symbols used in the Chapter, together with their definitions. Chapter 7 contains
concluding remarks and discussion.
8
Chapter 2
A simulation model of
territorial emergence
2.1
Modelling and simulation methods
To understand how territories emerge from animal movements and interactions, I constructed a simulation environment based on scent-mediated conspecific avoidance. Animals are modelled as nearest-neighbour lattice random
walkers who deposit scent at each lattice site they visit. The scent lasts for a
finite time, the active scent time TAS , after which it is no longer present (see
Table 2.1 for a definition of TAS and the other symbols used in this chapter).
An animal’s territory is defined as the set of lattice sites that contain scent
that it has deposited in the past TAS timesteps, and the animal is constrained
to move within areas that are not part of another’s territory.
In 2D, this means that no animal can move into a lattice site containing
scent of another individual. Figure 2.1 gives a graphical illustration of this
mechanism. In 1D, upon visiting a lattice site that contained foreign scent,
the individual moves back to the site it came from. Individuals with these
movement and interaction mechanisms are called territorial random walkers.
To perform the 2D simulations, the Gillespie algorithm is used (Gillespie,
1976), modelling each individual as a continuous-time random walker with its
own internal clock determining when its next jump should occur. In 1D, the
walkers are discrete time random walkers with synchronous updating, since
this gives identical results to the Gillespie algorithm method, but requires less
simulation time (Figure 2.2). In all simulations, the lattice spacing is a. The
jump rate between nearest neighbours for the Gillespie algorithm is F and the
discrete time-steps in 1D are ∆t = 1/(2F ).
In 1D, the simulations consist of 2 animals on a finite lattice with periodic
9
6
Scent of animal 1
Scent of animal 2
5
Position of animal 1
Position of animal 2
4
3
2
1
1
2
3
4
5
Figure 2.1: Representation of the simulation model. This represents a
hypothetical snapshot in time of the position of two territorial random walkers
(animals), the red and blue dots, and their territories, represented by the red
and blue open circles, respectively. If a red (blue) open circle is present at a
lattice site, it means that the red (blue) animal has been in that location some
time in the past TAS timesteps. The absence of any scent marks at coordinates
(5,1), (2,3) and (2,4) implies that no animal has occupied those coordinates
within a time TAS , i.e. this is interstitial area. The next time the blue animal
moves, it can go to any of the four adjacent lattice sites with equal probability,
whereas the red animal is constrained to move either up or right.
boundary conditions and N sites, so that the population density is ρ = 2/L
where L = N a is the length of the lattice. Since each animal has identical
random movement statistics, it is sufficient to use just 2 animals to capture
the properties of the emergent dynamics in 1D. The 2D simulations took place
on a square lattice of N × N sites with periodic boundary conditions. They
consisted of 25 animals, so that ρ = 25/L2 . Results in 2D were obtained by
averaging over 100 runs. Since 1D simulations were much faster than 2D, and
only contained 2 animals rather than 25, the results were found by averaging
over 10,000 simulation runs. The simulations were coded in C and compiled
on Windows XP OS.
In 1D, I also ran simulations to calculate the marginal probability distribution of a territorial random walker. This requires ensuring that each animal
already has a territory at the beginning of each simulation run. A biologi10
cally relevant initial scent profile is a curve with minimum values at the scent
boundaries and a maximum at the lattice site where the animal was initially
present. Since territories can only move when the scent profile at the boundary of at least one of two neighbouring animals is equal to zero, an initial
scent profile with this feature is used. The shape of the curve, which interpolates between zero at the boundaries and the maximum corresponding to the
animal’s position, was obtained by averaging over stochastic simulations that
were run starting with a spatially uniform distribution of animals and with
no territories present. The moment when this average was computed was just
after the border mean square displacement (MSD) had reached its asymptotic
regime (see section 2.2.1).
0.16
Dimensionless MSD
0.14
0.12
0.1
0.08
2
0.06
T ρ D=1
0.04
T ρ2D=1.4
0.02
TASρ D=1.8
AS
AS
2
0
0
0.5
1
1.5
Dimensionless time
2
2.5
6
x 10
Figure 2.2: Gillespie algorithm versus synchronous simulations. The
dimensionless territory border MSD ρ2 ∆x2b = ρ2 h(xb −hxb i)2 i is plotted against
dimensionless time 2F t = t/∆t, simulated in 1D using the Gillespie algorithm (solid lines) and discrete-time random walks with synchronous updates
(dashed lines). The notation h. . . i represents an average over the stochastic
realizations of multiple (in this case 10,000) trajectories starting with the same
initial conditions. The discrete-time random walk gives similar results to the
Gillespie algorithm, whilst taking about half the computational time.
11
2.2
2.2.1
Results of the 1D system
Single file diffusion of the territorial border
For each pair of parameters (ρ, TAS ), the MSD of a single territory border
√
position xb is asymptotically proportional to t. This is because the border
movement is a type of single file diffusion process, i.e. the two borders cannot
be in the same place at the same time, the system is 1D and overdamped
(Nelissen et al., 2007). At long times
∆x2b (t)
∼K
r
t
,
2F
(2.1)
where K depends on both TAS and ρ, and ∆x2b = h(xb − hxb i)2 i is the border
MSD. The time t is divided by 2F = 1/∆t in equation (2.1) to ensure that K
has the dimensions of a diffusion constant (length2 time−1 ), so can be readily
compared with the animal’s diffusion constant D = a2 F .
The quotient K/D decreases exponentially as the dimensionless quantity
Z1D = TAS ρ2 D is increased (Figure 2.3a). This universal trend occurs regardless of the choices of TAS , ρ and D that were made when running the simulations. To explain this, notice that the normalising time 1/[ρ2 D] is closely
related to the first passage time for an animal to cross its territory from one
border to another, and then return to where it started, the so-called boundary
return time TBR . If an animal is unable to cross its territory and return within
a time TAS then one or other of the borders will move. Therefore as TAS /TBR
is increased, one would expect the border diffusion constant K to decrease.
Section 2.2.2 makes precise the relation between 1/[ρ2 D] and TBR .
The MSD of the territory width also obeys a trend dependent on Z1D ,
universal amongst choices of TAS and ρ (Figure 2.3b). In this case the MSD
reaches a saturation value, denoted by s∗ , as t → ∞. Therefore at long times,
the probability distribution of the territory width is independent of time, with
mean 1/ρ and variance dependent on Z1D .
2.2.2
First passage times to cross the territory
Though an exact calculation of TBR for all values of TAS is complicated by
the non-Markovian nature of the model, the two extreme situations, TAS = 0
and TAS = ∞, can be computed analytically.
If TAS = 0, each territory consists only of the lattice site where the animal
is located. This allows the animals to be modelled as two particles on a
periodic lattice that cannot cross one another. Their positions are denoted
x and y and it is assumed that both the number of lattice sites, N , and the
12
−0.5 b)
a)
Best fit
aρ=0.020
aρ=0.022
aρ=0.025
aρ=0.029
aρ=0.033
aρ=0.040
0
−1
−1.5
10
log10(K/D)
0.5
log (s*/L2)
1
−0.5
−2
−1
−1.5
0
−2.5
0.5
1
1.5
2
2.5
3
0.6
Z1D
0.8
1
1.2
1/4
Z1D
Figure 2.3: Universal curves for border movement in 1D. Panel (a)
shows the dependency of the territory border diffusion constant K, normalised
by dividing by the animal’s diffusion constant D, on the dimensionless product
Z1D = TAS ρ2 D. Notice that the various points all lie on the same universal
curve, regardless of the value of ρ used. Panel (b) shows the how the longtime MSD of the territory width s∗ depends on Z1D . A similar universality is
observed, regardless of the population density in the simulations.
normalised difference (y − x)/a are even, to ease calculations (the cases where
either or both are odd give similar results, stated at the end of the calculation).
Using a diffusion graph transform (Burioni et al., 2002), I write
y+x
,
2a
y−x
s=
(mod N).
2a
c=
(2.2)
The variable c is the centroid of x and y, normalised by dividing by a, whereas
s is half the number of lattice sites from x to y when measured from the left
to the right. The advantage of this transformation is that TBR becomes twice
the first-passage time for s to go from 0 to N − 1, and then jump to N next
step, where N = 0 (mod N ).
If x 6= y so that s > 0 then the possible movements of (x, y) are as follows,
each taking place with probability 1/4. To the right of each possibility is the
corresponding movement of the pair (c, s).
13
(x, y) 7→ (x + a, y + a),
(c, s) 7→ (c + 1, s),
(x, y) 7→ (x − a, y − a),
(c, s) 7→ (c − 1, s),
(x, y) 7→ (x − a, y + a),
(c, s) 7→ (c, s + 1),
(x, y) 7→ (x + a, y − a),
(c, s) 7→ (c, s − 1),
(2.3)
where addition and subtraction are assumed to be modulo L for the variables
x and y, and modulo N for c and s. If x = y so that s = 0 and the previous
move was
(x + a, y − a) 7→ (x, y),
(c, s − 1) 7→ (c, s),
then the next move is
(x, y) 7→ (x + a, y − a),
(c, s) 7→ (c, s − 1).
If x = y (s = 0) and the previous move was
(x − a, y + a) 7→ (x, y),
(c, s + 1) 7→ (c, s),
then the next move is
(x, y) 7→ (x − a, y + a),
(c, s) 7→ (c, s + 1).
To perform the aforementioned first-passage calculation for s, let Ps (n, t) be
the probability of having s = n at time t. I adopt the notation that Ps (0, t)
means the probability of having s = 0 given that s = 1 on the previous step,
whilst Ps (N, t) means the probability of having s = N (N > 0) given that
s = N − 1 on the previous step. This notation is convenient for performing
the first-passage time calculation and gives rise to the following discrete-time
master equation:
1
Ps (0, t + ∆t) = Ps (1, t),
4
1
1
Ps (1, t + ∆t) = Ps (0, t) + Ps (2, t) + Ps (1, t),
4
2
1
1
1
Ps (n, t + ∆t) = Ps (n − 1, t) + Ps (n + 1, t) + Ps (n, t), if 2 ≤ n ≤ N − 2,
4
4
2
1
1
Ps (N − 1, t + ∆t) = Ps (N, t) + Ps (N − 2, t) + Ps (N − 1, t),
4
2
1
Ps (N, t + ∆t) = Ps (N − 1, t).
(2.4)
4
14
Employing techniques from Redner (2007, section 2), a direct calculation (Appendix A) shows that for TAS = 0,
TBR = L(2L − a)/D.
(2.5)
In the case N is odd, equation (2.5) is replaced by TBR = (L − a)(2L − 3a)/D.
If N is even but y − x is odd then TBR = (L − 2a)(2L − 5a)/D.
In the other extreme case, where TAS = ∞, the territories stay stationary
once formed, so TBR is determined entirely by that initial formation. For a
random walk restricted to move on a line segment, the mean first-passage time
to go from one (reflecting) edge to the other edge is the square of the length
of the segment (section 2.4 from Redner (2007)). Assuming that one of the
territories has N − m lattice sites and the other N + m, then the average mean
first-passage time is equal to [(N + m)2 + (N − m)2 ]/(4F ) = (N 2 + m2 )/(2F ).
Let PT (m) be the probability that the initial territories have N −m and N +m
sites. Then
TBR
L
1 X 2
1
=
(N + m2 )PT (m) =
F
F
m=0
"
N2 +
N
X
m=0
#
m2 PT (m) .
(2.6)
Using numerical simulations, averaged over 10,000 different runs, for various
P
2
2
2
L, it turns out that N
m=0 m PT (m) ≈ 0.036N so TBR ≈ 1.036L /D. For
intermediate values of TAS , simulation analysis revealed that unless TAS and
ρ are both small, TBR is roughly L2 /D (Figure 2.4). Since L = 1/ρ, it follows
that 1/[ρ2 D] . TBR . 2/[ρ2 D], with TBR ≈ 1/[ρ2 D] unless Z1D is very small.
2.2.3
Territorial animal movement and home range overlap
Simulation analysis revealed that the MSD of an animal was diffusive at short
times and subdiffusive at long times (see Figure 2.5). Whilst mathematical
analysis of animal movement in subdiffusing borders is given in Chapter 3,
here I compare both border and animal MSDs measured directly from the
simulations. As the borders move, the total area that the animal has occupied
over a time window T∗ gradually increases. This area is often referred to as
the home range of the animal. Meanwhile the overlap between home ranges
increases, so there is a time TC after which all of the animal’s home range is
overlapping with its neighbours. In this section, I show how to calculate TC
and its dependence on TAS .
The home range is determined by the probability distribution of the animal.
Whilst the distribution of the border at any point in time is Gaussian, as
15
2
ρ’=0.05
ρ’=0.02
ρ’=0.01
1.8
T’BR
1.6
1.4
1.2
1
0.8
0
2000 4000 6000 8000 10000 12000 14000
T’AS
Figure 2.4: Boundary return times. This shows the dimensionless bound′
ary return times TBR
= TBR ρ2 D for various dimensionless values of ρ′ = ρa
′
and TAS = TAS /∆t, as measured from the 1D simulations (averaged over
10,000 runs). TBR is the mean first passage time for an animal to cross from
one side of its territory to the other and back.
documented for exclusion processes in 1D (Harn & Kärger, 1995), I fitted a
flat-topped curve with exponential tails to the animal’s distribution (see Figure
2.5b) given by
Q(xa ) =

λeλ(xt +xa )


,

 2+2xt λ
if x ≤ −xt ,
λ
if −xt ≤ x ≤ xt ,
2+2xt λ ,



λ(xt −xa )
λe

if xt ≤ x,
2+2xt λ ,
(2.7)
where xa is the position of the animal, λ is a parameter to be determined from
the simulations and 2xt is the width of the flat-top. This expression allows the
home range size S to be inferred from the animal MSD, ∆x2a = h(xa − hxa i)2 i.
Since the distribution Q(xa ) has tails that extend to infinity, sampling
from it, as is done when measuring home ranges from location data, yields a
home range that increases indefinitely with the number of samples. Such a
home range is often referred to as a 100% minimum convex polygon (MCP)
16
a)
0.08
0.06
10
b)
2
8
1.5
6
1
4
0.5
2
0
0
0.1
0.2
0.3
0.05
0.5
0.6
0.7
0.8
0.9
1
0
Animal 1
Animal 2
Borders
0.03
0.02
T /T
0.01
C
0.5
1
1.5
*
2
3
15
c)
2
10
1
5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
Border probability distribution
0.04
0
0
0.4
X position
Animal probability distribution
Normalised MSD
0.07
2.5
Border probability distribution
Animal probability distribution
0.09
X position
Time/T
*
Figure 2.5: Mean square displacement of the animal and territorial
border locations. Borders are represented by green lines, animal 1 by solid
blue lines and animal 2 by dashed blue lines. In (a) I have plotted the time
dependence of the MSD of an animal, ∆x2a = h(xa − hxa i)2 i, and the sum
of the left and right boundaries, each ∆x2b , adjusted to correspond to a 90%
minimum convex polygon home range estimation (see section 2.2.3). Both
animals exhibit the same time-dependent MSD so only one is plotted. The
choice of the observation time span, from zero up to time T∗ , determines the
expected amount of overlap between home ranges (see section 2.2.3). In panel
(b) the probability distributions of the boundaries’ and animals’ locations at
time T∗ are plotted, as functions of the position X divided by the lattice width
L. Panel (c) is similar to (b) except that after retreating from scent, the movement is a correlated random walk, where the probability of continuing straight
is 0.5+ 0.49n , where n is the number of steps since the animal last encountered
foreign scent. This plot illustrates the role the type of movement performed
by the animals may have on the shape of their probability distributions.
(Harris et al., 1990). However, the middle C% of the area under the probability density curve, i.e. the C% MCP, is of finite width when C < 100. Therefore
sampling from the truncated distribution that consists of the central part of
the distribution function (2.7) will give an increasing sequence of home range
estimates with a finite limit (Moorcroft & Lewis, 2006).
For any C, it turns out that the limit as λxt → ∞ of the quotient
p
√
2 ∆x2a /S(C), where S(C) is the C% MCP size, is 1/ 3. The value of C
that minimises the difference between the minimal and maximal values of
p
2 ∆x2a /S(C) is C = 90 to the nearest integer (Figure 2.6). Furthermore,
p
√
2 ∆x2a /S(90) is within 5% of 1/ 3 for the range of simulation values used
to construct Figure 2.3a. Therefore I used S = S(90) and assumed that
p
√
2 ∆x2a /S = 1/ 3 to infer the home range size from the animal MSD.
17
1
C=80
C=85
C=90
C=95
C=99
0.9
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2SD / home range size
2SD / home range size
0.8
5
0.5
0
0
10
λ xt
50
λ xt
15
100
150
20
Figure 2.6: Standard deviation and home range size. This shows
p how the
standard deviation of an animal, divided by its home range size (2 ∆x2a /S(C)
in the notation of the main text) varies with the dimensionless quantity λxt ,
where xt is the width of the top of the animal’s probability distribution function (PDF) and λ represents how steeply the tails of the PDF decay. The
home range size is measured using a C% minimum convex polygon, for various C. The value of C that has the smallest difference between the minumum
and maximum of its curve is C = 90. The inset shows the same curves, but
for a larger
√ range of λxt , demonstrating that they all converge to the same
value 1/ 3.
As the distribution of the border locations is Gaussian,
q the width of the
central 90%, that is the 90% MCP overlap, is O = 3.29 ∆x2b . Here, 3.29 is
the number of standard deviations that constitutes the width of the central
90% of a Gaussian distribution, which can be derived from Normal distribution
tables (Wasserman, 2004). It follows that
s
2O
≈ 1.899
S
∆x2b
.
∆x2a
(2.8)
When 2O/S < 1, the animal has some area of exclusive use. However, this
disappears once 2O/S ≥ 1. In Figure 2.5a, I have multiplied the MSD of a
√
single border ∆x2b by 3.608 ≈ (3.29/ 3)2 to ensure that the two curves cross
precisely when 2O/S = 1, i.e. at time TC . The dependence of TC on the ratio
18
between TAS and TBR is shown in Figure 2.7. If TC is less than TBR then by
the time the animal has traversed its territory and returned, the home ranges
will have overlapped, meaning that any attempt to measure home ranges in
the field will likely result in no exclusive area being observed.
4
3.5
)
C
2
10
1.5
BR
2.5
Log (T /T
3
Position density
4.5
2.5
2.5
a)
2
2
2.5
b)
1
1
1
0.5
0.5
0.5
0
0
0.5
1
0
c)
2
1.5
1.5
1.5
0
0.5
1
0
0
0.5
1
Position
1
0.5
0
−0.5
−1
0
0.5
1
1.5
T
2
2.5
3
/T
AS
BR
Figure 2.7: Exclusivity of space use. This shows the cross-over from territories with an area of exclusive use to ones without. The open circles represent
cases where TC is so low compared to TBR that by the time a data gatherer has
observed sufficient animal locations to determine a home range, the territory
border would have moved so much that the animal would likely appear to have
no exclusive area. The closed circles represent cases where exclusively used
area is more likely to be observed. For a fixed observation time T∗ the insets
indicate the probability distribution of the two animals as a function of the
spatial position relative to the box size. The degree of overlap between territorial neighbours diminishes as TAS /TBR increases, as indicated by sequentially
inspecting the insets (a), (b) and (c). The ratios T∗ /TAS , moving from inset
(a) to (c), are 3.5, 1.75, 1.17, respectively.
19
2.3
2.3.1
Results of the 2D system
Varying the animal movement process
As well as the nearest neighbour lattice random walk (NNRW) described at
the start of the chapter, I examined the effect of using a variety of different
animal movement rules in the agent-based model. Three of these, ballistic
walks (BW), correlated random walks (CRW) and Lévy walks (LW), have an
underlying movement process that takes place in continuous space. In these
cases, the animal deposits scent marks at spatial intervals of a on the nearest
lattice site. Model animals are not permitted to be in a position where the
nearest lattice site contains scent of a conspecific. If they are about to move
into such a position, they instead turn at random to avoid this situation. The
velocity, v, of each animal was kept constant, and is equal to 4aF in the
discrete space models, where F is the jump rate and a the lattice spacing.
These movement processes have all been used in the past to model animal
movement. The CRW paradigm has been used to model a variety of animals,
from small invertebrates like butterflies (Pieris rapae) (Kareiva & Shigesada,
1983), bark beetles (various species) and nematodes (Steinernema carpocapsae) (Byers et al., 2001), to large mammals such as caribou (Rangifer tarandus) (Bergman et al., 2000) and seals (Callorhinus ursinus) (Johnson et al.,
2008). In this paradigm, moving animals are less likely to make lots of turns
through large angles, since they possess some inertia. Correlation in the movement process might therefore be advantageous from the point of view of minimising energy use. Rather than sampling turning angles from a uniform
distribution, as per a diffusive random walk, the angle distribution is peaked
around zero (Kareiva & Shigesada (1983) contains a good explanation of such
walks in an ecological context).
In the model of the CRW movement process, step lengths are sampled
from an exponential distribution with mean step-length a and turning angles
from a wrapped double exponential (Laplace) distribution. From this, the
correlation time parameter T = −a/v ln[hcos(θ)i] is calculated, where hcos(θ)i
is the mean of the cosines of the possible turning angles θ (Viswanathan et al.,
2005). T measures the amount of time an animal has directional persistence
and has the following practical interpretation. If position fixes from a CRW
with correlation time T are taken at time intervals greater (less) than T then
analysing the turning angles between the position fixes will likely suggest that
the animal is moving in an uncorrelated (correlated) fashion.
I also ran simulations with a lattice CRW (LCRW) movement process that
takes place in discrete-space. The step length is fixed at a and the turning
20
angles are either π, ±π/2 or 0. The probability at each step of continuing
forward is the integral between −π/4 and π/4 of the wrapped Laplace distribution used in the CRW movement process. The probability of turning π/2
to the right is the integral between π/4 and 3π/4, turning left by π/2 has the
same probability as turning right, and if none of these three options are chosen
then the animal turns by π. As in the continuous-space case, the nature of
the movement can be captured by the correlation parameter T .
Figure 2.8: Contour plot of animal utilisation distributions. This
shows the frequency distributions of animal locations over a period of time
for 16 animals in 2D with periodic boundary conditions, observed up to time
T∗ = 2.5TAS (density is ρa2 = 0.0016). The positions X and Y are spatial
coordinates normalized to the width L of the box. On moving away from
foreign scent, the animals perform a lattice correlated random walk (LCRW)
with turning angles constructed from a Laplace distribution (see main text for
details of the contruction) with parameter proportional to 0.9n , where n is the
number of steps since last encountering foreign scent. The coloured crosses
represent the initial animal locations from which their trajectories started to
be recorded.
In recent years, the idea that an animal may sometimes exhibit LWlike movement statistics, rather than CRW, has generated significant interest
(Reynolds & Rhodes, 2009). Much of this interest seems to have arisen from
the apparent optimality in Lévy search strategies under certain assumptions
21
(Viswanathan et al., 1999, 2011). Reports have been published claiming that
several animals exhibit LW movement, including albatrosses (Diomedeidae exulans) (Viswanathan et al., 1996), mussels (Mytilus edulis) (de Jager et al.,
2011) and humans (Rhee et al., 2008).
However, the Lévy paradigm has
not been without controversy (Edwards et al., 2007; Plank & Codling, 2009;
Auger-Méthé et al., 2011; Jansen et al., 2012) and the optimality question has
not been tested when animals are confined to move within a territory.
When constructing an LW, the turning angle distribution is uniform, whilst
the step length distribution is a Lévy distribution of order 0 < α < 2, possessing a power law tail l−1−α where l is the length travelled between random
turns. Fulger et al. (2008) constructed a method for sampling from the Lévy
distribution, which I used in order to generate step lengths in the model.
In the BW case, the animal moves in a straight line until it reaches the
edge of its territory. At this point it turns at random back inside its territory
and continues in a straight line. Whilst this is perhaps the least biologically
realistic of all the movement processes, it is the basis of the ideal gas model
of physical chemistry, which has occasionally been used as a model of animal
movements and interactions (Jetz et al., 2004; Hutchinson & Waser, 2007).
The final movement process I used is based on NNRW, but each time the
animal encounters scent of a conspecific, it switches its movement to an LCRW
such that the underlying Laplace distribution has parameter proportional to
0.9n , where n is the number of steps since the animal last encountered foreign
scent. In other words, once the animal reaches the border of its territory, it will
tend to retreat in a correlated fashion, gradually becoming less correlated the
more time it spends away from foreign scent. This retreating tendency models
the idea that an animal may wish to avoid costly confrontations on the edge
of its territory (White & Harris, 1994). I used it to plot Figure 2.8, whereas a
1D version is shown in Figure 2.5c. The outcome of these simulations is that
each animal’s probability distribution becomes less flat-topped, demonstrating
how slightly more complicated movement processes can generate very different
animal space-use distributions.
2.3.2
Subdiffusive territory border movement
Territories in the 2D simulations can be considered as particles undergoing
an exclusion process. Consequently, the MSD of a territory border is asymptotically proportional to t/ ln(t) (Landim, 1992). The generalised diffusion
constant, K, of a border is defined through the following equation
∆x2b ∼ 2Kt/ ln(t/τ ),
22
(2.9)
where τ is a characteristic time, τ = 1/(4F ) in the discrete-space simulations,
since there are four possible options each time the animal jumps, and τ = a/v
in continuous-space.
For each movement process tested (NNRW, LCRW, BW, LW), the value
of K depends solely on the single dimensionless parameter Z2D = TAS ρv 2 τ ,
regardless of the individual ρ or TAS values used (Figure 2.9). The nature
of this trend depends on the underlying animal movement process. For the
NNRW case, K/D decays exponentially as Z2D is increased. When the animals
move ballistically, an exponential decrease in K/D is also observed, but the
gradient is steeper. The LCRW and LW movement processes have universal
curves that lie between the NNRW and BW cases.
1 a)
1 b)
0
0
−1
−1
log10(K/v2T)
2
log10(K/v T)
−2
−3
−4
−5
−6
−7
−8
0
NNRW
LCRW: T/τ=0.84
LCRW: T/τ=1.65
LCRW: T/τ=2.86
BW
2
4
−2
−3
−4
−5
−6
6
Z2D
8
10
−7
0
12
NNRW
LW µ=1.0
LW µ=1.2
LW µ=1.5
LW µ=1.7
LW µ=1.9
BW
2
4
6
Z2D
8
10
12
Figure 2.9: Universal curves for border movement in 2D. The normalised border diffusion constant K/D depends solely on the dimensionless
parameter Z2D = TAS ρv 2 τ . In panel (a) this dependency is observed for the
LCRW movement processes, as compared to NNRW and BW. The solid line
is a best fit for the NNRW case log10 (K/v 2 τ ) = 0.085 − 0.247Z2D , the dashed
line the best fit for the BW case log10 (K/v 2 τ ) = 0.467−0.709Z2D . The dotted
−S
lines give the best fit for the LCRW movement process, K/D = Q(T /τ )−R Z2D
where Q ≈ 12.0, R ≈ 2.10, and S ≈ 4.48. Panel (b) shows the curves for the
LW movement processes, again compared to NNRW and BW. The solid lines
show best fits to the simulation output.
For the LCRW movement process, when the population density is low,
K/D decreases with Z2D almost as fast as for the BW movement process. In
this case, the territories are relatively large so the animal is likely to make many
turns when moving from one territory border point to another. This means
the distribution of the total angle turned during one traverse of the territory
is highly peaked around 0, so the time to cross the territory is similar to the
BW case. As the population density increases, the distribution of the total
23
angle turned when going from one border point to another becomes wider.
Therefore the time to traverse the territory becomes increasingly dissimilar
to the BW case, whilst always remaining shorter than in NNRW simulations.
This causes the curves of K/D against Z2D for the LCRW movement processes
to lie between the analogous curves for the BW and NNRW situations, almost
parallel to the BW case for low Z2D , but with a shallower gradient for higher
Z2D .
For fixed Z2D , when the distribution of turning angles in the LCRW case
is highly peaked, in other words when T /τ is large, the movement process is
closer to a ballistic walk than when T /τ is small. Therefore as T /τ increases,
the curve of K/D against Z2D is further away from the NNRW curve and
closer to BW. This is evident from the three LCRW curves in Figure 2.9a.
0.65 b)
0.6
0.6
0.55
2
a
Dimensionless animal MSD, 〈 x 〉/R
2
a)
0.5
0.5
NON−MONOTONIC MSD
0.45
0.4
a
〈 x2 〉/R2
0.2
ξ
1.2
0.3
1
0.4
0.35
0.8
0.3
0.6
0
0.1
1000
2000
tv/a
3000
0.25
4000
0.2
MONOTONIC MSD
0
0
10
20
30
40
Dimensionless time, tv/a
0.15
0
50
5
10
15
Z
2D
Figure 2.10: Oscillations in the animal MSD when the movement
process is correlated. Panel (a) shows how the animal MSD varies with
time for various different values of Z2D and ξ = T v/R. The solid line has
ξ = 1.51, Z2D = 15.56; the dotted line ξ = 1.51, Z2D = 2.22; the dashed
line ξ = 0.80, Z2D = 2.22; the dot-dashed line ξ = 0.34, Z = 2.22. Before
beginning to measure the animal MSD, the simulation was run until such a
time as the borders have an MSD proportional to t/ ln(t). Then each animal
was placed in the centre of its territory. In the region where the MSD is nonmonotonic, as Z2D and ξ are increased, the oscillatory behaviour becomes
more pronounced. The inset demonstrates the long-time behaviour of the
MSD, which is to increase in proportional to 2Kt/ ln(t), where K depends on
both Z2D and ξ. For the Z2D = 15.56 case, K is so low that the line looks flat,
since K decreases exponentially as Z increases. Panel (b) show the threshold
value of ξ for various Z2D , below which the MSD is monotonic and above
which it is not.
The curves of K/D against Z2D for the LW case have steeper decay for
lower α (Figure 2.9b). The limit as α → 2 is a diffusive walk, so the curves
24
roughly tend towards the NNRW case. As α is decreased, the Levy walks
will tend to exhibit a higher number of long steps. This makes the walk, and
consequently the nature of the border movements, less like the NNRW case
and more like the BW case.
2.3.3
The effect of correlation on the animal’s MSD
The CRW movement process was used to examine the effect of correlation
on the animal MSD at intermediate times, as predicted by analytic calculations in Giuggioli et al. (2012). Predictions point to correlated random walks
within slowly moving territory borders displaying MSDs that exhibit damped
oscillations. The persistence of the oscillations was shown to depend on the
ratio of the correlation distance between successive steps to the size of the
territory. This is given by the dimensionless parameter ξ = T v/R where R is
√
the mean territory radius, estimated to be R = 1/ πρ. I ran simulations to
demonstrate that this phenomenon also occurs in the agent-based system described in the present chapter, with the CRW movement process. The results
are shown in Figure 2.10.
The inset in panel (a) has been plotted to show that the solid curve, corresponding to large Z2D and ξ values, appears close to constant whereas the
other curves show a clear increase at long times. Although one would expect
that an increase in ξ would generate damped oscillations in the MSD, as ξ becomes too large, the animals move so quickly that they are, on average, able
to re-mark all or most of their territory boundaries before the scent becomes
inactive. In such a scenario the territories move much less. As a consequence,
there is no set of parameters for which there is both a strongly oscillating MSD
at intermediate times and a rapidly increasing MSD at long times.
25
2.4
Glossary
Symbol
TAS
v
a
F
∆t
D
τ
N
L
ρ
T
R
α
xa
xb
∆x2a
∆x2b
K
TBR
s∗
T∗
O
S
TC
Z1D
Z2D
ξ
Table 2.1: Glossary of the key symbols used in Chapter 2
Explanation
Active scent time: the time during which a scent mark is
considered by conspecifics as a fresh territory message
Animal velocity
Lattice spacing
Mean jump rate between adjacent lattice sites in
continuous time simulations
Time between successive jumps in discrete-time
simulations
Animal diffusion constant D = a2 F = a2 /(2∆t)
Characteristic time τ = 1/(4F ) in the 2D discrete-space
simulations and τ = a/v in continuous-space
Number of lattice sites in 1D simulations; 2D
simulations contain N × N sites
Width of simulation area (L = aN )
Population density (ρ = 2/L in 1D, ρ = 25/L2 in 2D)
Correlation time, used for correlated random
walk (CRW) simulations (see section 2.3.1)
√
Mean territory radius, approximated as R = 1/ πρ
Parameter used for Lévy walk (LW) simulations (see
section 2.3.1)
Animal position
Territory border position
Animal mean square displacement (MSD), averaged over
all animals and multiple simulations ∆x2a = h(xa − hxa i)i
Territory border MSD
Territory border diffusion constant
Boundary return time, i.e. the mean first passage time in
1D for an animal to traverse its territory and return
Long-time MSD of the territory width in 1D
Time-window over which home ranges are measured
Home range overlap
Home range size
Maximum T∗ for which we are likely to see each animal
maintaining areas for exclusive use
Composite parameter Z1D = TAS ρ2 D in 1D
Composite parameter Z2D = TAS ρv 2 τ in 2D
Composite parameter ξ = T v/R from CRW simulations
I/O
I
I
I
I
I
I
I
I
I
I
I
I
I
O
O
O
O
O
O
O
O
O
O
O
C
C
C
Explanation of the various symbols used in Chapter 2. The third column
denotes whether the symbol is an input to the simulation model (I), an
output of the model (O) or a composite of various input and output
parameters (C).
26
Chapter 3
An analytic model of
territorial animal movement
The complications inherent in measuring movement and scenting information
from neighbouring animals at the same time makes it difficult, if not impossible, to infer territory border movement directly from location data. In field
studies, by the time sufficient location fixes have been obtained to measure
territory size (Harris et al., 1990), borders may well have changed. Therefore
it is not possible to have perfect knowledge of the precise location of a territory
at any point in time. To obtain information about the mechanisms underlying territory formation requires building a theory that does not rely on this
knowledge.
In this chapter, I show how to infer the border movement simply by
analysing location data of individual animals. Analysis of the simulation
model (Chapter 2) has uncovered that territorial borders move subdiffusively,
with a generalised diffusion constant K that is dependent on the dimensionless
parameter Z1D = TAS ρ2 D in 1D or Z2D = TAS ρD in 2D (see Table 3.1 for definitions of these parameters). Here, a theory of animal movement within such
subdiffusing territory borders is constructed. The theory is initially described
in 1D, modelling the animal as a Brownian particle. This is then extended to
the 2D situation, where the animal movement has some directional persistence
as well as random movement, which is an approximate model of a correlated
random walk.
3.1
A Fokker-Planck approach in 1D
Due to the exclusion process evident in the 1D simulation model (section
2.2), the two borders of a territory are modelled as particles whose mean
27
Figure 3.1: Cartoon of a model animal within subdiffusing territory
borders. The borders are modelled as subdiffusive particles, connected together by a spring, which represents the territory. The animal is modelled as
a Brownian particle that is free to move within its territory.
√
square displacement (MSD) increases as K t. Also, territories whose width is
smaller than average tend to grow, whereas those larger than average shrink.
To model this aspect, one of my supervisors (LG) proposed to connect the
two borders by a spring (Figure 3.1), represented by a quadratic potential
U (L1 , L2 ) = γ(L2 − L1 − L)2 /4 where L1 , L2 are the positions of the territory
borders. This leads to the following Fokker-Planck expression for the probability distribution, Q(L1 , L2 , t), for the territory borders to be in position L1
and L2 at time t
(
∂Q(L1 , L2 , t)
∂2
∂2
= Kϕ1d (t)
+
Q(L1 , L2 , t)+
∂t
∂L21 ∂L22
)
∂
∂
γ
−
[(L2 − L1 − L)Q(L1 , L2 , t)] ,
2 ∂L2 ∂L1
(3.1)
where L is the average width of the territory and, as is standard for FokkerPlanck equations (see e.g. Risken (1996)), the quadratic potential U (L1 , L2 )
appears in its differentiated form, γ(L2 −L1 −L)/2. A non-constant ϕ1d (t) represents the anomolous diffusive nature of the borders in the Fokker-Planck formalism (Giuggioli et al., 2007). In the present case, ϕ1d (t) = (1/2)(t/ζ)−1/2 ,
where ζ is a characteristic time constant, so that the border MSD is propor√
tional to t.
28
Equation (3.1) can be solved exactly by using the variable transformation
λ = L2 −L1 and L = (L2 +L1 )/2, so that λ is the territory width and L the po-
sition of the territory’s centre (Giuggioli et al., 2011b). This enables the equa-
tion to be decoupled as Q(L1 , L2 , t) = Q1 (L, t)Q2 (λ, t). To ensure that the
~
borders cannot cross, the boundary condition ∇Q(L
1 , L2 , t) · n̂ = 0 is imposed,
where n̂ is the normal to the line of points where L1 = L2 (Ambjörnsson,
2008). Using initial conditions Q(L1 , L2 , 0) = δ(L1 + L/2)δ(L2 − L/2), where
δ is the Dirac delta, so that the border L1 starts at −L/2 and L2 at L/2,
the solution to equation (3.1) with the aforementioned boundary condition is
(Giuggioli et al., 2011b)
Q(λ, L, t) = H(λ)
(λ−L)2
b(t)
−
e
−
+e
p
πb(t)
(λ+L)2
b(t)
2
L
− c(t)
e
p
πc(t)
,
(3.2)
n
h
io
Rt
where b(t) = 4(K/γ) 1 − exp −2γ 0 ds ϕ1d (s)
is twice the MSD of the
Rt
territory width λ, c(t) = 2K 0 ϕ1d (s)ds is twice the MSD of the territory
centroid L, and H(y) is the Heaviside function, H(y) = 1 (H(y) = 0) if y > 0
(y < 0). In the case relevant to this chapter, where ϕ1d (t) = (1/2)(t/ζ)−1/2 ,
the specific values of b(t) and c(t) are b(t) = 4(K/γ) 1 − exp −2γ(t/ζ)1/2
and c(t) = 2K(t/ζ)1/2 .
Equation (3.2) is the product of the following expressions for Q1 (L, t) and
Q2 (λ, t)
2
L
− c(t)
e
Q1 (λ, t) = p
Q2 (λ, t) = H(λ)
−
e
πc(t)
(λ−λ̄(t))2
b(t)
,
(3.3)
−
+e
p
πb(t)
(λ+λ̄(t))2
b(t)
.
(3.4)
My task was to use these results, derived by LG, to calculate both the MSD
and the marginal distribution of the animal, as well as comparing the results
with the agent-based model of Chapter 2.
3.2
The mean square displacement of an animal
The starting place for computing the animal’s MSD is to calculate the joint
probability distribution P (x, L1 , L2 , t) of the animal to be at position x at
time t, whilst its territory borders are at positions L1 and L2 . This is done
by assuming that the movement of the animal is much faster than that of the
29
borders, by using an adiabatic approximation
P (x, L1 , L2 , t) ≈ Q(L1 , L2 , t)Wx0 (x, t|L1 , L2 ),
(3.5)
where Wx0 (x, t|L1 , L2 ) is the probability density function of an animal to be in
position x at time t, starting from x0 at t = 0, given that its territory borders
are fixed in positions L1 and L2 .
The distribution of a Brownian walker within fixed borders was calculated by Montroll & West (1987) using the method of images (Chandrasekhar,
1943), and implies that Wx0 (x, t|L1 , L2 ) can be expressed as the following infinite sum of Gaussian distributions
Wx0 (x, t|L1 , L2 ) = [H(x − L1 ) − H(x − L2 )] gx0 (x, t|L1 , L2 ),
gx0 (x, t|L1 , L2 ) =
+∞
−
X
e
[x+2n(L2 −L1 )−x0 ]2
w(t)
−
[x−2L1 +2n(L2 −L1 )+x0 ]2
w(t)
+e
p
πw(t)
n=−∞
(3.6)
, (3.7)
where w(t) = 4Dt and D is the diffusion constant of the animal. The MSD of
the animal can therefore be calculated as
h(x − x0 )2 i =
Z
∞
−∞
dL2
Z
∞
dL1
−∞
Z
L2
L1
dx(x − x0 )2 gx0 (x, t|L1 , L2 )Q(L1 , L2 , t).
(3.8)
To perform the integration with respect to x, the Poisson summation formula
(Montroll & West, 1987) is used to transform gx0 (x, t|L1 , L2 ) into the following
expression
∞
X
1
nπ
gx0 (x, t|L1 , L2 ) =
exp −Dt
×
L − L1
L2 − L1
n=−∞ 2
πn[x0 − L1 ]
iπn[x − L1 ]
cos
.
exp
L2 − L1
L2 − L1
The integral from equation (3.8) with respect to x is therefore
30
(3.9)
Z
L2
L1
(L2 − x0 )3 + (x0 − L1 )3
+
3(L2 − L1 )
2 !
∞
nπ
[x0 − L1 ]πn
4 X 1
exp −Dt
cos
×
π 2 n=1 n2
L2 − L1
L2 − L1
(L2 − L1 )2 (−1)n − (x0 − L1 )(L2 − L1 )((−1)n − 1) .
dx(x − x0 )2 gx0 (x, t|L1 , L2 ) =
(3.10)
By rewriting this equation in terms of L and λ, plugging the result into equation (3.8) and performing the integration with respect to L, the following
equation is obtained
2
h(x − x0 ) i
Z +∞
L2 b(t) c(t)
+
dλZ(λ, t)×
+
+
+
12
24
2
0
(
+∞
X
2nπx0 − (2n)2 π2 [c(t)+w(t)]
4λ2 (−1)n
4λ2
cos
e
2 (2n)2
π
λ
n=1
(2n − 1)πx0
4(−1)n
c(t) cos
+
π(2n − 1)
λ
)
π 2 (2n−1)2 [c(t)+w(t)]
2λx0
(2n − 1)πx0
4λ2
,
sin
e−
π(2n − 1)
λ
=x20
(3.11)
where
2
2
+L
− λ b(t)
e
Z(λ, t) = p
πb(t)
2 cosh
2λL
b(t)
.
(3.12)
In the limit K → 0, it turns out that c(t) → 0 and b(t) → 0, meaning that
Z(λ, t) can be written as the Dirac delta function δ(λ − L), reducing equation
(3.11) to a well known expression for the MSD of a Brownian particle inside
immobile borders, see e.g. Giuggioli et al. (2005). If x0 = 0, it is possible
to perform the integral in equation (3.11) and reduce the expression to a
single infinite sum of generalized hypergeometric function 2 F2 (see e.g. Paris
(2005) and references therein or Hardy (1999) for the properties of 2 F2 ). This
calculation is performed in Giuggioli et al. (2011b, Appendix D). However, the
reduced expression does not add any further intuition and is slower to evaluate
on Matlab (version R2009a) than expression (3.11), so is omitted here.
As t → ∞, the expressions inside the integral in equation (3.11) all tend
to 0, due to the w(t) term in the each exponent tending to ∞, whereas b(t) →
4K/γ and c(t) → ∞. Therefore at long times h(x − x0 )2 i ≈ x20 + L2 /12 +
31
√
K/6γ + c(t)/2 so the MSD of the animal scales as c(t)/2 = K t. At short
times, expanding the exponential up to second order in the even series and
up to first order in the odd series gives h(x − x0 )2 i ≈ x20 + w(t)/2, giving
the well-known MSD dependence ∆x2 ∼ 2Dt of an unconstrained Brownian
particle.
Examining equation (3.11) reveals that it depends on the dimensionless
parameters D ′ = D/(L2 γ), K ′ = K/(L2 γ), β = γζ and t′ = γt. D ′ and K ′
represent the average area covered in a time γ −1 by, respectively, the walker
and the boundaries, relative to the square of the average boundary separation
L. β, being proportional to γ, represents the dimensionless rate at which the
boundary separation returns to its average value.
0.15
〈 x 〉/L
2
0.1
2
D’=0.1, K’=0.035, β=0.1
D’=0.02, K’=0.007, β=0.5
D’=0.02, K’=0.005, β=0.5
D’=0.01, K’=0.007, β=0.5
D’=0.01, K’=0.005, β=0.5
0.05
0
0
10
20
30
t/ζ
40
50
60
Figure 3.2: Mean square displacement of 1D territorial animal. The
time dependence of the animal’s mean square displacement (MSD) is shown for
various parameter values. As D ′ is increased, so the gradient at short times
increases, whereas K ′ controls the gradient at long times. The parameters
used for the thicker solid curve are constructed from those of the dot-dashed
curve by increasing γ by a factor of five, causing the long time MSDs to be
parallel, with the solid curve higher than the dot-dashed curve.
In Figure 3.2, the MSD in expression (3.11) is plotted against time for
various values of D ′ , K ′ and β, in the case where x0 = 0. The movement
from the thicker solid curve to the dot-dash curve is made by increasing γ
32
by a factor of five. This makes the spring less flexible so the MSD of the
separation distance λ converges to a lower value, whereas the MSD of the
centroid L increases at the same rate. Therefore, at long times, these two
curves are parallel. The dashed and dotted curves only differ in their values of
D ′ , so their asymptotic long-time behaviours are the same. When D ′ is lower,
the animal takes longer on average to reach the territory borders, resulting in
a shallower gradient at short times. The dashed curve only differs from the
dot-dashed curve in its value of K ′ , the border diffusion constant. Therefore
they diverge at long times, but have the same short-time behaviour. The thin
solid curve also differs from the dotted curve in its value of K ′ so has different
long-time behaviour, but has the same D ′ , so the same short-time behaviour.
3.3
Comparison with the agent-based model
I compared the agent-based model of Chapter 2 with the analytic approximation constructed here, by examining the respective marginal distributions of
an animal’s position. For the analytic model constructed in this chapter, this
marginal distribution is
M1D (x, t) =
Z
+∞
0
dλ Z(λ, t)
Z
+∞
−∞
2
L
− c(t)
e
dL p
πc(t)
Wx0 (x, t|λ, L).
(3.13)
Making use of the Poisson summation formula (Montroll & West, 1987) for
the first series in Wx0 (x, t), equation (3.13) can be transformed into
M1D (x, t) =
Z
+∞
0
dλ Z(λ, t)
+∞
Z
x+λ/2
x−λ/2
x − x0
1X
cos nπ
λ
λ
n=1
+∞
−
X
e
n=−∞
2
e−
[x−2L+(2n+1)λ+x0 ]2
w(t)
p
πw(t)
Integration over L then gives
33
L
− c(t)
e
dL p
πc(t)
π 2 n2 w(t)
4λ2
)
.
(
1
+
2λ
+
(3.14)
("
!
!#
x + λ/2
x − λ/2
dλ Z(λ, t) erf p
− erf p
×
c(t)
c(t)
0
#
+∞
2 2
1X
x − x0
1
− π n w(t)
4λ2
+
cos nπ
e
+
2λ λ n=1
λ
"
#
[x+x0 +(2n+1)λ]2
+∞
−
X
4c(t)+w(t)
h(x, x0 , t) + λ qn− (t)
e
p
p
erf
−
π[4c(t) + w(t)]
4c(t) + w(t)
n=−∞

q
!)
c(t)
+ (t) + 4
h(x,
x
,
t)
−
λ
q
0
n
w(t) 

p
erf 
,
(3.15)

4c(t) + w(t)
1
M1D (x, t) =
2
"
Z
+∞
with
s
s
!
w(t)
c(t)
− 2x0
+2
c(t)
w(t)
s
s
w(t)
c(t)
1
± 4n
.
qn± (t) =
2 c(t)
w(t)
h(x, x0 , t) =x
s
c(t)
,
w(t)
(3.16)
Like the MSD expression (3.11), equation (3.15) depends solely on the dimensionless parameters D ′ , K ′ , β and t′ .
To compare equation (3.15) with simulation output, it is necessary to determine the dependence of K ′ and β on the input parameters to the simulation
model, i.e. the diffusion constant, D, the active scent time, TAS , and the
walker population density, ρ. In Figure 2.3 these dependencies are given for
a range of values of TAS , ρ and D, by plotting the dimensionless parameter
K/D and the asymptotic value s∗ = limt→∞ s(t)/L2 of the normalized MSD of
the boundary separation distance s(t) = h(λ − L)2 i. These parameters depend
on the dimensionless quantity Z1D = TAS ρ2 D introduced in section 2.3. The
fitting curves from Figure 2.3 provide the means for selecting the appropriate
parameter values K ′ , D ′ and β, which are present in the expression for the
marginal probability distribution of the walker in equation (3.15).
The function s(t) is found by multiplying expression (3.3) by (λ − L)2 and
then integrating with respect to λ, to give
b(t)
s(t) =
2
(
"
L2
4
L − b(t)
L2
1− √ p
1 − erf
+4
e
π b(t)
b(t)
34
L
p
b(t)
!#)
.
(3.17)
To find the limit s∗ , all occurences of b(t) are replaced with 4K/γ in expression
(3.17) and, dividing by L2 ,
2K
s∗ =
γL2
(
"
L2
L2
4
L
− 4K/γ
+4
e
1− √ p
1 − erf
4K/γ
π 4K/γ
L
p
4K/γ
!#)
. (3.18)
To measure K from the MSD of the border in the simulation, it is also necessary to choose a value for the time constant ζ, since the border MSD is
c(t)/2, which depends on t, K and ζ. Figure 2.3a was produced by picking
ζ = 1/2F so this value of ζ must be used in any comparisons between the simulation output and the reduced model. However, this choice is arbitrary, since
if ζ = C/2F for some C > 0 were chosen instead, each value of K measured
√
from the simulation output would be reduced by a factor of C, i.e. points in
√
Figure 2.3a would be shifted down by log(D C). Then, if the new values of
ζ and K were placed back into equation (3.1), the changes would cancel one
another out.
By means of Figure 2.3a, the value of K/D can be found for a given
set of simulation parameters TAS , ρ and D. Since D is an input into the
simulations, K can be derived from K/D. This, together with Figure 2.3b
and equation (3.18), enables γ to be calculated for the given value of Z1D . By
using these values of K, D, γ and ζ, together with the fact that L = 1/ρ, the
dimensionless quantities K ′ , D ′ and β are found, required to construct the
marginal distribution (3.15).
This procedure is used to compare the animal’s marginal distribution from
the simulation model of Chapter 2 with the analytic model of the current
chapter, for various values of Z1D (Figure 3.3). When Z1D is high, the border movement is very slow compared to that of the animal, so the analytic
model approximates the simulation model well (Figure 3.3a). However, as Z1D
is decreased, the adiabatic approximation becomes gradually less applicable,
therefore the difference between the marginal distributions becomes more pronounced (e.g. Figure 3.3d).
3.4
Extension to a 2D correlated walk
In Giuggioli et al. (2012), one of my supervisors (LG) extended the 1D analytic movement model to the case of a 2D correlated random walk (CRW) in
both square and circular territories with slowly moving borders. The generalisation from a Brownian particle to one approximating a CRW is made by
replacing the animal’s diffusion equation by a so-called telegrapher’s equation
35
Probability distribution
0.02
a)
0.02
0.015
b)
0.015
0.01
0.01
0.005
0.005
0
−1
−0.5
0
0.5
0
−1
1
−0.5
Probability distribution
x/L
0.02
c)
0.02
0.015
0.015
0.01
0.01
0.005
0.005
0
−1
−0.5
0
0
0.5
1
0.5
1
x/L
0.5
0
−1
1
x/L
d)
−0.5
0
x/L
Figure 3.3: Comparison of the simulation and analytic models. The
animal probability distribution from the simulated model (dashed lines), compared with the reduced model (solid lines) at time tF =5,525 for four values
of Z1D : a) Z1D =1.8, b) Z1D =1.4, c) Z1D =1.2 and d) Z1D =0.6. To plot the
probability distribution of the reduced model, M1D (x, t) in equation (3.15) is
multiplied by the lattice constant a. In all panels the density has been kept
fixed at ρa2 = 0.02, whereas TAS F = 4500, 3500, 3000, 1500 for the panel a),
b), c) and d), respectively. Running the simulations requires a choice of initial
scent spatial profile, explained in section 2.1.
(Goldstein, 1951). In Cartesian co-ordinates, used for the square geometry,
the equation governing the probability distribution, WT (x, t), of the animal
position in the x-direction at time t is
2
1 ∂WT (x, t)
∂ 2 WT (x, t)
2 ∂ WT (x, t)
+
=
v
,
∂t2
T
∂t
∂x2
(3.19)
where v is the speed of the animal and T the correlation time (see section
2.3.1 or Table 2.1). The equation in the y-direction is written by replacing x
with y throughout equation (3.19).
In polar co-ordinates, used for circular territories, the telegrapher’s equation for the probability, WT (r, θ, t), of the animal being at polar co-ordinates
(r, θ) at time t is
2
∂ 2 WT (r, θ, t)
∂
1 ∂WT (r, θ, t)
1 ∂
1 ∂2
2
+
=v
+
+
WT (r, θ, t).
∂t2
T
∂t
∂r 2 r ∂r r 2 ∂θ 2
(3.20)
Both equations (3.19) and (3.20) reduce to the respective diffusion equations,
in Cartesian and polar coordinates, by setting D = v 2 T and taking the limit
T → 0. Equations (3.19) and (3.20) were solved exactly by LG and used in
36
an adiabatic approximation to represent the joint distribution of the territory
borders and the animal (Giuggioli et al., 2012). My tasks were to characterise
the animal MSD for different parameters, as well as to calculate the marginal
distributions of the animal, used in Chapter 4 to fit to location data.
3.4.1
Circular territories
The probability of the animal to be at polar co-ordinates (r, θ) (or (x, y) in
Cartesian) inside a circular territory at time t, whilst the territory has its
centre at (rc , θc ) in polar co-ordinates (or (xc , yc ) in Cartesian) and has radius
R, is given by the following expression (Giuggioli et al., 2012)
Pr0 ,θ0 (r, θ, σ, rc , θc , t) =H(R − r̄)×
hr0 ,θ0 (r, θ, t|R, rc , θc )Q(R, rc , θc , t).
Here r̄ =
p
(3.21)
(x − xc )2 + (y − yc )2 is the radial position of the animal relative
to the territory centre, (r0 , θ0 ) is the position of the animal at time t = 0 in
polar co-ordinates (or (x0 , y0 ) in Cartesian) and
hr0 ,θ0 (r, θ, t|R, rc , θc ) =
t
− 2T
e
πR2
1
+
πR2 µn,m r̄ 2
µ
r0 Jn n,m
R
R
×
(µ2n,m − n2 )Jn2 (µn,m )
∞ X
∞ Cn µ2 Jn
X
n,m
n=0 m=0
tΩn,m
cos[n(θ − θ0 )] cos(tΩn,m ) +
2T Ωn,m
(3.22)
is the probability density function for an animal to be in position (r, θ), given
that it started at (r0 , θ0 ) and is inside an immobile circular territory of radius
R and centre (rc , θc ), where C0 = 1 and Cn = 2 for n > 0. The symbol
µn,m is the m-th zero of the derivative of the n-th Bessel function of the first
kind Jn (z), i.e. the values of µ satisfied by the implicit equation Jn′ (µ) = 0
(Watson, 1944), and
Ωn,m =
s
µ2n,m v 2
1
−
.
2
ρ
4T 2
(3.23)
The function Q(R, rc , θc , t) is the probability distribution of the territory having radius R and centre (rc , θc ) at time t, given by
37
Q(R, rc , θc , t) = R(R, t)Q(rc , θ, t),
2 /b̄(t)
R(R, t) =
e−(R−R̄)
p
2
2 /b̄(t)
+ e−(R+R̄)
π b̄(t)
,
e−rc /c̄(t)
Q(rc , θ, t) =
,
πc̄(t)
(3.24)
where R̄ is the average radius of the territory, which is also the initial territory
radius. The time-dependent parameter
2K
b̄(t) =
γ
Z t
ds ϕ2d (s)
1 − exp −2γ
(3.25)
0
represents the movement of the territory width, so is the circular-territory
analogue of b(t). The time-dependent parameter
c̄(t) = 4K
Z
t
ds ϕ2d (s)
(3.26)
0
is the circular-territory version of c(t), and
ϕ2d (t) =
1
t/T
−
ln(1 + t/T ) [ln(1 + t/T )]2 (1 + t/T )
(3.27)
ensures that the MSD of a point on the territory border at long time scales
as t/ ln(t/T ), as expected from a 2D exclusion process (Chapter 2). This is
analogous to the function ϕ1d (t) from the 1D scenario.
The marginal distribution of the animal, with arbitrary initial conditions,
is calculated by integrating Pr0 ,θ0 (r, θ, σ, rc , θc , t) against the parameters R, rc ,
θc , r0 and θ0 as follows
M2D (r, θ, t|Φ) =
Z 2π
Z R̄
Z 2π
Z ∞
Z ∞
1
dθ0 r0 rc Pr0 ,θ0 (r, θ, σ, rc , θc , t),
dr0
dθc
dR
drc
π R̄2 0
0
0
0
r̄
(3.28)
where Φ = (v, K, T , γ, R).
The integrals with respect to r0 , θ0 and θc can all be performed, giving
the following expression for the marginal distribution for the animal inside a
slowly moving circular territory
38
∞
∞
2
r + rc2
2rc r
2rc R(R, t)
I0
exp −
×
dR
drc
M2D (r, θ, t|Φ) =
c̄(t)
c̄(t)
c̄(t)
r̄
0
(
)
∞ J1 µ0,m R̄ J0 µ0,m r̄ −t/T
X
R
R
sin(tΩ0,m )
1
2e
cos(tΩ0,m ) +
+
,
πR2
2T Ω0,m
µ0,m J02 (µ0,m )
πRR̄ m=0
Z
Z
(3.29)
where In (z) is the n-th modified Bessel function of the first kind.
a)
0.32 b)
ξ=1 κ=0.01 γ’=0.1
ξ=0.2 κ=0.005 γ’=0.01
ξ=0.1 κ=0.01 γ’=0.01
γ’=0.001
0.3
1
Threshold ξ
Normalised animal MSD
1.2
0.8
0.6
0.28
γ’=0.1
0.26
γ’=1
0.24
0.22
NON−MONOTONIC MSD
0.2
0.18
0.4
0.16
0.2
0.14
MONOTONIC MSD
0.12
0
0
5
10
15
Normalised time
20
−5
−4
−3
log10(κ)
−2
Figure 3.4: Animal MSD in circular territory. Normalised mean square
displacement (hr 2 i/R̄2 ) of a walker starting at the origin inside a circular
territory with fluctuating boundaries, plotted against dimensionless time t/T .
Panel (a) shows the MSD as it varies with time for three different values
of ξ = vT /R̄. As ξ increases, the MSD moves from a regime where it is
monotonic to one where oscillations occur. Panel (b) shows the threshold ξ
values for passing from one regime into another, for different choices of the
dimensionless parameters κ = KT /R̄2 and γ ′ = γT . The various plots have
been constructed using equation (3.30).
To visualise the movement of the animal inside a circular territory with
slowly moving borders, it is convenient to plot the time dependence of the
MSD, which is given by the following expression (Giuggioli et al., 2012)
39
(
Z +∞
+∞
X
4R2
R̄2 b̄(t)
dR R(R, t)
+
+ c̄(t) +
×
hr i =
2
4
µ20m J0 (µ0m )
0
m=1
sin (tΩ0m ) − µ20m c̄(t)
2µ1m c̄(t)
e 4R2 − 2
cos (tΩ0m ) +
×
2T Ω0m
(µ1m − 1)J1 (µ1m )
)
t
sin (tΩ1m ) − µ21m c̄(t)
e 4R2
cos (tΩ1m ) +
(3.30)
e− 2T ,
2T Ω1m
2
when the initial condition has both the animal and the territory centre at
the origin. For certain parameter values, damped oscillations in the MSD are
observed (Figure 3.4a). The values for which the MSD is monotonic depend
on the dimensionless parameters γ ′ = γT , κ = KT /R̄2 and η = vT /R̄ (Figure
3.4b). The oscillations occur if there is a sufficient amount of spatial correlation
in the walk, compared to the size of the territory. Therefore they are not seen
in the limit where the animal moves in a Brownian fashion.
3.4.2
Square territories
The probability of the animal to be at position (x, y) inside a square territory
at time t, whilst the territory has its centre at (Lx , Ly ) and is of width λ, is
given by the following expression (Giuggioli et al., 2012)
Px0 ,y0 (x, y, λ, Lx , Ly , t) =[H(x − Lx − λ/2) − H(x − Lx + λ/2)]×
[H(y − Ly − λ/2) − H(y − Ly + λ/2)]×
hx0 (x, t|λ, L)hy0 (y, t|λ, L)R(λ, t)Qx (Lx , t)Qy (Ly , t).
(3.31)
Here, (x0 , y0 ) is the position of the animal at time t = 0,
"
t
∞
1 e− 2T X
x − x0
hx0 (x, t|λ, Lx ) = +
cos nπ
+
λ
λ
λ
n=1
# sin(θn t)
x
−
x
−
2L
0
x
n
(3.32)
cos(θn t) +
(−1) cos nπ
λ
2T θn
is the probability of the animal being at position x along the x-axis at time t
given that the territory centre is at x-position Lx and is of width λ, T is the corp
relation time for the animal (see Table 2.1) and θn = n2 π 2 v 2 /λ2 − 1/(4T 2 ).
Similarly, for the movement along the orthogonal direction,
40
"
t
∞
y − y0
1 e− 2T X
cos nπ
+
hy0 (y, t|λ, Ly ) = +
λ
λ n=1
λ
# sin(θn t)
y
−
y
−
2L
0
y
n
(3.33)
cos(θn t) +
(−1) cos nπ
λ
2T θn
is the probability of the animal being at position y along the y-axis at time
t given that the territory centre is at y-position Ly and is of width λ. The
function
2 /b̃(t)
Rs (λ, t) =
e−(λ−L)
q
2 /b̃(t)
+ e−(λ+L)
(3.34)
π b̃(t)
is the probability of the territory width being λ at time t, the function
2
e−Lx /c̃(t)
Qx (Lx , t) = p
πc̃(t)
(3.35)
is the probability of the territory centre being at position Lx along the x-axis
at time t, and the function
2
e−Ly /c̃(t)
Qy (Ly , t) = p
πc̃(t)
(3.36)
is the probability of the territory centre being at position Ly along the y-axis
at time t. The time-dependent parameter
4K
b̃(t) =
γ
Z t
ds ϕ2d (s)
1 − exp −2γ
(3.37)
0
represents the movement of the territory width, so is the square-territory analogue of b(t). The time-dependent parameter
c̃(t) = 2K
Z
t
ds ϕ2d (s)
(3.38)
0
is the square-territory version of c(t), representing the movement of the territory centroid. The constant parameters Θ = (v, K, T , γ, L) represent, re-
spectively, the speed of the animal (v), the territory border diffusion constant
(K), the correlation time (T ), the tendency for a territory to move towards
an average width (γ), and the average territory width (L).
The marginal distribution of the animal, with arbitrary initial conditions,
is calculated by integrating Px0 ,y0 (x, y, λ, Lx , Ly , t) against the parameters
λ, Lx , Ly , x0 and y0 as follows
41
M2D (x, y, t|Θ) =
Z x+ λ
Z L
Z ∞
Z y+ λ
Z L
2
2
2
2
1
dλ
dx
dy
dL
dLy Px0 ,y0 (x, y, λ, Lx , Ly , t)
0
0
x
L2 − L
0
−L
x− λ
y− λ
2
2
2
2
(3.39)
The integrals in x0 , y0 , Lx and Ly can all be performed using standard tech-
niques to give the following expression
1
M2D (x, y, t|Θ) = 2
L
Z
∞
dλR(λ, t)V(x, λ, t)V(y, λ, t)
(3.40)
0
where
V(σ, λ, t) =
∞
X
t
L
sin(θn t)
e− 2T cos(θn t) +
U (σ, λ, t) +
×
2λ
2T θn
n=1
nπσ nπL
U (σ, λ, t)
n
sin
sin
,
(−1) Sn (λ, σ, t) −
nπ
λ
λ
(3.41)
and
!
!
λ − 2σ
λ + 2σ
p
p
U (σ, λ, t) = erf
+ erf
,
2 c(t)
2 c(t)
2 2
nπL
n π c(t)
1
sin
×
exp −
Sn =
nπ
2λ
λ2
"
!
inπσ
2λσ − λ2 + 2inπc(t)
p
−
Im erf
exp
λ
2λ c(t)
!
#
inπσ
2λσ + λ2 + 2inπc(t)
p
exp
,
erf
λ
2λ c(t)
(3.42)
where erf(z) is the complex error function, an extension of the real error func√ Rx
2
tion erf(x) = (2/ π) 0 dt e−t , and Im(a + ib) = b returns the imaginary part
of a complex number.
Despite the formidable form of equations (3.29) and (3.40), they can be
readily fitted to data using maximum likelihood techniques, as will be shown
in the next chapter. However, it turns out that fitting equation (3.40) to
data is about three or four orders of magnitude faster than using equation
(3.29). This is due to the extra integral in equation (3.29), as well as the use
42
of Bessel functions, which are slower to compute than the error functions used
in equation (3.40). For this reason, I use the square territory paradigm in the
next chapter to analyse location data.
Appendix E contains Matlab code for calculating the function (3.40), using an efficient numerical algorithm for computing complex error functions
(Weideman, 1994). The attached CD ROM contains a movie of the evolution
of the marginal distribution through time, whereas the utilisation distribution
is plotted in Figure 4.1 (Appendix F.2).
0.45
a)
0.4
0.35
γ’=0.001
γ’=0.1
γ’=1
0.12
0.115
0.3
0.25
0.2
0.15
0.11
NON−MONOTONIC MSD
0.105
0.1
0.095
0.1
0.09
0.05
0.085
0
0
b)
0.125
Threshold η
Normalised animal MSD
0.13
η=1 κ=0.01 γ’=0.1
η=0.15 κ=0.005 γ’=0.01
η=0.1 κ=0.01 γ’=0.01
5
10
15
20
Normalised time
25
0.08
−5
30
MONOTONIC MSD
−4.5
−4
−3.5
log10(κ)
−3
−2.5
Figure 3.5: Animal MSD in square territory. Normalised mean square
displacement (hx2 + y 2 i/L2 ) of a walker starting at the origin inside a square
territory with fluctuating boundaries, plotted against dimensionless time t/T .
Panel (a) shows the MSD as it varies with time for three different values of
η = vT /L. As η increases, the MSD moves from a regime where it is monotonic
to one where oscillations occur. Panel (b) shows the η values separating the
monotonic (upper region) from the nonmontonic (lower region) regime, for
various choices of the dimensionless parameters κ = KT /L2 and γ ′ = γT .
The various plots have been constructed using equation (3.43).
As in the circular case, I plot the time dependence of the MSD to visualise
the animal’s dynamics. When the initial condition has both the animal and
the territory centre at the origin, the MSD expression is (Giuggioli et al., 2012)
43
(
Z +∞
+∞
X
8λ2 (−1)n
L2 b(t)
dλRs (λ, t)
+
+ c(t) +
×
hx + y i =
6
12
π 2 (2n)2
0
n=1
sin(Θ2n t) − (2n)2 π22 c(t) 8(−1)n c(t)
4λ
+
cos(Θ2n t) +
×
e
2T Θ2n
π(2n − 1)
)
sin(Θ2n−1 t) − (2n−1)22π2 c(t) − t
4λ
e
cos(Θ2n−1 t) +
e 2T ,
2T Θ2n−1
2
2
(3.43)
which depends upon the dimensionless parameters γ ′ = γT , κ = KT /L2
and ξ = vT /L. As in the circular case, damped oscillations in the MSD are
observed if the amount of spatial correlation in the walk is sufficiently high
compared to the mean territory width (Figure 3.5).
A peculiar characteristic, which is evident from Figure 3.5, is the presence
of very sharp maxima in the non-monotonic regime. This feature can be
readily explained by examining the limit case K = 0. When η is in the nonmonotonic regime and n is sufficiently large, Θn ≃ nπη, so the first series in
equation (3.43) can be written as
∞
X
(−1)n n−2 {cos(2nπηt/T ) + sin(2nπηt/T )/[4nπηT ]} ,
(3.44)
n=1
whilst the second series vanishes since K = 0. The dominant contribution in
the infinite series (3.44) comes from the cosine terms, which display maxima at
t/T = (k + 1/2)/ζ with k = 0, 1, 2, ... These values correspond to the locations
of the maxima in Figure 3.5. To understand the apparent non-smooth nature
of the MSD at these maxima, the cosine series in (3.44) is differentiated to
give
∞
X
(−1)n n−1 sin(2nπηt/T ) = − arctan{sin(2πηt/T )/[1 + cos(2πηt/T )]},
n=1
(3.45)
which has discontinuous jumps at t/T = (k + 1/2)/η with k = 0, 1, 2, ..,
proving that the MSD is not smooth at the maxima. A similar argument
helps explain the appearance of sharp peaks in the circular case (Figure 3.4)
(Giuggioli et al., 2012).
44
3.5
Glossary
Table 3.1: Glossary of the key symbols used in
Explanation
Animal position at time t
Initial animal position
Position of left territory border
Position of right territory border
Animal diffusion constant
Territory border diffusion constant
Territory width (L2 − L1 in 1D)
Territory centre (L2 + L1 )/2
Rate at which territories tend to return
to an average size
ζ
Characteristic time-scale
ϕ1d
Time derivative of the border MSD
L
Mean territory width
′
D
Normalised D, D ′ = D/(L2 γ)
′
K
Normalised K, K ′ = K/(L2 γ)
β
Normalised γ, β = γζ
(r, θ)
Position of the animal at time t
(r0 , θ0 ) Initial position of the animal
(rc , θc ) Territory centre
R
Territory radius
R̄
Mean territory radius
ϕ2d
Time derivative of the border MSD
T
Correlation time (see section 2.3.1)
v
Animal velocity
(x, y)
Animal position at time t
(x0 , y0 ) Initial animal position
Lx
x-position of territory centre
Ly
y-position of territory centre
s∗
Long-time MSD of the territory width in 1D
a
Lattice spacing
F
Mean jump rate between adjacent lattice
sites in continuous time simulations
TAS
Active scent time (see table 2.1)
ρ
Population density
Z1D
Composite parameter Z1D = TAS ρ2 D in 1D
Symbol
x
x0
L1
L2
D
K
λ
L
γ
Chapter 3
Model
A1
A1
A1
A1
A1,S
A1,AC,AS,S
A1,AS
A1
A1,AC,AS
A1,AC,AS
A1
A1,AS
A1
A1
A1
AC
AC
AC
AC
AC
AC,AS
AC,AS
AS
AS
AS
AS
A1,S
S
S
S
S
S
Explanation of the various symbols used in Chapter 3. The third column
shows whether the parameter is used in the 1D analytic model (A1), the 2D
model with circular territories (AC), the 2D model with square territories
(AS) or the 1D simulation model of Chapter 2 (S).
45
Chapter 4
Application to data on urban
red foxes
Almost 80 years ago, Huxley (1934) hypothesised that territories are like elastic
discs, whose sizes fluctuate due to changes in neighbour pressure over time.
Whilst his original observations were of coot (Fulica atra) populations, such
fluctuations were also observed in Bristol’s foxes (Vulpes vulpes) in 1994-6,
when the population was decimated due to an epizootic of sarcoptic mange
(Baker et al., 2000). In this chapter, I re-analyse the data from that period,
using the theory developed in Chapters 2 and 3. This enables quantification
of both the elasticity in territorial patterns and the changes in fox behaviour
elicited by rapid population decline.
The data were split into the period before the outbreak of mange and the
period after. These are called the pre-mange and post-mange data sets, respectively. By fitting the probability distribution of territorial animals from
the model of Chapter 3, i.e. equation (3.40), to each of these two data sets, I
quantify the changes in the movement and interaction mechanisms employed
by the animals as the population density decreased. In particular, I examine
variations in the border movement (K), active scent time (TAS ), mean territory size (L2 ), the amount of directional persistence in the animal movement
process (T ), and the animal’s velocity (v) (see Table 4.2 for details of the
parameters).
I also examine the possibility of using evolutionary invasion analysis to
determine an evolutionarily stable strategy (ESS) for scent-marking behaviour
(Maynard Smith, 1982; Parker & Maynard Smith, 1990). If the TAS of an
invader is made different to the rest of the population, we observe differences in
both the territory size of the invader, compared with the other animals, and the
number of encounters with neighbouring animals. However, the evolutionary
47
costs and benefits of the two quantities may not be equal. For example, in
the Bristol fox population, encounters with neighbours tend to be aggressive
(White & Harris, 1994), even fatal (Harris & Smith, 1987). This suggests that
minimising encounters is of great evolutionary importance, whereas increasing
territory size may be of comparatively little importance. For other species
or environments, the relative benefits of maximising territory size compared
to minimising encounters are likely to be different. Though I was not able
to obtain any concrete results are from this analysis, I give some pointers to
future directions.
4.1
Data collection and analysis
Location data were taken from a long-term study of the red fox population
in the Bristol urban area. These were gathered between 1990 and 1995 by
previous members of one of my supervisor’s (SH) lab. Radio fixes with a
spatial resolution of 25m × 25m were taken every 5 minutes between 20:00
and 04:00 GMT, which encompasses most fox activity (Saunders et al., 1993),
so throughout this chapter ‘1 day’ is equal to 8 hours of fox location data.
Radio telemetry data from 22 different territorial adult foxes (i.e. > 1 year
old) were analysed, recorded between 1990 and 1995. Data from both males
and females were used because the space use distributions of a male and a
female from the same group are very similar (White et al., 1996). Each fox
was tracked during one, two or three seasons (spring, March-May; summer,
June-August; autumn, September-November; winter, December-February; see
Table 4.1). The last fox in the study area died in spring 1996 (Baker et al.,
2000).
The log maximum likelihood method was employed to fit data to the theoretical probability distribution M2D (x, y, t|Θ) (equation 3.40). In particular, the Nelder-Mead simplex algorithm (Nelder & Mead, 1965; Lagarias et al.,
P
1998) was used to find the maximum of L(Θ) = n ln[M2D (xn , yn , tn |Θ)] for
each set of parameter values Θ, where the sum is taken over 99% of position-
time locations (xn , yn , tn ) that attain the highest M2D (xn , yn , tn |Θ) values. I
excluded 1% of outliers to ensure the results were not biased by anamolous
behaviour (Harris et al., 1990; Kenward et al., 2001).
The Nelder-Mead algorithm is useful in situations where the data contain a lot of variation between individual agents (i.e. is noisy), as is the
case with our fox location data (Bortz & Kelley, 1998). Furthermore, the algorithm is quick to converge to a solution. The complicated nature of the
function M2D (x, y, t|Θ) means that computing it numerically many times over
48
Fox
number
2308
2310
2316
2319
2320
2332
1800
2308
2318
2320
2332
2337
2345
2543
1800
2308
2554
2570
2581
2332
2748
2751
2796
2843
2993
2748
2750
2827
2748
2750
Table 4.1: Details of fox radio-tracking data
Season
Nights Sex Location Pre/Posttracked
fixes
mange
Spring 1990
2
F
191
Pre
Spring 1990
3
F
279
Pre
Spring 1990
3
M
285
Pre
Spring 1990
3
F
281
Pre
Spring 1990
2
M
185
Pre
Spring 1990
2
M
191
Pre
Spring 1991
4
M
388
Pre
Spring 1991
9
F
930
Pre
Spring 1991
3
F
281
Pre
Spring 1991
6
M
570
Pre
Spring 1991
8
M
862
Pre
Spring 1991
4
F
400
Pre
Spring 1991
4
F
373
Pre
Spring 1992
8
F
771
Pre
Spring 1993
3
M
290
Pre
Spring 1993
3
F
291
Pre
Spring 1993
1
M
97
Pre
Spring 1993
1
F
90
Pre
Spring 1993
4
F
387
Pre
Spring 1994
4
M
388
Pre
Spring 1994
3
M
290
Pre
Spring 1994
3
M
291
Pre
Spring 1994
3
F
289
Pre
Spring 1994
3
F
283
Pre
Autumn 1995 4
M
337
Post
Autumn 1995 7
M
383
Post
Autumn 1995 7
M
428
Post
Winter 1995
5
M
344
Post
Winter 1995
4
M
226
Post
Winter 1995
12
M
595
Post
Details of fox radio-tracking data: the seasons during which each fox was
tracked, the number of nights tracked and number of radio location fixes
gathered during that season, and whether the season was before or well after
the summer 1994 outbreak of sarcoptic mange. In total, pre-mange there are
8693 data points, post-mange 2313.
for different parameter values can be highly time-consuming, so using a fast
algorithm is very important from a practical perspective.
Making use of this algorithm requires finding a starting value for Θ, called
Θ0 = (v0 , K0 , T0 , γ0 , L0 ), that is expected to be close to the maximum to which
the algorithm converges. The value v0 was set to be the total distance moved
49
by the foxes divided by the total time moved. T0 was obtained by the formula
T0 = −5/ ln[hcos(θ)i] minutes, where hcos(θ)i is the mean of the cosines of the
turning angles θ from the data (Viswanathan et al., 2005). The factor of 5
comes about since location measurements were taken every 5 minutes. Since
the long-time MSD of the animal is 2Kt/ ln(t/T ) (equation 3.43), K0 was
obtained by fitting a curve A + 2K0 t/ ln(t/T0 ) to the fox MSD against time t
for t > 1 day, using the least-squares method, where A is a fitting constant.
L0 was found by taking the square root of the mean 100% minumum convex
polygon home range area (Harris et al., 1990). To find γ0 , the maximum of
L(Θ) was calculated for Θ = (v0 , K0 , T0 , γ, L0 ) as γ varies across parameter
space from γ = 10−3 to γ = 104 . Error bars for the best fit were obtained
using the bootstrap algorithm for variance calculation, see e.g. Wasserman
(2004), by resampling each data set 100 times.
Once the Nelder-Mead algorithm had been used to find the mean value of
Θ, the standard deviation was calculated by using the bootstrap algorithm,
see e.g. Wasserman (2004). This involved sampling M = 100 times from
each data set, pre- and post-mange. For each i from 1 to M , a total of N
points x1,i , . . . , xN,i were picked from the data set, with replacement, where
N is the total number of location fixes (N = 8693 pre-mange and N = 2313
post-mange). Then the Nelder-Mead algorithm was used to find the maximum
likelihood parameter value Θi of the sample x1,i , . . . , xN,i , for each i. The error
bars for Θ were obtained by finding the standard deviation of Θ1 , . . . , ΘM .
Each value of Θi takes approximately 2 hours of CPU time to calculate using
Matlab (version R2009a) on a 2.96 GB RAM desktop computer with a 3.0
GHz processor (see Appendix E for Matlab code to calculate M2D (x, y, t|Θ)).
4.2
Territorial dynamics pre- and post-mange
To investigate the effect of population decline on the animal’s movements and
interactions, both the pre-mange and post-mange data sets were fitted to the
theoretical probability distribution M2D (x, y, t|Θ) of territorial animal movement (equation 3.40). Table 4.2 details the parameters Θ = (v, K, T , γ, L)
that give the best fit of each data set to M2D (x, y, t|Θ). Figure 4.1 shows
all the pre-mange fox positions, super-imposed on the utilisation distribution
from the model, i.e. the expected distribution of animal fixes across a seaRT
son. This distribution is given by the integral 0 ∗ dt M2D (x, y, t|Θ), where
T∗ = 84.0 days is the maximum time between the first and the last location
fix for any of the seasons during which data were gathered. To find a video of
how the data and distribution vary over time, see Appendix F.2.
50
1.5
Normalised Y Position
1
0.5
0
−0.5
−1
−1.5
−1.5
−1
−0.5
0
0.5
Normalised X Position
1
1.5
Figure 4.1: Fox data with theoretical utilisation distribution. The
utilisation distribution of the analytic model of animal movement inside a
slowly moving territory, with location fixes of Bristol foxes before the 1994
outbreak of sarcoptic mange, and parameter values fitted to the pre-mange
data. The fixes have been normalised so that the centre of each seasonal home
range is at position (0, 0) and the distances from the centre have been divided
by L = 435 metres, the average territory width. Different symbols denote
different foxes. The contours are placed at deciles (1/10, 2/10, 3/10 etc.) of
the distribution height, except for the outer two, which are placed at 1/100
and 1/1000 of the height. To view the evolution of the probability distribution
over time, see Appendix F.2.
As well as the foxes having had a much larger average velocity after the
mange outbreak, the value of K increased more than eightfold, meaning that
territory borders moved much more rapidly after the population density declined. This gives a quantitative basis to the elastic disc hypothesis (Huxley,
1934), which conjectures that territories will expand as a result of reduced
population density. As the foxes died out and territories were vacated, neighbouring foxes took over the newly-free areas, causing the borders to move
and the territories to enlarge, as evidenced by an increase in average territory
size from L2 = 0.189 km2 pre-mange to L2 = 0.857 km2 post-mange. Table
4.2 shows the values of the territory width L, which is the square root of its
51
area. The increase in γ after the mange outbreak suggests that territories
were pushed towards an average size faster when the population density was
lower, in keeping with the idea that territories were more ‘elastic’ during the
post-mange period.
Whilst the Bristol foxes increased v as the population density dropped,
surprisingly they did not increase T . That is, the turning angle distribution
had similar statistics both before and after the mange outbreak. However,
due to their increased velocity, the distance for which they would persist in
roughly the same direction, vT , was 2.5 times higher post-mange. In other
words, the mean step length between turns was higher. The lack of change
in turning angle distribution indicates an inbuilt species- or habitat-specific
behavioural strategy.
The amount of home range overlap, HRO, between neighbours is calculated
by assuming that the two territories share a common edge. The MSD of this
edge in the perpendicular direction is Kt/ ln(t/τ ), due to the 2D exclusion
process (see section 2.3.2). This increases with time, so to obtain an estimate of
the home range, it is necessary to pick a time T∗ to measure the displacement of
the common edge. In other words, T∗ represents the time window during which
location measurements are taken. The width of the overlapping strip between
the two neighbouring home ranges is equal to the mean absolute displacement
p
p
√
KT∗ / ln(T∗ /τ ). The width of the home range is 1/ ρ + KT∗ / ln(T∗ /τ ).
Therefore the fraction of the home range that overlaps with this particular
neighbour is
HRO = 1 −
1
1+
p
KT∗ ρ/ ln(T∗ /τ )
.
(4.1)
Notice that as T∗ increases, the fraction of overlap increases towards the theo-
retical maximum value of 1, where the two home ranges coincide. For the fox
data, T∗ = 84 days, which is the maximum time between the first and the last
location fix for any of the seasons during which data were gathered.
Using equation (4.1), the mean overlaps in the fox population turn out to be
HRO = 24.7% ± 3.3% pre-mange and HRO = 30.6% ± 2.5% post-mange (error
bars are 1 standard deviation throughout this chapter). This apparent increase
in percentage overlap is not statistically significant (p = 0.07, assuming HRO is
normally distributed (Wasserman, 2004)). Previous studies using this data set
(Baker et al., 2000) applied minimum convex polygon techniques to measure
the overlaps directly. Similarly, there appeared to be a small increase that was
not statistically significant.
Were TAS , v and T to have remained constant as the population density
52
0
Nearest Neighbour RW
Ballistic walk
−2
10
−4
10
−6
10
−8
10
0
Percentage overlap
Normalised territory border movement
10
60
40
20
0
0
2
4
6
−2
Population density (km )
5
10
Normalised active scent time
15
Figure 4.2: Home range overlap due to border movement. Simulation output showing the dependence of territory boundary movement on
the active scent time for nearest-neighbour random walks (NNRW) and ballistic walks (BW), also shown in Figure 2.9 but repeated here for convenience. The vertical axis is K/v 2 τ and the horizontal is Z2D = TAS v 2 τ ρ
(see Table 4.2 for definitions of terms). The crosses and circles show values
from the simulation output. The solid line is a best fit for the NNRW case
log10 (K/v 2 τ ) = 0.085 − 0.247Z, the dashed line the best fit for the BW case
log10 (K/v 2 τ ) = 0.467 − 0.709Z. The inset shows the percentage of each home
range that overlaps with neighbouring ranges. The solid (NNRW) and dashed
(BW) lines use fixed values of TAS , v, T∗ and τ = T taken from the pre-mange
data, whereas the dotted (BW) and dot-dashed (NNRW) lines use post-mange
data.
decreased, there would have been a dependency of the home range overlap
on the population density of the type shown in the inset of Figure 4.2 (solid
curve). In such a case, unless the density is very low, the percentage of overlap decreases as the population density increases. However, for extremely low
densities, neighbouring animals would be so far apart that they are unlikely
to encounter one another’s territorial borders, meaning the ranges would overlap little. This trend in home range overlap implies that the home range size
p
HRS = L2 [1 + KT∗ ρ/ ln(T∗ /τ )]2 is proportional to ρ−α where α > 1, as-
suming fixed TAS , v and T . The home range size is estimated by taking the
p
square of the mean home range width L[1 + KT∗ ρ/ ln(T∗ /τ )]. Using the
53
pre-mange values of these parameters, a straight line was fitted to the values
of log(HRS) against log(ρ) (using linear least-squares) to find that α ≈ 1.74.
4.3
Inferring active scent time from location data
The value of TAS for the fox population is found by using the best-fit values of
K from the analytic model together with the nearest neighbour random walk
(NNRW) trend curve from the simulation output (Figure 4.2). Since animals
that move with a correlation time T appear to be uncorrelated random walkers
when sampled at a temporal resolution lower than or equal to T , the time it
takes to move between adjacent lattice sites is set equal to the correlation
time, i.e. τ = T (see Table 4.2).
For the pre-mange data, the dimensionless value K/v 2 T was 4.89 ± 0.54 ×
10−3 , whereas post-mange K/v 2 T = 6.19 ± 0.56 × 10−3 . The best-fit curve
from the NNRW simulation output, log10 (K/v 2 T ) = 0.085 − 0.247Z2D , gives
Z2D = 10.5 ± 0.2 pre-mange and Z2D = 10.0 ± 0.2 post-mange. To link
the simulation parameters to the analytic model, the population density was
assumed to be ρ = 1/L2 , and was 5.29 ± 1.02 fox territories per km2 pre-
mange and ρ = 1.16 ± 0.12 km−2 post-mange. Therefore TAS = Z2D /(v 2 T ρ)
was 5.07± 0.55 days pre-mange and 3.37± 0.16 days post-mange, a statistically
significant decrease (p = 0.002, assuming TAS is normally distributed).
The active scent time was also used to infer how long it took for a neighbour
to seize a territory once it had been vacated, the so-called territory acquisition
time Tat , by running NNRW simulations. Once the simulations had run for
long enough so that the territory borders had MSDs proportional to t/ ln(t),
one animal was removed from the simulation. The territorial acquisition time
Tat is the time it takes before 90% of the removed animal’s territory has been
scented by other animals, so claimed as their territory. See Appendix F.3 to
view a movie of this situation. Figure 4.3 shows Tat , for the various different
values of TAS and ρ that had been simulated.
For this analysis, I used dimensionless active scent times ranging between
TAS v/a = 500 and TAS v/a = 9000, together with dimensionless population
densities ρa2 = 9, 16, 25, 50, 100 × 10−4 , where TAS is the active scent time, ρ
the population density, a the distance between lattice sites and v the speed of
the animal. Except when the population density was very low, the dimensionless value (Tat − TAS )ρv 2 τ tended to range between about 3.5 and 5.5 (Figure
4.3). For the post-mange case of Z2D = 10.0, the value of (Tat − TAS )ρv 2 τ
was found to be 4.50, by averaging over various values of ρ, v, τ and TAS
such that Z2D = 10.0. This implies that the territory acquisition time Tat was
54
T v/a=500
9
AS
Time−lag for territorial acquisition
TASv/a=1000
TASv/a=2000
8
TASv/a=3000
TASv/a=4000
TASv/a=5000
7
TASv/a=6000
TASv/a=7000
6
TASv/a=8000
TASv/a=9000
5
4
3
1
2
3
4
5
6
7
Dimensionless population density
8
9
10
−3
x 10
Figure 4.3: Time lag for territory acquisition. The dimensionless timelag, (Tat − TAS )v 2 τ ρ, between all scent from a vacated territory becoming
non-active and 90% of the territory being acquired by conspecifics. The dimensionless population density is a2 ρ.
approximately 5 days.
4.4
Evolutionary invasion analysis
As a step towards performing evolutionary invasion analysis with my model,
simulations of a population of territorial NNRW animals are run whereby each
animal retreats from neighbouring scent that had been present for less than a
time Tpop , except for one animal, the invader, who retreats from neighbouring
scent up to a time Tinv 6= Tpop . Therefore, the active scent time is Tpop for the
population and Tinv for the invader.
The invader might be a bold invader, so that Tinv < Tpop or a shy invader, Tinv > Tpop . A bold invader would tend to end up with a larger territory than the other animals, which may give it an evolutionary advantage
(Iossa et al., 2008). However, such an invader is also more likely to encounter
conspecifics directly. For red foxes, inter-group encounters tend to be aggres55
sive (White & Harris, 1994), so a higher encounter rate is evolutionarily disadvantageous. Shy invaders, on the other hand, tend to have fewer encounters
than the rest of the population, but smaller territories.
To perform a full invasion analysis of this system would first require determining the respective evolutionary advantages of increasing territory size
and avoiding encounters (Parker & Knowlton, 1980), which may vary from
species to species. Then, by varying both Tinv and Tpop over a wide range
of values, it would be possible to uncover which Tinv strategies perform better than the population from an evolutionary perspective, therefore which are
able to invade the population. If it turns out that there is a Tpop strategy for
which no Tinv is evolutionarily more advantageous then the population strategy is, by definition, an ESS. Any attempt by another strategy to invade such
a population will ultimately fail.
The inherent difficulty in performing this analysis is two-fold. First, the
simulations are very computationally intensive. It takes approximately 68
hours CPU time, using a 3.0 GHz processor in a 2.96 GB RAM desktop computer, to determine the ratios Sinv /Spop and Rinv /Rpop for just one set of
parameters, where Sinv is the size of the invader’s territory, Rinv is the rate at
which the invader encounters conspecifics, Spop is the average territory size of
the other animals in the population and Rpop the average rate at which any
other animal in the population encounters conspecifics.
Second, a population dynamics model involving birth, death and reproduction parameters, similar to López-Sepulcre & Kokko (2005), would be required to determine the relative fitness benefits of increasing territory area and
avoiding aggressive encounters. By analysing the convergence of such a model
towards its stable states, it would be possible to demonstrate more clearly
the conditions under which a scent marking strategy would be evolutionarily
stable for a given population. Such an analysis would be a separate project
in itself so I do not embark on it here. However the methods outlined in
this section provide some initial pointers towards building and parameterising
such a model in the future, by relating the difference between the active scent
times of the population and the invader to their respective territory areas and
encounter rates.
56
4.5
Glossary
Symbol
K
T
L
γ
v
ρ
T∗
TAS
Tat
a
Spop
Sinv
Rpop
Rinv
Tpop
Tinv
τ
Z2D
Table 4.2: Glossary for Chapter 4 and best-fit values
Explanation
S/A Pre-mange Post-mange
Territory border diffusion
A
3.97 ± 0.09 32.8 ± 0.6
constant (m2 /min)
Correlation time (mins)
A
14.9 ± 0.5
14.8 ± 0.4
(see section 2.3.1)
Average territory width (m)
A
435 ± 37
926 ± 44
Territory renormalisation rate A
108 ± 2
140 ± 2
(min−1 ) (see Table 3.1)
Mean animal velocity
A,S 7.39 ± 0.16 18.9 ± 0.3
(m/min)
Population density (km−2 )
A,S 5.29 ± 1.02 1.17 ± 0.12
Time window over which
A,S 84.0
84.0
home range is measured
(days)
Active scent time (days)
S
5.07 ± 0.55 3.37 ± 0.16
Territory acquisition time
S
4.89 ± 0.16
(days)
Lattice site separation
S
(length)
Population mean territory
SI
size (length2 )
Invader territory size
SI
2
(length )
Population mean encounter
SI
−1
rate (time )
Invader encounter rate
SI
(time−1 )
Population TAS (time)
SI
Invader TAS (time)
SI
τ = a/v (time)
S
Z2D = TAS ρv 2 τ (none)
S
10.5 ± 0.2
10.0 ± 0.2
Glossary of parameters used this chapter, together with their values for the
fox data pre-mange and post-mange, if applicable. Error bars are 1 standard
deviation using the bootstrap algorithm (see main text for details). The
second column details whether the parameter is from the analytic model (A)
or simulation model (S). If the parameter was only used for the simulations
in the invasion analysis, then S is replaced by SI in the second column. The
final two rows display the composite parameters used in the chapter.
57
Chapter 5
Towards an analytic model of
territory border movement
The striking universal curves of Figure 2.3, relating the border diffusion constant K to the dimensionless parameter Z1D = TAS ρ2 D (see Table 5.1 for the
symbols used in this Chapter), beg the question as to whether it is possible
to derive this relation analytically. In this chapter, I outline a programme for
making such analysis, and detail some calculations that form the first steps
towards this end.
Figure 5.1: Reduced model of territorial dynamics. Diagram of a model
of territorial dynamics that reduces the interacting particle model described
on Chapter 2. Territories are modelled as springs, joined together by borders
that are modelled as diffusive particles. Zooming in on a border point reveals
that it consists of two boundaries, each of which is moving randomly but with
a drifting tendency towards the other, given by the jump probability p > 1/2.
I begin by sketching a mathematical representation of the territorial sys59
tem (Figure 5.1). Each spring represents a territory, whose width fluctuates
around a mean length equal to the inverse of the animal population density.
Each border is a particle whose movement is intrinsically random, though also
constrained by the presence of the connected springs. Consequently, since this
is a form of symmetric exclusion process, the resultant movement of a tagged
border particle is subdiffusive (Lizana et al., 2010).
However, by zooming in, it becomes apparent that each border is actually
made of two boundaries, one for each of the two adjacent territories. In this
chapter, I study the process by which the movement of these two boundaries
gives rise to the intrinsic random movement of the border. Each boundary
has a probability p > 1/2 of moving towards the other. Therefore I studied a
system of two random walkers on an infinite 1D lattice whereby each walker
has an intrinsic bias, but the bias of one is anti-symmetric to the other. That
is, the probability of the left-hand particle jumping right (left) at each step is
0 < p < 1 (resp. 1 − p) and the probability of the right-hand particle jumping
left (right) is also p (resp. 1 − p).
The diffusion constant of a tagged particle in this system turns out to be
proportional to 1 − p, as long as p > 1/2. Simulation analysis reveals that
1 − p decays exponentially as Z1D is increased, suggesting that understanding
p is key to the analysis of the universal curves of Figure 2.3. I show that p
is related to the first passage time for an animal to cross its territory, and
conclude the chapter by outlining the rest of the programme for deriving an
approximate analytic theory of territory border movement.
5.1
An anti-symmetric exclusion process
The starting point of my investigation consists of writing a master equation
for the joint occupation probability Pn,m (t) of the two particles being at site n
and m at time t. In relation to the territoriality problem, the two particles are
the two boundaries that constitute a border, disregarding the presence of other
territories. Both cannot occupy the same site at the same time, but unless
impeded by this constraint, at each hop the left-hand (right-hand) particle
moves right (left) with probability p and left (right) with probability 1 − p. In
other words, the two particles exhibit anti-symmetric movement processes.
The hopping rate, i.e. hopping probability per unit time, is denoted by W
and the lattice spacing by a. As particles may only hop to nearest-neighbour
sites, Aslangul’s construction (Aslangul, 1999) can be used so that
60
dPn,m
(t) =2W [Pn+1,m (t) + Pn,m−1 (t)](1 − p)(1 − δn,m−1 )(1 − δn,m )
dt
+ 2W [Pn−1,m (t) + Pn,m+1 (t)]p(1 − δn,m+1 )(1 − δn,m )
− 4W (1 − δn,m+1 )(1 − δn,m )(1 − p)Pn,m (t)
− 4W (1 − δn,m−1 )(1 − δn,m )pPn,m (t).
(5.1)
The term 1 − δn,m , where δ is the Kronecker delta, δn,m = 0 if n 6= m and
δn,n = 1, represents the fact that two particles cannot hop from the same
lattice site, whereas 1 − δn,m±1 represent the situations where both particles
occupy adjacent lattice sites and so neither can move towards the other on the
next hop.
To seek the exact solution of (5.1), it is convenient to use the generating
function (Montroll, 1964) for Pn,m (t), which is
f (φ, ψ, t) =
∞
X
Pn,m (t)einφ eimψ .
(5.2)
n,m=−∞
The master equation (5.1) implies the following relation for the generating
function
df (φ, ψ, t)
(t) = − 4W f (φ, ψ, t)
dt
Z 2π
dφ′ ′
′
′
−4W
1 + cos1−p (φ − φ ) f (φ , φ + ψ − φ , t)
2π
0
+2W (cosp φ + cos1−p ψ)f (φ, ψ, t)
Z 2π
dφ′
f (φ′ , φ + ψ − φ′ , t) [cosp φ + cos1−p (ψ)]
−2W
2π
0
Z 2π
dφ′
−2W
f (φ′ , φ + ψ − φ′ , t) cosp (φ′ ) + cos1−p (φ + ψ − φ′ ) ,
2π
0
(5.3)
where I have introduced the notation cosp (θ) = peiθ + (1 − p)eiθ so that cos 1 ≡
2
cos.
At time t = 0 the particles occupy two lattice sites, denoted by N1 and N2 .
Without loss of generality, I assume N1 < N2 and since the particles cannot
cross, the particle starting at N1 is referred to as the left-hand particle, the
other is the right-hand particle. By using this initial condition and setting
61
θ = φ + ψ, the Laplace transform of (5.3) is
f˜(φ, θ − φ, ǫ) = g(θ, φ) +
where f˜(φ, θ − φ, ǫ) =
able ǫ, and
R∞
0
3
X
ai (θ, φ)
i=1
Z
2π
0
dφ′ bi (φ′ )f˜(φ′ , θ − φ′ , ǫ),
(5.4)
dtf (φ, ψ, t)e−ǫt is the Laplace transform with vari-
ei(θ−φ)∆N eiN1 θ
,
ǫ + 2W [2 − cosp φ − cos1−p (θ − φ)]
2W [2 − cosp φ − cos1−p (θ − φ)]
,
a1 (θ, φ) =
ǫ + 2W [2 − cosp φ − cos1−p (θ − φ)]
g(θ, φ) =
a2 (θ, φ) =
a3 (θ, φ) =
2W (1 − p)[2eiφ − (1 + eiθ )]
,
ǫ + 2W [2 − cosp φ − cos1−p (θ − φ)]
2W p[2e−iφ − (1 + e−iθ )]
,
ǫ + 2W [2 − cosp φ − cos1−p (θ − φ)]
b1 (φ) =
1
,
2π
b2 (φ) =
e−iφ
,
2π
b3 (φ) =
eiφ
,
2π
(5.5)
where ∆N = N2 − N1 . For a given value of θ, by setting h(φ) = f˜(φ, θ − φ, ǫ)
and writing ai (φ) = ai (θ, φ), g(φ) = g(θ, φ) to ease notation, (5.4) can be
written in terms of the single variable φ as follows
h(φ) = g(φ) +
3
X
ai (φ)
i=1
Z
2π
dφ′ bi (φ′ )h(φ′ ).
(5.6)
0
This is a Fredholm integral equation with degenerate kernel, so can be solved
exactly (Polyanin & Manzhirov, 1998). After some lengthy algebra (see Appendix B), the following solution is eventually found
∆N
2
φ−ψ
φ
+
ψ
p
u
(1−∆N
)u
i∆N
2
e
cos
f˜(φ, ψ, ǫ) = e
)
(e − e
e
2
1−p
∆N
1 )
2
2
ψ+φ
p
p
×
− ei∆N ψ
+ e(1−∆N )u eiψ ei(∆N −1) 2
1−p
1−p
(
1 )−1
2
φ+ψ
p
u
ǫ + 2W (2 − cosp φ − cos1−p ψ) e cos
−
.
2
1−p
iN1 (φ+ψ)
(
i∆N ψ
(5.7)
62
Here, u is defined by the equation
eu =
Z +1+
q
(Z + 1)2 − 4p(1 − p) cos2
1
2[p(1 − p)] 2 | cos θ2 |
θ
2
,
(5.8)
where Z = ǫ/4W and the branch of the square root function used here, and
elsewhere throughout the chapter, is the one that takes real positive values
when the argument is a positive real number.
5.2
Asymptotic analysis
For any initial condition, there is a non-zero probability of the two particles
being at adjacent lattice sites at some point in time, which I call t0 . Therefore,
for the purposes of examining the asymptotics, the initial condition is taken to
be the configuration at time t0 , by additionally, and without loss of generality,
assuming that the left particle is at position 0 at time t0 . That is, the initial
conditions are N1 = 0, N2 = 1, giving rise to a simpler form for (5.7)
f˜(φ, ψ, ǫ) =
eiψ | cos φ+ψ
2 |
1
4W [Z + 1 − 2 (cosp φ + cos1−p ψ)]
×
where R(θ, Z) = Z + 1 −
R(φ + ψ, Z)ei
φ−ψ
2
− 2(1 − p)| cos φ+ψ
2 |
R(φ + ψ, Z) − 2(1 − p) cos2
q
φ+ψ
2
,
(5.9)
(Z + 1)2 − 4p(1 − p) cos2 2θ . This readily reduces
to a result of Aslangul (1999, equation 2.11) when p = 21 .
The marginal distribution for the left-hand (resp. right-hand) particle
can be calculated by setting ψ = 0 (resp. φ = 0). For −π < φ < π, the
following expression for the generating function of the distribution of the left-
hand particle in Laplace domain is found, when the right-hand particle can be
anywhere else
φ
cos φ2
R(φ, Z)ei 2 − 2(1 − p) cos φ2
f˜(φ, 0, ǫ) =
.
.
4W (Z + 21 − 12 cosp φ) R(φ, Z) − 2(1 − p) cos2 φ2
(5.10)
This allows the mean position hx1 (ǫ)i of the left-hand particle to be calculated
in Laplace domain, by differentiating (5.10) with respect to φ, multiplying by
63
−ai and setting φ = 0, with the result
a
hx1 (ǫ)i =
4ǫ
1p 2
ǫ + 8W ǫ + 16W 2 (1 − 2p)2
1−
ǫ
+
aW (2p − 1)
.
ǫ2
(5.11)
Differentiating (5.10) twice with respect to φ, multiplying by −a2 and again
setting φ = 0 gives the second moment of the distribution
hx21 (ǫ)i
a2
=
4ǫ
1p 2
8W
2
2
−
1+
ǫ + 8W ǫ + 16W (1 − 2p)
ǫ
ǫ
p
a2 (1 − 2p) 2
2 + 8W ǫ + 16W 2 (1 − 2p)2 .
ǫ
4W
(1
−
2p)
+
W
+
ǫ3
(5.12)
By using the fact that L−1 [(ǫ2 + 2bǫ + b2 − a2 )−1/2 ] = e−bt I0 (at), where L−1
denotes the inverse Laplace transform and Iν (z) a modified Bessel function
of order ν, expressions (5.11) and (5.12) can be inverted exactly to give the
respective formulae in time domain
hx1 (τ )i =
a
4
Z
p
4(2p − 2)τ + 8 p(1 − p)
τ
ds
0
p
τ − s −4s
e I1 [8 p(1 − p)s] ,
s
(5.13)
hx21 (τ )i = a2 (2 − 2p)τ + 2(1 − 2p)(2 − 2p)τ 2
Z τ
p
p
τ − s − 2(1 − 2p)(τ − s)2 −4s
ds
+ 2 p(1 − p)
e I1 [8 p(1 − p)s] ,
s
0
(5.14)
where τ = tW is dimensionless time and x1 (τ ), x2 (τ ) are the positions of the
left- and right-hand particle, respectively. Let d(τ ) = hx2 (τ ) − x1 (τ )i be the
mean separation distance between the particles. Since the second moments of
the particles coincide and hx1 (τ )i = −hx2 (τ )i, it is convenient to denote by
hx2 (τ )i the second moment of either particle and by ∆x2 (τ ) = hx2 (τ )−hx(τ )i2 i
the mean-square displacement.
If p =
1
2
then the integrals in (5.13) and (5.14) can be computed exactly
p
R∞
(Aslangul, 1999). For p 6= 21 , the integrals 0 dssn e−4s I1 [8 p(1 − p)s] for
p
n = −1, 0, 1 are the Laplace transforms of tn I1 [8 p(1 − p)t] evaluated at the
point where the Laplace variable is equal to 4, that is
64
p
1 − |1 − 2p|
L{t−1 I1 [8 p(1 − p)t]}(ǫ)|ǫ=4 = p
,
2 p(1 − p)
p
1 − |1 − 2p|
,
L{I1 [8 p(1 − p)t]}(ǫ)|ǫ=4 = p
8 p(1 − p)|1 − 2p|
p
p
p(1 − p)
L{tI1 [8 p(1 − p)t]}(ǫ)|ǫ=4 =
.
8|1 − 2p|3
a)
2
10
p<0.5
p=0.5
p>0.5
200
b)
1
10
150
τ’
(<x2>−<x>2)/a2
250
100
0
10
50
0
0
50
τ
100
150
0.495
0.5
p
0.505
Figure 5.2: Divergence timescales of three regimes. Panel (a) shows the
MSD as it varies through time for values of p close to 12 , demonstrating when
the MSD begins to split into three regimes, p < 21 , p = 12 , p > 21 . Values
of p from the top curve to the bottom are p = 0.45, 0.49, 0.499, 0.5, 0.501,
0.505, 0.51. Panel (b) shows the timescale τ ′ beyond which the MSD curves
for different values of p diverge by more than 1% from the curve for p = 21 .
Each of these three terms is finite for p 6=
1
2,
so this allows the asymptotic
expressions for (5.13) and (5.14) to be calculated, yielding the following expressions for τ ≫ 1:

p


a,
if 21 < p < 1,

 2p−1 q
√
d(τ ) ≈ 12 a + π8 a τ ,
if p = 21 ,



 1−p a + 4a(1 − 2p)τ, if 0 < p < 1 .
1−2p
2
2
hx (τ )i ≈

(1−p)2 2


a + 2a2 (1 − p)τ,

 2(1−2p)2
1 2
2
4 a + 2a τ,



 −p2 a2 +
2(1−2p)2
if
(5.15)
1
2
< p < 1,
if p = 21 ,
2a2 (1 + p)τ + 4a2 (1 − 2p)2 τ 2 , if 0 < p < 12 .
(5.16)
65

(1−p)2 2
2


2 a + 2a (1 − p)τ,

4(1−2p)

√
3 2
∆x2 (τ ) ≈ 16
a − √12π a2 τ + 2a2 (1 − π1 )τ,



 −3p2 a2 + 2a2 τ,
4(1−2p)2
if
1
2
< p < 1,
if p = 12 ,
(5.17)
if 0 < p < 21 .
The different qualitative behaviours in both the MSD and the mean separation
distance are now evident. The limits p → 12 and t → ∞ do not commute, so
the asymptotic diffusion constant is very different in the case p = 12 from
the cases where p is either just above or just below 21 . Figure 5.2 shows the
timescales in which the three regimes diverge from one another.
30
〈 x 〉/a
2
6
2
p=0.4
p=0.5
p=0.6
asymptotics
〈(x−〈 x〉)〉2/a2τ
8
2
40
d(τ)/a
c)
b)
a)
10
20
4
1
10
2
0
0
5
τ
10
0
0
1.5
5
τ
20
10
40
τ
60
80
100
Figure 5.3: Comparison with simulations. The asymptotic expressions
from (5.15), (5.16) and (5.17) are compared with average values of 106 stochastic simulations of the system for p = 0.4, 0.5, 0.6. Panel (a) demonstrates how
the mean distance between particles d(τ ) exhibits qualitatively different behaviour in the three regions p < 12 , p = 21 and p > 12 when plotted against
dimensionless time τ . Panel (b) shows the quadratic nature of the asymptotic
second moment of a tagged particle when p > 12 , as compared with p = 21 or
p < 21 when the second moments are asymptotically linear. In panel (c), the
particles are shown to reach their asymptotic diffusion constants.
For d(τ ), the different qualitative dependencies occur in the exponent of
time so that for p <
1
2
the displacement saturates, whereas for τ ≥
increases. Furthermore, this increase is linear for p >
p=
1
2.
1
2
1
2
it
but sublinear when
Figure 5.3 compares the various asymptotic expressions with simulation
output for various p.
Conversely, at short times the behaviour of the system depends continuously on p. For τ ≪ 1, considering only terms that are linear in τ , the
following expressions are found
66
d(τ ) ≈ 1 + 4a(1 − p)τ,
(5.18)
hx (τ )i ≈ 2a (1 − p)τ.
(5.19)
2
2
The second moment expression at short times differs from the corresponding
long time expression by a constant for p >
1
2
but by order τ for p <
1
2.
Consequently, the shape of the second moment’s evolution over time is very
1
2
different for the two regions p <
and p > 12 , despite their identical short-time
approximations (see Figure 5.4).
a)
2
p=0.7
p=0.8
long−time limit
short−time limit
p=0.2
p=0.4
long−time limit
short−time limit
2
<x >/a
2
<x >/a
2
15
2
1.5
20 b)
1
5
0.5
0
0
10
0.5
1
1.5
τ
2
2.5
0
0
3
0.5
1
1.5
τ
2
2.5
3
Figure 5.4: Comparison with asymptotic regimes. The exact analytic
expressions for the second moment (5.14) are compared with with short-time
(5.19) and long-time (5.16) approximations. Panel (a) shows cases where p > 21
and both approximate expressions are parallel. As p increases towards 1, the
distance between the two approximations decreases and the curves converge
faster towards the long-time expression. Panel (b) shows cases where p <
1
2 . The short-time approximations are linear whereas the long-time ones are
quadratic.
5.3
The continuum limit
The transition to continuous space is made by taking the limits as a → 0, W →
∞, N1 → ∞, N2 → ∞ and p →
1
2
such that D = a2 W , x1,0 = aN1 , x2,0 = aN2
and v = 2aW (2p − 1). Here, D represents the diffusion constant, x1,0 and x2,0
the start positions of the left- and right-hand particles, respectively, and v the
velocity of one particle towards the other, the latter of which may be positive,
zero or negative. Also denote by ∆x0 = x2,0 − x1,0 the distance between the
two starting positions.
67
By setting φ = k1 a and ψ = k2 a, the aforementioned limit is found for
(5.7) and denoted by Q̃(k1 , k2 , ǫ):
ei∆x0 k2 eix1,0 (k1 +k2 )
+
2
2
ǫ + i(k2 − k1 )v + D
2 [(k1 + k2 ) + (k2 − k1 ) ]
q
k +k
i∆x0 1 2 2 ix1,0 (k1 +k2 )
e
i D
2 (k2 − k1 )e
×
2
2
ǫ + i(k2 − k1 )v + D
2 [(k1 + k2 ) + (k2 − k1 ) ]
q
2
∆x
v
v
D
0
2
exp √2D √2D − ǫ + 2D + 2 (k1 + k2 )
q
.
v2
2 − √v
ǫ + 2D
+D
(k
+
k
)
1
2
2
2D
Q̃(k1 , k2 , ǫ) =
(5.20)
This reduces to a result of Aslangul (1999, equation 3.1) by setting v = 0,
x1,0 = 0 and x2,0 = 0. By using the identity
L−1
"
#
√
e−A( ǫ+C−B)
√
ǫ+C−B


 eAB− A4t2

2Bt − A
−Ct
B2 t
√
√
=e
+ B 1 + erf
,
e


πt
2 t
(5.21)
from Roberts & Kaufman (1966), where erf(z) is the error function, (5.20) can
be Laplace inverted to give the following expression
D
2
2
Q(k1 , k2 , t) = e−i(k2 −k1 )vt− 2 [(k1 +k2 ) +(k2 −k1 ) ]t ×
(
r
D i∆x0 k1 +k2
i∆x0 k2
2
(k2 − k1 )×
e
+
ie
2


)
(∆x0 −2vs)2
Z t
−
8Ds
D
e
v
∆x
−
2vs
2
0
 e−(i(k2 −k1 )v− 2 (k2 −k1 ) )s ×
√
√
ds 
+ √ erfc
πs
2D
8Ds
0
eix1,0 (k1 +k2 ) ,
(5.22)
where erfc(z) is the complementary error function, erfc(z) = 1 − erf(z). In
order to Fourier invert (5.22) it is convenient to perform the double integral
in the coordinates K = k1 + k2 and k = k2 − k1 . This procedure yields the
joint probability distribution in continuous space and time
68
Q(x1 , x2 , t) =
e−
(x1 −x1,0 −vt)2
4Dt
√
4πDt
e−
(x2 −x2,0 +vt)2
4Dt
√
4πDt
+
−
t
e−
(x1 −x1,0 +x2 −x2,0 )2
8Dt
√
8πDt
[x2 −x1 +2v(t−s)]2
8D(t−s)
[x2 − x1 + 2v(t − s)]e
p
×
ds
4D π(t − s)3
0


(∆x0 −2vs)2
−
8Ds
v
∆x0 − 2vs 
e √
√
+ √ erfc
,
πs
2D
8Ds
Z
×
(5.23)
where Q(x1 , x2 , t) is the inverse Fourier transform of Q(k1 , k2 , t). The first
summand in (5.23) displays the short-time behaviour whereby the probability distribution of the left (right) particle can be approximated as a narrow
Gaussian travelling right (left) at speed v and the interaction between the two
particles is minimal. This interaction, represented by the second summand in
(5.23), becomes more pronounced as time increases.
It turns out (Appendix C) that (5.23) is a solution to the following FokkerPlanck equation that is obtained by taking the continuum limit of the discretespace master equation (5.3) in the region |n − m| > 1
∂Q
(x1 , x2 , t) = D
∂t
∂2
∂2
+
∂x21 ∂x22
Q(x1 , x2 , t) + v
∂
∂
−
∂x1 ∂x2
Q(x1 , x2 , t).
(5.24)
This continuum limit is only valid for x1 6= x2 . Since the particles cannot
cross, and therefore the probability density along x1 = x2 must be zero, one
can interpret this physically by imposing a zero-flux boundary condition along
the line x1 = x2 (Ambjörnsson, 2008), that is
D
∂
∂
−
∂x2 ∂x1
Q(x1 , x2 , t) + 2vQ(x1 , x2 , t) = 0.
(5.25)
x1 =x2
This boundary condition is also satisfied by (5.23). As such, the solution
reduces to a result of Ambjörnsson (2008) in the case v = 0, as well as Aslangul
(1999) when additionally x1,0 = 0 and x2,0 = 0.
To find expressions for the mean separation and MSD, an identical proce69
dure to the discrete case is pursued (Appendix D), giving the following results
d(t) = ∆x0 − vt erfc
Z t
2vt − ∆x0
2v 2 s + ∆x0 v − 4D − (∆x0 −2vs)2
1
8Ds
√
√
ds
,
−√
e
π 0
8Dt
8Ds
(5.26)
v 2 t2 + ∆x0 vt
2vt − ∆x0
√
∆x (t) = 2Dt − ∆x0 vt +
erfc
+
2
8Dt
Z t
(∆x0 −2vs)2
1
√
dse− 8Ds ×
π 0
v 2 (2t − s)(2vs + ∆x0 ) − 8Dv(t − s) + ∆x0 (2v 2 s + ∆x0 v − 4D)
√
−
4 2Ds
2
Z t
vt
2vt − ∆x0
2v 2 s + ∆x0 v − 4D − (∆x0 −2vs)2
1
8Ds
√
√
erfc
ds
,
e
+√
2
π 0
8Dt
4 2Ds
(5.27)
2
where ∆x2 (t) is the MSD of either particle (∆x21 (t) = ∆x22 (t)). In the case
v = 0, the integrals in (5.26) and (5.27) can be calculated exactly to give the
following
d(t) =
r
r
8D − ∆x0 √
π
e 8Dt t + ∆x0 − ∆x0 erfc ∆x0
,
π
2Dt
(5.28)
r
∆x0
1 − ∆x0
2Dt − ∆x0 2
e 8Dt erf √
∆x (t) =2Dt 1 − e 8Dt − ∆x0
+
π
π
8Dt
∆x0
∆x0
∆x0 2
erfc √
2 − erfc √
.
(5.29)
4
8Dt
8Dt
2
For v 6= 0 on the other hand, the infinite integrals
R∞
0
dssn/2 e−
(∆x0 −4vs)2
8Ds
for
n = −1, 0, 1 are finite, so calculating them allows the asymptotic expressions
for (5.26) and (5.27) to be calculated, yielding the following expressions for
t≫1
d(t) ≈

D



v,
q
if v > 0,
8Dt
π ,



∆x −
0
if v = 0,
∆x v
− D0
D
ve
70
− 2vt, if v < 0.
(5.30)

D2


+ Dt,
if v > 0,

4v
 2√
∆x2 (t) ≈ π−4π 8 ∆x0 2 + 2D(1 − π1 )t,
if v = 0,


∆x0 v
2
2

∆x0 D
e− D
− ∆x40 − 3D
+ 2Dt, if v < 0.
v
4v2
(5.31)
This contrasts with the small-time limit t ≪ 1, whereby d(t) ≈ ∆x0 − 2vt and
∆x2 (t) ≈ 2Dt for any v.
Notice that the v > 0 (v = 0, v < 0) cases of (5.30) and (5.31) are simply
the continuous-space limits of the p >
1
2
(p = 21 , p < 12 ) cases in the discrete-
space expressions (5.15) and (5.17). For example, in the case p >
setting
a2 τ
1
2
from (5.15),
= Dt and v = 2aW (2p − 1) reveals that d(t) = 2Dp/v and by
taking the limit p →
1
2
the continuous asymptotic result d(t) ≈ D/v reported
in (5.30) is recovered. Likewise, setting a2 τ = Dt in the case p >
and by taking the limit p →
1
2
1
2
from (5.17)
one recovers the continuous asymptotic result
∆x2 (t)
≈ Dt from (5.31).
5.4
Anti-symmetric exclusion and territory border
movement
In the territorial random walk system, p is the probability that, if there is
a gap between two adjacent boundaries, that gap will decrease in length the
next time a boundary moves. Such a probability is clearly always greater
or equal to
1
2
on average, otherwise the territories would fail to maintain a
positive average width. For such values of p, equation (5.17) shows that the
asymptotic diffusion constant of a boundary is proportional to 1 − p. When
the value of 1 − p is measured directly from the simulations, it also decays
exponentially as Z1D increases (Figure 5.5), suggesting that calculating p is of
fundamental importance in understanding why the border diffusion constant
decays exponentially as Z1D increases.
This relationship between p and Z1D can be explained as follows. First
observe that the probability of a boundary decaying is likely to be closely
related to the probability of an animal traversing its territory within a time
TAS . To this end, the first-passage probability F(t) for an animal to traverse
a territory of average length L = 1/ρ is calculated, where L corresponds in the
present lattice system to an integer M = 1/aρ sites. Since this is equivalent to
the situation where the animal starts at a reflecting boundary has to traverse
to the other, absorbing boundary, the asymptotic value of this first-passage
probability is calculated in Redner (2007) to be F(t) ∼ e−π
71
2 F t/4M 2
. There-
0
M=20
M=25
M=30
M=35
M=40
M=45
M=50
Best fit
ln(P (Z ))
−2
F
−6
1D
−3
−8
−10
−12
0
ln(V(S)/M2)
ln(1−p)
−4
−4
−5
−6
0
0.5
1
2
Z1D
1
3
1.5
2
Z
2.5
3
3.5
4
1D
Figure 5.5: Anti-symmetric exclusion and territory border movement.
This shows the relationship between the value of p measured from simulations
of a system of 1D territorial random walkers and the dimensionless quantity
Z1D defined in section 5.4. This is compared with the probability PF (Z1D )
that the animal fails to traverse a territory of average width 1/ρ within a
time TAS . In order to measure p from the simulations, the number of times a
boundary moved towards the adjacent boundary was counted, and divided by
the total number of times that the boundary moved. The equations for the
2
curves are 1 − p = 0.49e−2.7Z and PF (Z1D ) = e−π Z1D /4 , where π 2 /4 ≈ 2.5.
The inset shows how the variance V (S) of the territory size S decays as Z1D
increases, used in the main text to explain the discrepancy between the rate
of the exponential decays of the two curves in the main plot.
fore the probability PF (Z1D ) ∝
R∞
TAS
dtF(t) of failing to traverse the territory
within a time TAS is approximately e−π
2Z
1D /4
. In Figure 5.5, PF (Z1D ) is plot-
ted alongside the simulation measurements for 1 − p showing that both decay
exponentially with increasing Z1D and with similar exponents. In particular,
1 − p ≈ (1/2)PF (Z1D ).
To explain the small discrepancy in the two exponents, notice that as Z1D
is increased, the variance in the territory width decreases in an approximately
exponential fashion (inset Figure 5.5). Because of the N 2 dependence of the
mean first passage time to cross the territory (Redner, 2007), the mean first
72
passage time increases as the variance in the territory width increases. Therefore for a fixed ρ, the actual mean first passage time to traverse a territory
decreases as Z increases, whereas above I have assumed that the first passage
probability is always equal to that of a territory of average width. This has
the effect of causing the probability 1 − p to decrease with Z1D slightly faster
than in the analytic estimation. In other words the curve of PF (Z1D ) decays
slightly slower than the curve of simulation measurements of 1 − p.
5.5
Analysing territory border movement in 1D: a
way forward
In relation to analysing territory border movement, this chapter contains two
key results. First, the diffusion constant of a territory border is 2a2 W (1 − p)
(equation 5.17), so can be derived exactly from the probability, p, that if
a gap exists between two adjacent boundaries, it will decrease in width the
next time a boundary moves. Second, p is related to the first passage time
to cross a territory, via 1 − p ≈ (1/2)PF (Z1D ). Putting these together, the
diffusion constant of a border, if the presence of other borders is disregarded,
is approximately a2 W [2 − PF (Z1D )]. In other words, the border movement
can be derived from the input parameters TAS , ρ, a and F of the simulation
model, together with W .
Two further steps are required to give an analytic theory of the border
movement.
1. A method for deriving W from the parameters TAS , ρ, a and F .
2. Studying the effects of having a sequence of interlinked borders, as
sketched in Figure 5.1, each of which have an intrinsic diffusion constant of a2 W [2 − PF (Z1D )].
These results would give a complete method for deriving the border subdiffusion constant, without needing to run further simulations. This would help
speed up further analysis of the 1D model, and extensions thereof, as well as
making clearer the relationship between the movement of the borders and the
first passage time for an animal to cross a territory.
73
5.6
Glossary
Symbol
Explanation
p
Probability that the left (right) particle jumps right (left)
next hop, where the particles model territory boundaries
a
Lattice spacing
W
Mean jump rate between adjacent lattice sites for the
particles
Pn,m (t)
Probability that particles 1 and 2 are at positions n
and m, respectively, in discrete space
f (φ, ψ, t)
Generating function of the distribution Pn,m (t)
˜
Laplace transform of f (φ, ψ, t)
f (φ, ψ, ǫ)
τ
Dimensionless time τ = tW , where t is time
x1 (τ ), x2 (τ ) Positions of particles 1 and 2 at time τ , respectively
N1 , N2
Initial positions of particles 1 and 2, respectively, in
discrete space
∆N
The discrete space initial difference ∆N = N2 − N1
hxi i
Mean displacement of particle i
2
hxi i
Second moment of the position distribution for particle i
d(τ )
Separation distance d(τ ) = x2 − x1 between the two
particles at time τ
2
∆xi
Mean square displacement of particle i,
∆x2i = h(x2i − hxi i)i
D
Diffusion constant of unconstrained particle D = a2 W
v
Drift velocity of unconstrained particle v = 2aW (2p − 1)
x1,0 , x2,0
Initial positions of particles 1 and 2 in continuous space,
respectively; x1,0 = aN1 , x2,0 = aN2
∆x0
The continuous space initial difference ∆x0 = x2,0 − x1,0
Q(x1 , x2 , t) Probability that particles 1 and 2 are at positions x1
and x2 , respectively, in continuous space
M
Number of lattice sites in a territory of average width
L
Width of average territory
ρ
Population density ρ = 1/L
F
Mean jump rate between adjacent lattice sites for the
animals in the simulation model
TAS
Active scent time
Z1D
Composite parameter Z1D = TAS ρ2 D
PF (Z1D )
Probability of failing to traverse the territory within
a time TAS , for the input parameter Z1D
The various symbols used throughout this chapter. The parameters above
the horizontal line refer to the anti-symmetric exclusion calculation (sections
5.1, 5.2, 5.3). Those below the line relate to the 1D simulation model
described in Chapter 2 and referred to in section 5.4.
74
Chapter 6
Home range formation in
territorial central place
foragers
So far in this thesis, territorial animals have been modelled as either randomly moving, or ballistic, particles. However, fidelity to particular places for
the purposes of foraging or mating (Greenwood, 1980), may cause the resultant movement to incorporate intrinsic directionality (Moorcroft et al., 2006;
Giuggioli & Bartumeus, 2012).
In this chapter, I study a modified version of the territorial random walk
model where each animal is a random walker with an attraction towards a
central place (CP), such as a den or nest site where the animal returns occasionally (Moorcroft & Lewis, 2006), or a core area where the animal tends
to spend most of its time (White & Harris, 1994). Similar to the original
agent-based model of Chapter 2, territories emerge whose borders are continually fluctuating. However with CP attraction, the mean square displacement
(MSD) tends towards a finite value. This causes stable home range patterns
to emerge from the territorial dynamics.
Such stable home ranges have been reported in a number of species (see
e.g. Börger et al. (2008)), from wolves (Canis lupus) and coyotes (Canis latrans) (Moorcroft & Lewis, 2006) to hispid cotton rats (Sigmodon hispidus)
(Spencer et al., 1990), cane mice (Zygodontomys brevicauda) (Giuggioli et al.,
2005) and Baird’s tapirs (Tapirus bairdii) (Foerster & Vaughan, 2002). This
has prompted questions as to whether general mechanisms exist for home
range formation (Börger et al., 2008). The results of this chapter demonstrate
one plausible mechanism, that of CP attraction, in the context of territorial
animals.
75
As well as extending the agent-based model of Chapter 2, I follow the
approach of Chapter 3 to construct an approximate analytic model that incorporates CP attraction. This requires taking the appropriate continuum
limit of the discrete-space Holgate-Okubo localising tendency model (Holgate,
1971; Okubo, 1980) used in the agent-based simulations. I construct a programme for quantifying the underlying movement and interaction processes
of territorial central place foragers by examining the steady-state utilisation
distribution (i.e. home range) of the animals. In particular, I show how to
extract the active scent time from positional data. These results are used to
give insights into the mechanisms behind allometric scaling laws of animal
space use.
I also compare the results with previous approaches, which incorporate
conspecific avoidance and central place attraction using a reaction-diffusion
formalism (Moorcroft & Lewis, 2006). It turns out that there is a particular
limit of my model which gives results that are numerically and qualitatively
very similar to the reaction-diffusion model. I show how to parametrise the
reaction-diffusion model so that it is an approximation of my model in this
limit.
6.1
Agent-based simulations of territorial central
place foragers
Monte Carlo simulations of territorial central place foragers were performed
in both 1D and 2D where each animal has a bias of moving towards a central
place (CP). The 1D simulations consisted of 2 animals on a finite lattice with
periodic boundary conditions. The central places (CPs) for each animal were
uniformly distributed at a distance L = aN apart, where a is the lattice spacing and N a positive integer (see Table 6.1 for definitions of all the parameters
used in this Chapter). In 2D, 30 animals in a rectangular terrain with periodic
boundary conditions were simulated. The CPs were placed at the centroids
of a hexagonal lattice, modelling the fact that animal territories tend to be
roughly hexagonal in shape (Barlow, 1974). Adjacent CPs were separated by
a distance of L.
In the simulation environment, animals deposit scent at every lattice site
they visit, which remains for a time TAS . Animals are unable to visit sites
that contain scent of another animal. Besides that constraint, at each step
an animal moves to an adjacent site at random but its movement is biased
towards the CP. In 1D, this means that there is a probability of p > 1/2 of
moving towards the CP and 1 − p of moving away. In 2D, the movement
76
probabilities are as follows
i
1h
m − mc
,
1 + (2p − 1) p
4
(m − mc )2 + (n − nc )2
i
1h
m − mc
Right:
,
1 − (2p − 1) p
4
(m − mc )2 + (n − nc )2
i
n − nc
1h
1 − (2p − 1) p
Up:
,
4
(m − mc )2 + (n − nc )2
i
n − nc
1h
p
,
1 + (2p − 1)
Down:
4
(m − mc )2 + (n − nc )2
Left:
(6.1)
where (m, n) is the position of the animal and (mc , nc ) the position of the CP.
These probabilities are chosen so that, in the continuum limit, they reduce
to the form that gives the correct localising tendency in the Holgate-Okubo
model (see section 6.2.2). In particular, they are independent of the distance
the animal is away from the den site. This can be shown by replacing m − mc
by C(m − mc ) and n − nc by C(n − nc ) in equations (6.1), for some non-zero
constant C, and noticing that all the C-values cancel.
The MSD of the territory border eventually reaches a saturation value
that depends on both the strength of attraction towards the CP and the
dimensionless quantity Z1D = TAS /TD1 in 1D or Z2D = TAS /TD2 in 2D,
where TAS is the active scent time, TD1 = [Dρ2 ]−1 (TD2 = [Dρ]−1 ) is the
diffusive time in 1D (2D) representing the time it takes for an animal to
move around its territory, D is the animal diffusion constant and ρ the animal
population density. The positive parameter α = vL/D is used to measure the
dimensionless strength of CP attraction, where v is the drift velocity towards
the CP and L the distance between CPs of two adjacent territories.
For a fixed α, the amount of border movement depends on Z1D in 1D or
Z2D in 2D (Figure 6.1). Increasing α has the effect of reducing the animal’s
tendency to move into interstitial regions and claim extra territory. This causes
the borders to move less on average, as each animal keeps to a small core area
well within its territory most of the time. Consequently, when plotting the
MSD saturation value against Z1D or Z2D , the curves for higher values of α
lie below those for lower values (Figure 6.1).
To obtain these results, simulations were run until the MSD of the border
had reached its saturation value. Each 1D simulation result is an average of
1,000 simulation runs. In 2D, it is only necessary to average over 100 runs
owing to the fact that 15 times as many animals were simulated per run. The
simulations were coded in C and compiled on Windows XP OS. To obtain a
77
α=0.08
α=0.8
α=4
−1
Log10(Normalised border saturation MSD)
10
Log (Normalised border saturation MSD)
−0.5 a)
−1.5
−2
−2.5
−3
−3.5
0
2
4
6
Normalised active scent time, Z =T /T
1D
AS
8
D1
b)
0
−0.5
−1
−1.5
−2
−2.5
−3
α=0.08
α=0.4
α=0.8
α=1.6
α=2.4
α=4
2
4
6
8
10
Normalised active scent time, Z2D=TAS/TD2
Figure 6.1: Simulation output for systems of territorial central place
foragers. The dependence of the saturation mean square displacement (saturation MSD) ∆x2b = h(xb − hxb i)2 i (resp. ∆x2b = h(xb − hxb i)2 i) of the dimensionless territory border position xb (xb ) on the dimensionless parameters
α = vL/D and Z1D = TAS /TD2 (Z2D = TAS /TD2 ) from stochastic simulation output. The notation h. . . i denotes an ensemble average over stochastic
simulations. The border MSD is normalised by dividing by L2 , where L is
the average distance between central places of adjacent territories. Panel (a)
shows output from 1D simulations and panel (b) from 2D simulations. The
best-fit lines for the 2D plots are Log10 (∆x2b ) = 0.22 − 0.071Z2D for α = 0.08,
Log10 (∆x2b ) = 0.33 − 0.14Z2D for α = 0.4, Log10 (∆x2b ) = 0.28 − 0.18Z2D
for α = 0.8, Log10 (∆x2b ) = 0.08 − 0.19Z2D for α = 1.6, Log10 (∆x2b ) =
−0.54 − 0.20Z2D for α = 2.4, and Log10 (∆x2b ) = −0.93 − 0.19Z2D for α = 4.
single saturation MSD value for the 2D simulations takes an average of 4 hours
40 minutes CPU time using a 3.0GHz processor in a 2.96GB RAM desktop
computer.
6.2
An analytic model of a central place forager
within its territory
By taking into account the fact that the border movement is much slower
than that of the animal, an adiabatic approximation is employed, similar to
that in Chapter 3, to calculate the probability distribution of an animal inside
its fluctuating territory borders. The simulated animals are identical, so it is
sufficient just to model one animal. Since the MSD of each territory border
saturates at long times, the animal’s probability distribution reaches a steady
state. In this section, I calculate the steady state joint distribution of the
animal and its territory borders, in both 1D and 2D.
78
Figure 6.2: The 1D model of territorial central place foragers. The
CPs are fixed at positions A, B and C (left to right). The territory borders are
intrinsically subdiffusive and have positions L1 and L2 . Each animal moves
diffusively with a constant drift towards the CP and constrained to move
between the two territory borders to its immediate right and left. The position
of the animal studied in the main text is denoted by x. The animals at xA
and xC are drawn purely for illustrative purposes.
6.2.1
Movement in 1D
The adiabatic approximation implies that the joint probability distribution
of the animal and the borders can be decomposed as P1D (x, L1 , L2 , t) ≈
Q(L1 , L2 , t)W (x, t|L1 , L2 ) where Q(L1 , L2 , t) is the probability distribution
of the borders to be at positions L1 and L2 at time t, and W (x, t|L1 , L2 ) is
the probability distribution of an animal to be at position x at time t when
constrained to move between the borders at L1 and L2 .
The borders are modelled using a Fokker-Planck formalism, with a timedependent diffusion constant modelling the subdiffusive nature of the border movement, and quadratic potentials modelling the spring forces (Figure 6.2). Since the CP at B separates L1 from L2 , it is possible to write
Q(L1 , L2 , t) = Q1 (L1 , t)Q2 (L2 , t) where Q1 (L1 , t) and Q2 (L2 , t) are the probability distributions of L1 and L2 , respectively. These are governed by the
following equations
∂Q1 (L1 , t)
∂ 2 Q1 (L1 , t)
A+B
∂
= ϕ(t) K
γ
L
−
Q
(L
,
t)
,
+
1
1
1
∂t
∂L1
2
∂L21
∂Q1 (L1 , t) ∂Q1 (L1 , t) =
= 0,
(6.2)
∂L1
∂L1
L1 =A
L1 =B
79
∂ 2 Q2 (L2 , t)
B+C
∂Q2 (L2 , t)
∂
= ϕ(t) K
γ L2 −
Q2 (L2 , t) ,
+
∂t
∂L2
2
∂L22
∂Q2 (L2 , t) ∂Q2 (L2 , t) =
= 0,
(6.3)
∂L2
∂L2
L2 =B
L2 =C
where A is the position of the CP to the left of L1 , B is the position of the
CP between L1 and L2 , C is the position of the CP to the right of L2 , Kϕ(t)
is the time-dependent diffusion constant and γϕ(t)[L1 − (A + B)/2]2 /4 (resp.
γϕ(t)[L2 − (B + C)/2]2 /4) is the quadratic potential for each spring connected
to L1 (resp. L2 ). It ensures that the border L1 (resp. L2 ) fluctuates around
an average position of [A + B]/2 (resp. [B + C]/2).
Notice that there are two springs connected to L1 (resp. L2 ), so that the
total resulting potential is γϕ(t)[L1 − (A + B)/2]2 /2 (resp. γϕ(t)[L2 − (B +
C)/2]2 /2). As usual for Fokker-Planck equations (see e.g. Risken (1996)), this
potential appears in equation (6.2) (resp. 6.3) after having been differentiated
with respect to L1 (resp. L2 ), to give γϕ(t)[L1 − (A + B)/2] (resp. γϕ(t)[L2 −
(B +C)/2]). The boundary conditions in equations (6.2) and (6.3) ensure that
the borders cannot cross over the CPs, since each CP must remain within its
own territory.
In principle, K and γ can be measured directly from the simulation model,
as in Chapter 3. However, different to Chapter 3, it is possible to construct
steady state solutions to equations (6.2) and (6.3), which will be derived
shortly (equations 6.9 and 6.10). In these, K and γ collapse to a single dimensionless parameter κ = K/(γL2 ). The κ parameter can be derived by first
measuring the boundary’s saturation MSD from the simulations, and then
using equation (6.28), as explained in the next section.
Equations (6.2) and (6.3) can be solved in the Fourier domain using the
method of characteristics (Moon & Spencer, 1969). The general solution to
(6.2) is
exp
Q1 (L1 , t) =
(L1 −L1 )2
b(t)
p
πb(t)
,
(6.4)
where L1 (t) = (A + B)/2 + exp(−G(t)[L1,0 − (A + B)/2]), b(t) = (4K/γ){1 −
Rt
exp[−2γ 0 dsϕ(s)]}, G(t) is defined so that G′ (t) = γϕ(t) and L1,0 is the initial
value for L1 at time t = 0. By using the method of images (Montroll & West,
1987) to take account of the boundary condition and assuming, for simplicity,
that L1,0 = (A + B)/2, the following solution is found
80
Q1 (L1 , t) =[H(L1 − A) − H(L1 − B)]g1 (L1 , t),
g1 (L1 , t) =
∞
−
X
e
[L1 +2n(B−A)−(A+B)/2]2
b(t)
−
[L1 −2A+2n(B−A)+(A+B)/2]2
b(t)
+e
p
πb(t)
n=−∞
.
(6.5)
Similarly,
Q2 (L2 , t) = [H(L2 − B) − H(L2 − C)]g2 (L2 , t),
g2 (L2 , t) =
∞
−
X
e
[L2 +2n(C−B)−(B+C)/2]2
b(t)
−
[L2 −2B+2n(C−B)+(B+C)/2]2
b(t)
+e
p
πb(t)
n=−∞
. (6.6)
By making use of the Poisson summation formula (Montroll & West, 1987),
equations (6.5) and (6.6) can be re-written as follows
∞
X
πn(2L1 + A − B)
π 2 n2 b(t)
1
exp −
cos
1+
+
g1 (L1 , t) =
B−A
4(B − A)2
2(B − A)
n=1
πn(2L1 + 3A + B)
cos
,
2(B − A)
(6.7)
∞
X
π 2 n2 b(t)
πn(2L2 + B − C)
1
exp −
cos
1+
+
g2 (L2 , t) =
C −B
4(C − B)2
2(C − B)
n=1
πn(2L2 + 3B + C)
cos
.
2(C − B)
(6.8)
Since the territories move as tagged objects in a single file diffusion process
Rt
(section 2.2), 0 dsϕ(s) ∼ t1/2 in the 1D system (Harris, 1965). Therefore the
limit as t → ∞ of b(t) is b(t = ∞) = 4K/γ. Taking this limit in equations
(6.7) and (6.8) gives steady state solutions. Furthermore, by setting B = 0,
B − A = C − A = L, and using dimensionless variables L̄k = Lk /L, x̄ = x/L,
Q̄k (L̄k ) = LQk (Lk , t = ∞), ḡk (L̄k ) = Lgk (Lk , t = ∞) for k = 1, 2 and κ =
K/(γL2 ), the following dimensionless steady state density functions for the
81
border distributions is found
ḡ1 (L̄1 ) = 1 + 2
∞
X
(−1)n e−4π
2 n2 κ
n=1
ḡ2 (L̄2 ) = 1 + 2
∞
X
(−1)n e−4π
cos[2πn(L̄1 − 1)],
2 n2 κ
cos[2πnL̄2 ],
n=1
−1 ≤ L̄1 ≤ 0, (6.9)
0 ≤ L̄2 ≤ 1.
(6.10)
To calculate the animal probability distribution W (x, t|L1 , L2 ), I begin by
finding the continuous-space limit of the simulation model in the case where
the animals and their CPs are infinitely far apart, so that they never interact.
After solving this, I factor in the boundary conditions corresponding to the
existence of territory borders at L1 and L2 .
The limit where there are no interactions is W (x, t|L1 = ∞, L2 = ∞),
written as W∞ (x, t) to ease notation. The discrete-space master equation for
an animal in this limiting case is
∂U (n, t)
=2F p[U (n + 1, t) − U (n, t)]−
∂t
2F (1 − p)sgn(n − nc )[U (n, t) − U (n − 1, t)],
(6.11)
where U (n, t) is the probability of the animal being at position n at time t,
nc is the position of the CP, F is the jump rate between adjacent lattice sites,
and sgn(z) = 1 (sgn(z) = 0, sgn(z) = −1) if z > 0 (z = 0, z < 0). This can be
written as
∂U (n, t)
U (n + 1, t) − U (n, t) U (n, t) − U (n − 1, t)
2 1
=F a
−
+
∂t
a
a
a
U (n, t) − U (n − 1, t)
+
sgn(n − nc )aF (2p − 1)
a
U (n + 1, t) − U (n, t)
sgn(n − nc )aF (2p − 1)
.
(6.12)
a
The continuum limit of (6.12) can be found by taking the limits as a → 0,
F → ∞, n → ∞, nc → ∞ and p →
1
2
such that D = a2 F , x = an, xc = anc
and v = 2aF (2p − 1) (see Kac (1947) or section 5.3). Physically, D is the
diffusion constant of the animal, v the drift velocity towards the CP, x the
position of the animal and xc the position of the CP.
This procedure leads to the 1D Holgate-Okubo localising tendency model
82
(Holgate, 1971; Okubo & Levin, 2002)
∂W∞ (x, t)
∂2
∂
= D 2 W∞ (x, t) − v [x̂W∞ (x, t)],
∂t
∂x
∂x
(6.13)
where x̂ = −1, 0 or 1 if x > xc , x = xc or x < xc respectively. This has
a non-trivial steady state solution that is proportional to exp(−v|x − xc |/D)
(Moorcroft & Lewis, 2006).
Since the animal is constrained to move between the borders at L1 and
L2 , the probability distribution must be zero for x < L1 and x > L2 . As the
solution is a steady state, the flux across L1 and L2 is automatically zero, so it
suffices to ensure that the integral of the probability distribution between L1
and L2 is equal to 1. This leads to the steady state solution W (x, t = ∞|L1 , L2 )
for the Holgate-Okubo localising tendency model within fixed borders
W (x, t = ∞|L1 , L2 ) = [H(x − L1 ) − H(x − L2 )]h(x, ∞|L1 , L2 ),
c|
v exp − v|x−x
D
.
(6.14)
h(x, t = ∞|L1 , L2 ) = v(L1 −xc )
v(L2 −xc ) D 2−e D
− e− D
Using dimensionless variables α = vL/D, x̄ = x/L, L̄1 = L1 /L, L̄2 = L2 /L,
W̄ (x̄) = LW (x, t = ∞), h̄(x̄) = Lh(x, t = ∞) and setting xc = 0 for simplicity,
the following expression is found
h̄1D (x̄|L̄1 , L̄2 ) =
α exp (−α|x̄|)
.
2 − exp(αL̄1 ) − exp(−αL̄2 )
(6.15)
The steady state 1D dimensionless joint probability P̄1D (x̄, L̄1 , L̄2 ) of the left
(right) border being at dimensionless positions L̄1 = L1 /L (L̄2 = L2 /L) and
the animal being at position x̄ = x/L at long times, where L1 , L2 and x
are dimensional parameters and L is the distance between CPs of adjacent
territories, is then
P̄1D (x̄, L̄1 , L̄2 ) ≈[H(L¯1 + 1) − H(L̄1 )][H(L¯2 ) − H(L¯2 − 1)]×
[H(x̄ − L̄1 ) − H(x̄ − L̄2 )]ḡ1 (L̄1 ) ḡ2 (L̄2 ) h̄1D (x̄|L̄1 , L̄2 ),
(6.16)
where H(z) is the Heaviside step function (H(z) = 0 if z < 0, H(z) = 1 if
z ≥ 0).
83
6.2.2
Movement in 2D
In 2D, each territory is modelled as a circle with fluctuating radius and the
CP at the centre of the circle, assumed to be at the origin for simplicity. As
in the 1D case, an adiabatic approximation is used so that P2D (r, θ, σ, t) ≈
Q(σ, t)W (r, θ, t|σ), where P2D (r, θ, σ, t) is the joint probability distribution of
the animal to be at position (r, θ) in polar coordinates at time t and the
territory radius to be σ, Q(σ, t) is the probability of the territory radius to be
σ at time t and W (r, θ, t|σ) is the probability of the animal to be at position
(r, θ) at time t in a territory of fixed radius σ.
Similar to the 1D scenario, Q(σ, t) is modelled using a Fokker-Planck formalism with the radius fluctuating around an average value of L/2
∂Q(σ, t)
∂ 2 Q(σ, t)
∂
L
= ϕ(t) K
+
γ σ−
Q(σ, t) ,
∂t
∂σ 2
∂σ
2
∂Q(σ, t) ∂Q(σ, t) =
= 0,
∂σ
∂σ
σ=0
σ=L
(6.17)
where L is the distance between adjacent CPs. As such, it can be calculated
using the methods of the previous subsection to be
Q(σ, t) =[H(σ) − H(σ − L)]g(σ, t)
2 2
∞
X
π n b(t)
1
exp −
×
1+
g(σ, t) =
L
4L2
n=1
πn(2σ − L)
πn(2σ + L)
cos
+ cos
.
2L
2L
As the territories are tagged particles in a 2D exclusion process,
(6.18)
Rt
0
dsϕ(s) ∼
t/ln(t) (Landim, 1992). Taking the limit t → ∞ in equation (6.18) gives
a steady state solution, which can be expressed in dimensionless parameters
σ̄ = σ/L, Q̄(σ̄) = LQ(σ, t = ∞), ḡ(σ̄) = Lg(σ, t = ∞), as follows
ḡ(σ̄) = 1 + 2
∞
X
(−1)n e−4π
2 n2 κ
cos[2πn(σ̄)].
(6.19)
n=1
Following the methods of the 1D section 6.2.1, I calculated W (r, θ, t|σ) by first
taking the continuum limit of the discrete space master equation governing the
movement of an animal unconstrained by other territories (i.e. σ = ∞), then
factoring in the zero-flux boundary conditions. The discrete space master
equation is constructed from the simulation’s movement probabilities (equa84
tion 6.1), and can be written as
#
(
"
∂Um,n (t)
m − mc
+
= F [Um+1,n − Um,n ] 1 + (2p − 1) p
∂t
(m − mc )2 + (n − nc )2
"
#
m − mc
[Um−1,n − Um,n ] 1 − (2p − 1) p
+
(m − mc )2 + (n − nc )2
"
#
n − nc
[Um,n+1 − Um,n ] 1 + (2p − 1) p
+
(m − mc )2 + (n − nc )2
"
#)
n − nc
[Um,n−1 − Um,n ] 1 − (2p − 1) p
,
(m − mc )2 + (n − nc )2
(6.20)
where Um,n (t) is the probability of the animal being at position (m, n) at time
t and (mc , nc ) is the position of the CP. To find the continuum limit, equation
(6.20) is re-written as follows
1 Um+1,n − Um,n
Um,n − Um−1,n
−
+
a
a
a
Um+1,n − Um,n
m − mc
p
+
aF (2p − 1)
a
(m − mc )2 + (n − nc )2
Um,n − Um−1,n
+
a
Um,n − Um,n−1
1 Um,n+1 − Um,n
2
a F
−
+
a
a
a
Um,n+1 − Um,n
n − nc
aF (2p − 1) p
+
a
(m − mc )2 + (n − nc )2
Um,n − Um,n−1
.
(6.21)
a
∂Um,n (t)
=a2 F
∂t
Then the limit as a → 0, F → ∞, m → ∞, mc → ∞, n → ∞, nc → ∞
and p →
1
2
such that D = a2 F , x = ma, xc = mc a, y = na, yc = nc a and
v = 2aF (2p − 1) is found. This procedure gives rise to the 2D Holgate-Okubo
localising tendency model
∂W∞ (r, θ, t)
= D∇2 W∞ (r, θ, t) − v∇[x̂W∞ (r, θ, t)],
∂t
(6.22)
where x̂ is the unit vector pointing from the animal at (x, y) towards the CP at
(xc , yc ), or the zero vector if (x, y) = (xc , yc ), and W∞ (r, θ, t) is the probability
distribution W (r, θ, t|σ) in the limit σ → ∞ where there is no interaction with
85
other animals.
As in 1D, (6.22) has a non-trivial steady state solution (Moorcroft & Lewis,
2006), which is proportional to E1 (vr/D), where En (z) is the special function
R∞
defined by En (z) = 1 ds[exp(−zs)]/sn . The limit as z → 0 of En (z) is
infinite for n = 1 and finite for n > 1. For large z, En (z) ∼ e−z /z so the
limit as z → ∞ is 0 for every n. Since the animal must be within its territory,
meaning r ≤ σ, a boundary condition is imposed by normalising the steady
state solution so that the integral over the circle, of radius r centred at 0, is
equal to 1. This leads to the following steady state solution W (r, θ, t = ∞|σ)
for W (r, θ, t|σ)
W (r, θ, t = ∞|σ) = [H(r) − H(r − L)]h(r, ∞|σ),
h(r, θ, t = ∞|σ) =
v 2 E1 (vr/D)
.
πD 2 [1 − 2(vL/D)E2 (vL/D) − 2E3 (vL/D)]
(6.23)
By using dimensionless variables α = vL/D, r̄ = r/L, h̄2D (r̄|σ̄) = L2 h(r, θ, t =
∞|σ), the following equation is found
h̄2D (r̄|σ̄) =
α2 E1 (αr̄)
,
π[1 − 2ασ̄E2 (ασ̄) − 2E3 (ασ̄)]
(6.24)
where the θ symbol is dropped since the expression is radially symmetric. The
steady state dimensionless joint probability density function P̄2D (r̄, θ, σ̄) for
the territory and the animal at long times is then
P̄2D (r̄, θ, σ̄) ≈ [H(σ̄) − H(σ̄ − 1)][H(r̄) − H(r̄ − σ̄)] ḡ(σ̄) h̄2D (r̄|σ̄).
6.3
(6.25)
The marginal distribution of the animal
Equations (6.16) and (6.25) enable the marginal probability distribution of the
animal to be calculated in both 1D and 2D scenarios, where the territory can
be anywhere else, by integrating over all possible positions for the territory
border. In 1D the dimensionless marginal distribution of the walker at long
times is
M1D (x̄) =
Z
min{x̄,0}
dL̄1
−1
Z
1
dL̄2 ḡ1 (L̄1 ) ḡ2 (L̄2 ) h̄1D (x̄|L̄1 , L̄2 ),
max{x̄,0}
86
(6.26)
where the limits of the integrals ensure that both the animal and its CP are
within the animal’s territory. In 2D, the corresponding equation is
0.5
0
−1
1.5
0
Position
b)
1
0.5
0
−1
0
Position
1
2.5 c)
0.5 e)
2
1.5
1
0.5
1.5
0.5 g)
0
−0.5
0
−1
1
0
Position
−0.5
1
1
0.5
0
Position
1
−0.5
0
0.5
X Position
0.5 h)
0
−0.5
−0.5
0
−0.5
0
0.5
X Position
0.5 f)
d)
0
−1
(6.27)
Y Position
1
dσ̄ ḡ(σ̄) h̄2D (r̄|σ̄).
r̄
Y Position
1.5
1
Y Position
α=0.8
α=4
Z
Y Position
Probability density
2.5 a)
2
Probability density
Probability density
Probability density
M2D (r̄) = H(r̄)
0
X Position
0.5
0
−0.5
−0.5
0
X Position
0.5
Figure 6.3: Comparison of the many-bodied simulation system and
the reduced analytic model. Saturation marginal probability distributions
from simulations of systems of territorial central place foragers are overlaid
on the same distributions (equations 6.26 and 6.27) from the reduced analytic
models. Panels (a-d) compare the two distributions for the 1D system. Dashed
lines denote the simulation output and solid lines the analytic approximation.
The animal’s central place (CP) is at position 0, whereas CPs of conspecifics
exist at positions -1 and 1. The distribution decays to 0 at the conspecific
CPs, where the animal cannot tread. The values used were (a) κ = 0.0017,
α = 4.0, (b) κ = 0.0014, α = 0.80 (c) κ = 0.00033, α = 4.0 and (d) κ = 0.0040,
α = 0.80. Panels (e-h) compare the two distributions for the 2D system. The
black contours show the deciles (i.e. 10%, 20%, 30% etc.) of the height of
the probability distribution for the simulation system. The red contours show
the same quantities for the analytic approximation. The values used were (e)
κ = 0.011, α = 0.80, (f) κ = 0.0014, α = 4.0, (g) κ = 0.022, α = 0.80 and (h)
κ = 0.016, α = 4.0. As either κ or α are increased, the effect of the adiabatic
approximation becomes more apparent, since each red contour is further away
from the respective black contour. This is due to the fluctuations of the
territory border being more pronounced for higher κ or α.
The effects that the two parameters α and κ have on the marginal distribution
(Figure 6.3) can be characterised by observing that α tends to govern the shape
of the density function towards the centre of the territory, whereas κ governs
the degree to which the distribution tails off sharply (high κ) or with a shallow
gradient (low κ).
87
Expressions (6.26) and (6.27) are directly compared with those measured
from territorial central place forager simulations. It turns out that the 1D case
gives an excellent agreement for all parameter values tested (Figures 6.3(a-d)).
In 2D, a qualitatively close fit is attained only when κ and α are sufficiently
low. For higher κ or α, the borders are moving too fast for the adiabatic
approximation to be accurate (e.g. Figure 6.3h). However for lower κ and α,
the terrain contains very little interstitial area at any point in time, so the
territories are forced to tesselate the plane. Therefore they each form more of
a hexagonal than a circular shape (e.g. Figure 6.3e).
6.4
Obtaining active scent time from animal position data
To apply the present theory to data, locations must be gathered over a sufficiently long period for the animal MSD to saturate. For certain species,
such as red foxes, the saturation value fails to be reached during the maximal biologically relevant time-window. Male red foxes may spend parts of
the autumn and winter moving outside their territories to cuckold or disperse
(Soulsbury et al., 2011), so territorial dynamics can only be measured reliably
from the animal positions during spring and summer when the males tend to
stay within their territories. During those two seasons, the tendency to return
to the CP is so weak that the animal MSD continues to increase slowly, never
settling. In such cases, it is necessary to use methods developed in Chapter 4
to analyse the animal territorial system.
However, if the animal MSD does saturate then the marginal distribution
(6.27) can be fitted to non-dimensionalised location data to obtain the parameters α and κ. This knowledge of κ enables the saturation MSD ∆σ̄ 2 (κ) =
h(σ̄ − hσ̄i)2 i of the territory radius to be calculated by using the equation
2
∆σ̄ (κ) =
Z
0
1
1
dσ̄ σ̄ −
2
2
∞
X (−1)n exp(−4π 2 n2 κ)
1
ḡ(σ̄) =
+
.
12
π 2 n2
(6.28)
n=1
The MSD of σ̄ is the analogue, in the analytic model, of the dimensionless
territory border MSD ∆x2b = h(xb − hxb i)2 i from the simulation model, so I
equate ∆x2b and ∆σ̄ 2 (κ). By using the appropriate curve from the simulation
output (Figure 6.1b) related to the value of α calculated from the data, a value
for Z2D = TAS Dρ is obtained, from which TAS can be derived.
In summary, the active scent time may be obtained from data on animal
locations by using the following programme:
88
1. Fit equation (6.27) to the data, for example using the methods of section
4.1, in order to obtain values of α and κ.
2. Use this value of κ to find the theoretically expected saturation value of
the MSD ∆σ̄ 2 (κ) via equation (6.28).
3. Note that ∆σ̄ 2 (κ) from the analytic model is equal to ∆x2b from the
simulation model.
4. Find the best-fit line from Figure 6.1b for the value of α found in step
1.
5. Use this line, together with the value of ∆x2b from step 3, to determine
the Z2D -value from Figure 6.1b for the data being studied.
6. Assuming the user also has values for D and ρ from the data, TAS can
then be derived from Z2D = TAS Dρ.
6.5
Home range patterns and relations to allometry
Since the animal probability distribution reaches a steady state, it is possible
to calculate both the size of the resulting home ranges and the degree to
which they overlap. By using the 95% MCP method (Harris et al., 1990), the
dimensionless radius of the home range, after dividing by the mean distance
between CPs, is given by R95% , implicitly defined by the following equation
2π
Z
R95%
dr̄ r̄ M2D (r̄) = 0.95.
(6.29)
0
This allows R95% to be plotted as various functions of κ, one for each α
(Figure 6.4a). Each of these can be approximated by a sigmoidal function
of K = Log10 (κ). Specifically, R95% ≈ Q + ν/{1 + exp(−ζ[K − η])}, where Q ≈
0.495 − 0.010α, ν ≈ 0.278 − 0.048α, ζ ≈ 3.18 and η ≈ 3.139 − 0.025α (Figure
6.4a). For certain values of κ and α, the value of R95% is less than 0.5, meaning
that gaps arise between adjacent territories. These so-called buffer zones have
been observed between wolf (Canis lupus) territories (Lewis & Murray, 1993)
as a safe place for wolf prey, such as white-tailed deer (Odocoileus virginianus),
to occupy.
The allometric predictions of Jetz et al. (2004) show that the fraction of
exclusively used area E is approximately proportional to M −ξ where ξ ≈ 1/4
2 ,
and M is the mass of a single animal. In my model E ≈ (1 − R95% )2 /R95%
since the dimensionless exclusive area is approximately π(1 − R95% )2 , so allometric studies predict (1 − R95% )/R95% ∝ M −ξ/2 . By using the values of ξ
89
40
α=0.08
α=0.4
α=0.8
α=1.6
α=2.4
α=4
0.7
0.65
0.6
Normalised active scent time, Z2D
95% MCP home range radius, R95%
a)
0.75
0.55
0.5
0.45
−4
10
−2
10
Normalised territory border MSD, κ
35
30
25
20
15
10
5
0
0
10
b)
0
10
1
2
10
10
E−4 ∝ Mass
3
10
4
10
Figure 6.4: Home ranges and allometry. Panel (a) shows how the radius
R95% of the normalised (by dividing by the mean distance between CPs) 95%
minimum convex polygon home range depends on κ and α in the 2D analytic
model. The various shapes (circles, squares, crosses etc.) show the exact
values and the solid lines show the least-squares best-fit sigmoidal curves.
Notice that whenever R95% < 0.5, a buffer zone appears between adjacent
2
territories. The proportion of exclusive area E = (1 − R95% )2 /R95%
scales
with mass (Jetz et al., 2004) so this value is plotted in panel (b) against the
dimensionless parameter Z2D = TAS Dρ for various α. Again, solid lines are
derived from the best-fit sigmoidal curves whilst the points denoted by various
shapes show exact values.
fitted from the large data sets in Jetz et al. (2004), the value of R95% can be
estimated for an animal of given mass. Using the trend lines from the simulation plots in Figure 6.1b and equation (6.28) allows κ and α to be related to
Z2D , thus estimating how Z2D scales with M , as shown in Figure 6.4b.
In Jetz et al. (2004), the tendency for larger animals to have a lower proportion of exclusive area in their home ranges was explained intuitively, by
noticing that they are less efficient than smaller animals in patrolling their
territory to deter conspecifics. That is, the time it takes for a larger animal to
get around its territory is greater than that of a smaller animal. In my model,
this means the diffusive time, TD2 , increases with mass. The results presented
in this chapter show that this ability to deter conspecifics is also driven by an
additional factor: the active scent time. The ability to maintain exclusive area
in fact arises from the ratio of TAS to TD2 . Figure 6.4b shows that a smaller
animal’s ability to maintain a higher proportion of exclusive space use arises
from maintaining a higher ratio of TAS to TD2 , not just a lower diffusive time.
90
6.6
6.6.1
Comparison with previous approaches
The reaction-diffusion approach
Territoriality in animals with central place attraction has been studied previously using a reaction-diffusion formalism, first proposed by Lewis & Murray
(1993), then developed further through a series of papers, culminating in
the book by Moorcroft & Lewis (2006). Although both that model and the
one presented here use conspecific avoidance mediated by scent marking as
the mechanism of territory formation, the present model is built from the
individual-level interaction processes, whereas the reaction-diffusion model relies on a mean-field approximation for the scent mark response. Despite the
very different natures of their construction and the resulting expressions, I
compare the two models by examining the conditions under which they are
numerically similar.
In the reaction-diffusion model, u(x̄, t) and w(x̄, t) are the dimensionless
probability density functions for the left and right animals, respectively. In
addition, p(x̄, t) and q(x̄, t) denote the dimensionless densities of the scent of
the left and right animals, respectively. The dimensionless diffusion constant
of each animal is given by d and the dimensionless advection coefficient controlling the strength of motion away from conspecific scent and towards the
CP is c. The model also contains a parameter controlling the over-marking
response rate: that is, the tendency for animals to scent-mark more having
encountered foreign scent. However, since the animals in the model described
in this thesis are counter-markers rather than over-markers (Hurst, 2005), that
is they mark next to conspecific scent but they do not increase marking rate
as a response to scent, this parameter is set to 0. With these conditions,
the reaction-diffusion system described in Moorcroft & Lewis (2006) has the
following dimensionless steady state solution
du(x̄)
= −βu(x̄)[2 − u(x̄)],
dx̄
dw(x̄)
= βw(x̄)[2 − w(x̄)],
dx̄
(6.30)
where 0 ≤ x̄ ≤ 1, together with the probability conservation conditions
Z
0
1
dx̄u(x̄) =
Z
1
dx̄w(x̄) = 1.
(6.31)
0
Equation (6.30) is equation (6.11) in Moorcroft & Lewis (2006). The dimensionless parameter β is a function of 5 dimensional parameters, β = c′ l/(µLD),
where L and D are the same values as used elsewhere in the present study,
l is the scent marking rate for the individual or pack, µ is the rate of scent91
mark decay and c′ is the strength of attraction towards the CP. The parameter
c′ is not the same as the drift velocity v from my model since it is proportional to the strength of foreign scent at x̄ (see equations (4.5) and (4.6) in
Moorcroft & Lewis (2006)), whereas, in the model studied in this thesis, the
magnitude of the drift velocity is constant throughout space.
The way the rate of scent deposition is modelled also differs between the
two approaches. In the reaction-diffusion model, scent-marking is a rate parameter, so is time-dependent. The biological implication being that as the
animal’s speed increases, consecutive scent marks will be deposited further
apart. In my model, the scent marks are deposited every time the animal has
moved a distance a (the lattice spacing), regardless of its speed. The reason
for my choice is that it is advantageous for animals to ensure that they deposit
territorial messages at regularly spaced intervals so that they leave no gaps in
the territory boundaries, which might allow conspecifics to intrude.
Scent decay is also modelled in different ways in the two models. In the
reaction-diffusion model the scent decays exponentially, whereas my model assumes scent is ignored after a fixed period of time (TAS ). Whilst exponential
decay of scent makes sense regarding the decay of the chemicals that produce
the odour, a conspecific may ignore a scent mark it can still smell, if the odour
suggests that the mark is old and the territory is no longer being defended.
For example, such behaviour has been reported for brown hyaenas (Hyaena
brunnea), whose scent marks, in the form of faeces pastes, may still be detectable by conspecifics over a month later, but who tend to ignore scent that
is more than about four days old (Maude, 2010).
6.6.2
A numerical comparison
Making numerical comparisons of my model with the reaction-diffusion model
requires a further reduction of my 1D analytic model, since the 1D reactiondiffusion model only represents animal movement in the right-hand (left-hand)
half of the left-hand (right-hand) territory. Focussing on the left-hand territory, this requires the analytic model presented here to be simplified by fixing
ḡ1 (L̄1 ) = δ(L̄1 ) where δ(z) is the Dirac delta function. The resulting marginal
distribution for the position of the animal in dimensionless coordinates is
MR (x̄) = H(x̄)
Z
1
dL̄2 ḡ2 (L̄2 )h̄1D (x̄|L̄1 = 0, L̄2 ).
(6.32)
x̄
This expression is compared with the distribution u(x̄) from the reactiondiffusion model, whereas w(x̄) is compared with MR (1 − x̄). To find the best
fit, the square of the difference between the curves of ln[u(x̄)] and ln[MR (x̄)]
92
0.5
Position
−1
0 −4
10
i)
2
0
0
Probability density
0.5
1
Probability density
κ
Probability density
Position
iii)
2
−2
0
0
0.5
1
ii)
2
1
0
0
0.5
Position
10
1
0.5
Position
10
10
5
1
b)
1
Probability density
Probability density
3 ii)
2
1
0
0
β
15
Probability density
20
2
1
0
0
10
α=0
α = 0.01
α=1
i)
Probability density
0
25 a)
2
iii)
1
0
0
0.5
1
Position
Position
iv)
2
0
0
0.5
1
Position
−3
−3
10
−1
κ
10
0
10
10
0
10
20
β
30
40
Figure 6.5: Comparison with a previous model of territory formation. The parameter β from the reaction-diffusion model introduced in
Moorcroft & Lewis (2006) (see also main text) is compared with the parameters α and κ from the 1D analytic model introduced here (equation 6.32).
Panel (a) shows the β-value that gives the best-fit animal marginal distribution
curve for each given value of α and κ. The insets compare the probability distributions for particular values of α and κ, where the solid lines represent my
model and the dashed lines the reaction-diffusion model. The values used are
(i) κ = 0.0007, α = 0.0001, (ii) κ = 0.0007, α = 1, (iii) κ = 0.04, α = 0.0001,
(iv) κ = 0.04, α = 1. Panel (b) shows the best fit κ-value for a given β. The
β-values used for the insets are (i) β = 3, (ii) β = 10, (iii) β = 35. Low
values of α always give a better fit to a given marginal distribution from the
reaction-diffusion model than higher values and do not affect the value of κ
that gives the best fit. Therefore I set α = 0 when performing the fitting for
panel (b). Low values of α and κ together with high values of β tend to give
rise to good fits, but outside this range the two models show quite different
results.
is minimised (Figure 6.5).
Though the two models are qualitatively very different, if α and κ are both
very small, it is possible to find a value of β that fits closely (Figure 6.5a).
However, if either α or κ are increased, even the best fit value of β gives a
qualitatively different curve. Conversely, for lower values of β, the best fit
curve to the model studied here becomes increasingly different to the curve
from the reaction-diffusion model (Figure 6.5b).
To explain the similarities in these parameter regimes, the limit case where
the scent marks never decay is examined, so that TAS → ∞ and κ → 0. If
in addition α → 0, the marginal distribution MR (x̄) tends towards a step
function MR (x̄) = 2 if x̄ ≤ 1/2 and MR (x̄) = 0 if x̄ > 1/2. The analogous
limit in the reaction-diffusion model is µ → 0 so that β → ∞. In this limit
case, u(x̄) and w(x̄) are step functions. By taking the limit numerically as
93
β → ∞, it turns out that u(x̄) = 2 if x̄ ≤ 1/2 and u(x̄) = 0 if x̄ > 1/2 so
that u(x̄) and MR (x̄) coincide. Similarly, w(x̄) and MR (1 − x̄) coincide in this
limit.
Whilst my model has two parameters, as opposed to one in the reactiondiffusion model, it is possible to collapse my model to one parameter by formally taking the limit α → 0 in equation (6.32), giving the following expression
MR (x̄) =[H(x̄) − H(x̄ − 1)]×
)
( ∞
X
2
2
2
(−1)n e−4π n κ [Ci(2πn) − Ci(2x̄πn)] − ln(x̄) ,
2
(6.33)
n=1
where Ci(z) = −
R∞
0
dt cos(t)/t is the cosine integral. This is precisely the
limit where the reaction-diffusion model tends to agree best with mine. Plots
of equation (6.33) can be found in the insets (solid lines) of Figure 6.5 for
those cases where α = 0.
6.6.3
An analytic comparison
To remove the need for fitting curves in comparing my model in equation (6.32)
with the reaction-diffusion model in equation (6.30), an analytic expression
relating β and κ is constructed, in the limit α → 0. The key observation is
that by replacing ḡ2 (L̄2 ) (equation 6.10) with the expression
ḡrd (L̄2 ) =
4ǫβ L̄2 e2β L̄2
,
(1 + ǫe2β L̄2 )2
(6.34)
where ǫ is a small constant, and letting L̄2 range from 0 to ∞, the marginal
distribution of the animal in the limit α → 0 in my model becomes
MRD (x̄) = lim H(x̄)
α→0
Z
∞
dL̄2 ḡrd (L̄2 )h̄1D (x̄|L̄1 = 0, L̄2 ) = H(x̄)
x̄
2
,
1 + ǫe2β x̄
(6.35)
so that u(x̄) = MRD (x̄) is a solution to the reaction-diffusion equation (6.30).
The integral in equation (6.35) is performed by noticing that the limit as
α → 0 of h̄1D (x̄|L̄1 = 0, L̄2 ) is simply 1/L̄2 . It is necessary to assume that L̄2
ranges from 0 to ∞ because the utilisation distibution in equation (6.30) does
not vanish at x̄ = 1, implying that the territory border may extend beyond
x̄ = 1.
To find the correct ǫ to use for a given β, I make use of the probability
94
−2
4
(i)
0
0
0.5
x/L
1
Border distribution
−3
10
8 (ii)
6
Border distribution
2
κ
Border distribution
10
15
(iii)
10
5
0
0
0.5
x/L
1
4
2
0
0
0.5
x/L
1
10
15
20
β
25
30
35
Figure 6.6: Comparing border movement with the reaction diffusion
model. The parameter β from the reaction-diffusion model is compared with
κ from my model by equating the border MSDs using equation (6.37), thus
relating the two models without requiring curve-fitting. The trend is qualitatively similar to that obtained by finding the best numerical fit (Figure 6.5).
The insets show the border’s dimensionless distributions for the reaction diffusion model MRD (x̄) (dashed) and my model MR (x̄) (solid). For each inset,
the MSDs of both distributions are equal. The parameters used are (i) β = 7.0
(ii) β = 16.5 and (iii) β = 29.0. The resultant values of κ are (i) κ = 0.00734,
(ii) κ = 0.00149 and (iii) κ = 0.000487. When β is high, the two curves
are close. However for low β, the two curves become slightly offset, since the
animal probability distribution vanishes at x̄ = 1 in my model, but does not
vanish at x̄ = 1 in the reaction-diffusion model.
conservation condition
R1
0
dx̄ MRD (x̄) = 1 to give the equation
β = ln(1 + ǫe2β ) − ln(1 + ǫ).
(6.36)
This means that by replacing ḡ2 (L̄2 ) with ḡrd (L̄2 ) for an appropriate choice
of β, it is possible to switch between the reaction-diffusion formalism and my
reduced analytic model, in the limit α → 0.
To find this appropriate β, notice that, despite their different algebraic
forms, the graphs of ḡ2 (L̄2 ) and ḡrd (L̄2 ) have qualitatively similar shapes (see
insets in Figure 6.6). Therefore I compare them by equating their respective
MSDs. That is, the relationship between β and κ is given by the following
95
equation
∞
1 X (−1)n exp(−4π 2 n2 κ)
+
=
12
π 2 n2
n=1
Z ∞
2
Z ∞
4ǫβ L̄22 e2βL2
4ǫβ L̄32 e2βL2
dL̄2
,
dL̄2
−
(1 + ǫe2β L̄2 )2
(1 + ǫe2β L̄2 )2
0
0
(6.37)
which makes use of equations (6.10) and (6.34). The expression on the lefthand side of equation (6.37) is obtained by multiplying equation (6.10) by
(L̄2 − 1/2)2 and integrating with respect to L̄2 between 0 and 1. The result is
identical to that in equation (6.28).
For a given β, equations (6.36) and (6.37) allows the value of κ in my model
to be found, thereby determining the amount of territory border movement
implicit in the reaction-diffusion model (Figure 6.6). Conversely, this procedure enables a reaction-diffusion approximation to my model to be calculated,
in the limit α → 0.
96
6.7
Glossary
Symbol
TAS
D
ρ
v
L
L1 , L2
K
γ
x
(r, θ)
σ
a
F
p
TD1
TD2
Z1D
Z2D
α
κ
L̄1 , L̄2
x̄
r̄
σ̄
Explanation
Model
Active scent time: time for which a scent mark
S1,S2
is avoided by conspecifics
Animal diffusion constant
S1,S2,A1,A2
Animal population density in dimension d
S1,S2
Animal drift velocity towards central place (CP) S1,S2,A1,A2
Distance between CPs of adjacent territories
S1,S2,A1,A2
Positions of the left and right borders
A1
Territory border generalised diffusion constant
S1,S2,A1,A2
Rate at which territories tend to a mean size
A1,A2
Position of the animal in 1D
A1
Position of the animal in 2D polar coordinates
A2
Radius of the territory.
A2
Lattice spacing.
S1,S2
Rate of jumping to the nearest neighbour
S1,S2
Jump probability of animal towards its CP
S1,S2
TD1 = [Dρ2 ]−1 is the 1D diffusive time
S1
−1
TD2 = [Dρ] is the 2D diffusive time
S2
Normalised TAS in 1D, Z1D = TAS /TD1
S1
Normalised TAS in 2D, Z2D = TAS /TD2
S2
Normalised drift velocity α = vL/D
S1,S2,A1,A2
Normalised territory border MSD, κ = K/(γL2 ) A1,A2
Dimensionless positions of the left and right
A1
borders, L̄1 = L1 /L and L̄2 = L2 /L
Dimensionless 1D animal position, x̄ = x/L
A1
Dimensionless radial component of the animal
A2
position in 2D, r̄ = r/L
Dimensionless radius of the territory, σ̄ = σ/L
A2
Glossary of the various symbols used throught Chapter 6. The third column
details whether the symbol is used in the 1D simulation model (S1), the 2D
simulation model (S2), the 1D analytic model (A1) or the 2D analytic model
(A2).
97
Chapter 7
Discussion and conclusions
I have constructed an agent-based model whereby territories form as a natural
consequence of the movements and scent-mediated interactions of individual
animals (Chapter 2). These territories never settle to a steady state, but are
constantly changing size and shape. Their sizes fluctuate around an average,
which is inversely proportional to the animal population density. The territory
borders exhibit slow, subdiffusive movement, with a (generalised) diffusion
constant, K, that depends exponentially on Z1D or Z2D , the ratio between the
active scent time and the time it takes the animal to move over the territory
in 1D or 2D, respectively.
These properties of the border movement, observed in simulation output,
enabled an analytic model of animal movement within slowly moving territories to be constructed (Chapter 3). An adiabatic approximation was employed,
assuming that the territory borders move much slower than animals, to obtain
analytic expressions detailing the time-evolution of animal location distributions, in both 1D and 2D. In the biologically realistic case of borders that
move much slower than the animal, the approximate model agrees very well
with the agent-based simulation model.
As well as removing the need for extensive simulation analysis to determine
the animal’s movement patterns, the analytic expressions derived in Chapter
3 enable straightforward fitting of the model to location data. In Chapter 4, I
made use of a natural experiment in a red fox (Vulpes vulpes) population which
was decimated by an epizootic of sarcoptic mange in the mid 1990s. I fitted
fox location data before and after the mange outbreak to the analytic model of
territorial animal movement. This enabled quantification of the behavioural
changes caused by rapid population decline.
Chapter 5 was concerned with understanding better the reasons underlying
the observed trend between the territory border diffusion constant and the
99
active scent time (Figures 2.3 and 2.9). In particular, I derived analytically
how the border movement emerges from the interplay of adjacent territory
boundaries. This represents the first step in constructing an analytic theory
relating border movement to scent-mark longevity. Completing this theory
would give a mathematical explanation of the features observed in simulation
analysis so obviate the need for running further time-consuming simulations
to obtain results in different parameter regimes.
In Chapter 6, I studied a modification of the models described in Chapters
2 and 3. There, rather than purely random movement, animals also had a
drift tendency towards a central place, such as a den or nest site, or a core
foraging area. This drift makes each animal’s space use distribution reach a
steady-state, often referred to as the animal’s home range. By integrating the
locations of territory borders over time, it is possible to relate the home range
overlap to the active scent time, and thus link population-level parameters
to individual-level characteristics. These results enabled a programme to be
constructed for using location data to quantify the underlying movement and
interaction processes of the animals, in the case where the utilisation distribution reaches a stable state.
7.1
Ecological implications
That dynamic territories with fluctuating borders naturally emerge from animal movement and interaction processes is perhaps the most important message of this thesis. Field studies of neighbouring territorial animals often
suggest that their home ranges overlap, despite the fact that territorial behaviour aims at maintaining exclusive use of an area. My results explain
this apparent paradox by showing that overlaps can arise as a direct consequence of moving territory borders, and may not be an implicit biological
phenomenon. As location data are gathered over time, so the territory borders move. Once sufficient location fixes have been obtained to measure a
territory size (Harris et al., 1990), the shift in borders may make it seem as if
the territories overlap, when in fact they do not.
This natural flexibility in borders means that even if a population of territorial animals is in decline, they are still able to keep their borders contiguous.
Such behaviour was observed when Bristol’s fox population was decimated by
the 1994-6 mange epizootic (Baker et al., 2000), and foxes would seek to acquire areas previously occupied by newly-deceased animals. If animals were to
keep the same movement and interaction processes after a decline in population, the theoretical results presented here suggest that the territory borders
100
would increase their movement exponentially (see Figure 2.9). Such a large
increase could mean that animals would become no longer able to maintain
areas of exclusive use.
The results summarised in Figure 2.9 also suggest that animals could, in
theory, mitigate against this exponential increase in border movement by raising either their speed, directional persistence or active scent time. Indeed,
Bristol’s foxes increased their speed when the population declined, also observed in Baker et al. (2000). However, they did not change their turning
angle distribution and the active scent time appeared to decrease rather than
increase. Though there was a large increase in the border movement (Table
4.2), the faster movements of the foxes ensured the border movement was not
enough to cause a significant increase in the percentage of overlap between
neighbouring home ranges.
The lack of change in turning angle distribution may indicate an inbuilt
behavioural strategy in urban foxes. Perhaps this is due to this distribution
being optimal for searching confined areas such as a territory. Alternatively, it
could be a result of the underlying topology of the urban environment through
which foxes move. For example, they are forced to navigate around buildings.
Also, they may preferentially tend to choose certain terrains that are energetically efficient to move through, such as alleys and small roads, to travel
between foraging patches. Such behaviour has been observed in badgers (Meles
meles) in the Bristol area (Cresswell & Harris, 1988).
The decrease in active scent time from 5 to just over 3 days (Table 4.2)
meant that foxes waited for a shorter time before attempting to acquire territorial area that they believed to be vacated, once the population was in
decline. This may have been due to the foxes being aware that the population
was declining, so primed to close up any gaps that appeared as a result of
territory vacation due to fox mortality. It is important for territorial animals,
such as foxes, to maintain contiguous borders in order to exclude potential invaders (Baker et al., 2000). Therefore it is advantageous for them to be swift
in closing gaps between territories during times such as the mange epizootic,
when deaths cause large unoccupied areas to appear in the terrain.
The apparent decrease in the foxes’ active scent time may have been due to
the auditory aspect of their territorial behaviour. In addition to scent, vocal
cues are used to inform neighbours of their presence (Newton-Fisher et al.,
1993). The absence of these cues after the death of a fox would suggest
to neighbours that the territory had been vacated. Consequently, after the
mange outbreak, they may have been more willing to venture into areas that
contained scent that was only 3 or 4 days old, causing the active scent time
101
to appear to reduce.
As well as these behavioural insights, the results of this thesis have great
implications for modelling epizootic spread, particularly when the disease is
terminal. Rather than assuming constant territory or home range sizes, as is
the trend in epizootic modelling of territorial animals (Smith & Harris, 1991;
Smith & Wilkinson, 2002; Kenkre et al., 2007; Salkeld et al., 2010), my results
demonstrate that rapid declines in population density elicited by terminal
disease spread cause large changes in the territorial dynamics. This in turn
affects the behaviour of the animals and the inter-animal contact rates. The
modelling approach of this thesis provides the necessary basis to enable future
epidemiological studies to take such behavioural and territorial fluctuations
into account, allowing for improved predictions of disease spread.
7.2
Comparisons with previous work
The models presented in this thesis are not the first to examine territory formation through scent-mediated conspecific avoidance. A number of studies have
modelled such a process using reaction-diffusion formalisms, beginning with
Lewis & Murray (1993) and catalogued in the book by Moorcroft & Lewis
(2006). These have helped explain a number of ecological phenomena, including prey corridors that appear between wolf territories (Lewis et al., 1997), the
effect of a territory dissolving due to removal of a coyote pack (Moorcroft et al.,
1999), home range formation without conspecific avoidance (Briscoe et al.,
2001), the influence of evolution on territory size and shape (Hurford et al.,
2006; Hamelin & Lewis, 2010) and the relative effects of resources and topography on space-use distributions of canids (Moorcroft et al., 2006).
Whilst these models build the animal movements from individual-level
processes (Moorcroft & Lewis, 2006, appendices), the conspecific avoidance
mechanism is constructed in a mean-field way, by coupling the spatial distribution of the animal’s locations with that of its scent marks. On the other
hand, the agent-based approach used here builds the conspecific avoidance
process mechanistically from interactions of individual agents. As well as circumventing the inherent problems with reaction-diffusion formalisms (section
1.4), the mechanistic approach of this thesis enables the link between location
data and the underlying interaction processes to be made clear (see Chapter
4 or section 6.4).
A further drawback of the reaction-diffusion models proposed so far is that
they tend to assume a priori the position of a central place, such as a den or
nest site, to which the animals return from time to time. In other words,
102
one of the outcomes of the territory formation process is predetermined. The
theoretical study of Briscoe et al. (2001) provides an exception to this, since
home ranges form due to memory traces left on the landscape. However, the
model also does not include any conspecific avoidance, so I refer to ‘home
range’ rather than ‘territory’ formation. Also, the study assumes a priori that
the animal is confined within a finite area.
On the other hand, my modelling approach makes no assumption about
the positions of den sites, and I use periodic boundary conditions to approximate infinite space. In certain circumstances, such as on a small island
(Zabel & Taggart, 1989) or a steep-sided valley (Moorcroft et al., 2006), spatial confinement may be an important feature. However, mainland Britain,
where Bristol’s fox population reside, is around 106 times as large as an average urban fox territory, so for modelling purposes it is effectively infinite in
extent. Furthermore, the generic agent-based framework provided here can be
easily adapted to cope with territoriality in highly confined regions, whereas
it is unclear whether the results of Briscoe et al. (2001) extend to unconfined
regions.
Recently, Van Moorter et al. (2009) proposed a model of home range formation using memory effects that does not assume spatial confinement a priori. Like Briscoe et al. (2001), this model does not include any conspecific
avoidance, rather it combines resource distribution with a cognitive map of
the landscape to show that, under certain circumstances, model animals appear to form stable home ranges. Similar to the model presented here, Van
Moorter’s is based on a random walk. As such, it has the potential to be combined with ours to model territoriality in populations where the movement is
highly influenced by resource distribution.
As well as the random aspect of the animal’s movement, Van Moorter’s
model incorporates a drift tendency towards areas where the animal expects
to find food, based on its memory of the resource patches it has recently
encountered. As the animal gathers information about an area sufficiently
large to meet its resource requirements, it will gain fidelity towards this area,
and form a home range there. Since it knows food can be found within the
home range at sufficiently frequent intervals to survive, it has no need to
venture away to forage.
Like the optimal foraging approaches discussed in the Introduction (Section
1.4), Van Moorter’s model assumes that an animal will have little or no desire
to occupy a home range larger than is necessary for resource requirements.
When animals are territorial, though, they may benefit from ensuring borders
are contiguous (Baker et al., 2000), so seek to increase territorial area if they
103
can. When the population density is low, or resources are very abundant,
this may enable animals to maintain larger territories than is necessary for
resource use (Ansell, 2004).
Territory size is positively correlated with reproductive success for a variety of animals (Nilsson, 1976; McCleery & Perrins, 1985; Seddon et al., 2004;
Iossa et al., 2008), suggesting that there is a tendency for animals to put pressure on their borders, in an attempt to enlarge their territories, as well as
spending time well within their borders for foraging purposes. Therefore models of territoriality that include foraging to keep an animal close to the centre,
must also take neighbour pressure into account, by including a tendency to
visit the borders occassionally. This is likely to result in a trade-off between
the time spent foraging versus snooping around the borders.
7.3
7.3.1
Possible future directions
Building an analytic theory of border movement
In Chapter 3, I described a programme for building an analytic theory of territory border movement. Currently, it is necessary to use simulation analysis
to relate active scent time to the border diffusion constant K. The universal
curves of Figures 2.3, 2.9 and 6.1 provide evidence that K depends upon the
quotient of active scent time and the time it takes for an animal to cover its
territory. However, each new movement process requires another curve to be
constructed, which both takes significant computational time (about 2 weeks
CPU time per curve) and only ever gives results that are valid within a finite
parameter range. An analytic theory would obviate the need for simulation
analysis, as well as giving insights into the observed trends.
7.3.2
Using approximate reaction-diffusion approaches
Our mechanistic movement-and-interaction model enables quantification of
the relations between individual processes and population-level patterns that
cannot be captured by the mean-field approach to interactions inherent in current reaction-diffusion formalisms, such as those of Moorcroft & Lewis (2006).
That said, such approaches have been very fruitful both in modelling territoriality and other biological phenomena, and the techniques have a long
history of development (Murray, 2003). Therefore it would be useful to derive
reaction-diffusion equations that are good approximations of the various models in this thesis, or extensions thereof. Though these equations will not be
the same as those of Moorcroft & Lewis (2006), the techniques employed there
104
and elsewhere could be used to help analyse modifications and extensions of
my models.
An advantage of the reaction-diffusion approach is that the models are
both analytic and take into account the spatial heterogeneity of territorial
patterns (Moorcroft et al., 2006). On the other hand, the models presented
here consist of (i) an agent-based model (Figure 2.1) that incorporates spatial
heterogeneity (Figure 2.8) but currently requires time-consuming simulation
analysis, and (ii) an approximate analytic model (equations 3.29 and 3.40)
that requires an a priori determination of the territory shape, i.e. square
or circular. Constructing the appropriate reaction-diffusion approximation
to the agent-based model, using techinques similar to those in section 6.6.3,
would give an approximate analytic model of territory formation that also
incorporates spatial heterogeneity.
7.3.3
Extensions to the model
As well as conspecific avoidance and fidelity to a central place, factors such as
resources (Gray et al., 2002), memory (Briscoe et al., 2001) and topography
(Moorcroft et al., 2006) have been shown to affect the space use of territorial animals. In section 7.2, I discussed how to factor resources and memory
into my model to quantify their effects on both individual movements and
territorial dynamics.
Topography may also affect animal movement, due to a desire either to
move away from steep or high ground, as observed in wolves (Moorcroft et al.,
2006), or towards steep-sided valleys to avoid predation, as seen in populations
of mountain goats (Pfitsch & Bliss, 1985). Therefore the topographical structure of mountainous regions may affect both territorial patterns and predatorprey dynamics. Placing features such as mountains and valleys into my model
would enable assessment of the topographical effects on both territorial and
predation interactions.
For the most part, the models in this thesis are examples of mechanistic
modelling, showing how animal behaviour causes population-level phenomena. To understand why animals have certain behavioural features requires an
evolutionary modelling approach. I have made some initial inroads into this
(section 4.4) but much more needs to be done. Specifically, the preliminary
methods of section 4.4 need to be extended and combined with population
models such as López-Sepulcre & Kokko (2005). Analysis of the attractors
and steady states of these models would determine which strategies, if any,
are evolutionarily stable.
105
7.3.4
Epidemiological applications
Calculating contact rates is very important for quantitative study of epidemics,
e.g. Zhang & Ma (2003). The model described in this thesis is the first to
understand how individual interactions relate to territorial structure, so has
great potential to improve models of epizootics in territorial populations, by
measuring contact rates directly from the model. Additionally, if the disease is
terminal, territories will dissolve, causing other animals to acquire the newlyfreed areas. Rather than assuming that territory sizes and locations are fixed,
my model takes into account the important aspect of territorial flexibility,
which also greatly affects contact rates. Letting diseases spread through my
simulations, for example using a classical SIR model or variant, would enable
more accurate predictions of epizootics in territorial populations than using
current models, which tend to assume fixed territories.
7.4
Final remarks
This thesis contains the first model of territory formation built up from the
movement and interaction mechanisms of individual animals. It represents a
step forward from previous mechanistic models, where only the movements
were constructed from individual-level processes (Moorcroft & Lewis, 2006).
Whilst the simple random-walk based models presented here already have
practical application to populations such as Bristol’s foxes, the generic framework of territorial random walkers can readily be extended to other, more
complicated ecological scenarios. I have already begun to extend my model in
this way, taking into account different random movement processes (Chapter
2), ballistic motion (Chapters 2 and 4) and central place attraction (Chapter
6). Other ways to extend the model, such as incorporating memory, resource
or evolutionary effects, have been discussed in this chapter. The modelling
framework described in this thesis thus provides the necessary basis for future
mechanistic studies of territory formation.
106
Appendices
107
Appendix A
Boundary return time
calculation
To compute the first passage probability for going from site 0 to site N using
the master equation (2.4) from chapter 2, I use the corresponding generating
function
P˜s (n, z) =
∞
X
Ps (n, m∆t)z m ,
(A.1)
m=0
where m = t/∆t is discrete, dimensionless time. Since there is no cause for
confusion, I will re-write P˜s (n, z) as Ps (n, z). Factoring in the initial condition
Ps (n, t) = δ0,n , where δ is the Kronecker delta, and writing Fs (N, z) for the
generating function of the first passage probability to N , gives the following
iterative equations
Ps (0, z) = 1 + z̄Ps (1, z),
Ps (1, z) = 4z̄Ps (0, z) + z̄Ps (2, z) + 2z̄Ps (1, z),
Ps (n, z) = z̄Ps (n − 1, z) + z̄Ps (n + 1, z) + 2z̄Ps (n, z),
if 2 ≤ n ≤ N − 2,
Ps (N − 1, z) = z̄Ps (N − 2, z) + 2z̄Ps (N − 1, z),
Fs (N, z) = z̄Ps (N − 1, z),
(A.2)
where z̄ = z/4. I then introduce the sequence
f1 =4z̄ 2 ,
fn =
z̄ 2
1 − fn−1 − 2z̄
109
,
(A.3)
so that the following relations hold
4
1
f2 . . . fn+1 + fn+1 Ps (n + 1, z),
s
z̄
z̄
4
Ps (N − 1, z) = N −1 f2 . . . fN ,
z̄
4
Fs (N, z) = N −2 f2 . . . fN .
z̄
Ps (n, z) =
Let fn =
gn
hn
if 1 ≤ n ≤ L − 2,
(A.4)
so that
fn =
z̄ 2 hn−1
.
hn−1 − gn−1 − 2z̄hn−1
(A.5)
Let gn = z̄ 2 hn−1 Cn and hn = (hn−1 −gn−1 −2z̄hn−1 )Cn for some Cn . Since the
quantities of interest are the ratios of gn to hn , let Cn = 1 so that gn = z̄ 2 hn−1
and hn = hn−1 − gn−1 − 2z̄hn−1 . It follows that hn = (1 − 2z̄)hn−1 − z̄ 2 hn−2 .
√
Therefore hn = Aλn+ + Bλn− where λ± = (1 − 2z̄ ± 1 − 4z̄)/2 and A, B are
to be found by factoring in boundary conditions.
Since f1 = z̄ 2 and f1 = z̄ 2 h0 /4h1 , it is possible to choose h1 = 1, h0 = 4.
√
√
Then A + B = 4 and 1 = 2A(1 − 2z̄ + 1 − 4z̄) + 2B(1 − 2z̄ − 1 − 4z̄). It
√
√
follows that A = 2 − 1 − 4z̄ and B = 2 + 1 − 4z̄. Putting all this together
gives
4z̄ N
√
.
(A.6)
N
(2 − 1 − 4z̄)λN
+ + (2 + 1 − 4z̄)λ−
√ z−1
since this means that λ± = z̄e±iφ .
It is convenient to let φ = arctan 1−
z
Fs (N, z) =
√
2
Then it is straightforward to show that
Fs (N, z) =
2
√
.
2 cos(N φ) − z − 1 sin(N φ)
(A.7)
To find the mean first-passage time to go from site 0 to site N, find the derivative of Fs (L, z) evaluated at z = 1 and multiply it by the time ∆t it takes to
perform one jump, giving the result from equation (2.5) in the main text.
110
Appendix B
Solution of the Fredholm
integral equation from section
5.1
Since (5.6) from the main text is a Fredholm equation with degenerate kernel (Polyanin & Manzhirov, 1998), its solution is a linear combination of the
R 2π
quantities γi = 0 dφ′ bi (φ′ )h(φ′ ) for i = 1, 2, 3, which satisfy the following
system of equations
γ1 (1 − α11 ) − γ2 α12 − γ3 α13 = β1 ,
−γ1 α21 + γ2 (1 − α22 ) − γ3 α23 = β2 ,
−γ1 α31 − γ2 α32 + γ3 (1 + α33 ) = β3 .
(B.1)
R 2π
The various βi ’s and αij ’s can be calculated as βi = 0 dφ′ bi (φ′ )g(φ′ ) and
R 2π
αij = 0 dφ′ bi (φ′ )aj (φ′ ) to yield the following expressions
∆N−1
2
p
,
1−p
i(∆N −1)θ
∆N
2
p
e−(∆N +1)u
eiN1 θ e 2
,
β2 =
θ
1−p
8F (1 − p) cos 2 sinh u
i(∆N +1)θ
∆N−2
2
e−(∆N −1)u
p
eiN1 θ e 2
,
β3 =
θ
1−p
8F (1 − p) cos 2 sinh u
i∆Nθ
eiN1 θ e 2 e−∆N u
β1 =
8F (1 − p) cos θ2 sinh u
111
1
α11 =
[p(1 − p)]− 2 − 2e−u cos 2θ
α21 =
e− 2 e−u {(1 − p)−1 − 2[p/(1 − p)] 2 cos θ2 cosh u}
α31 =
e 2 e−u {p−1 − 2[(1 − p)/p] 2 cos 2θ cosh u}
α12 =
e 2 {e−u [(1 − p)/p] − [(1 − p)/p] 2 cos 2θ }
α22 =
[(1 − p)/p] 2 − e−u cos( 2θ )
α32 =
[(1 − p)/p]eiθ e−u {[(1 − p)/p] 2 e−u − cos 2θ }
α13 =
e− 2 {e−u [p/(1 − p)] − [p/(1 − p)] 2 cos θ2 }
α23 =
[p/(1 − p)]e−iθ e−u {e−u [p/(1 − p)] 2 − cos 2θ }
α33 =
[p/(1 − p)] 2 − e−u cos( 2θ )
2 cos θ2 sinh u
,
iθ
1
2 cos 2θ sinh u
iθ
,
1
2 cos θ2 sinh u
iθ
,
1
2 cos 2θ sinh u
,
1
2 cos θ2 sinh u
,
1
2 cos 2θ sinh u
iθ
,
1
2 cos θ2 sinh u
,
1
2 cos θ2 sinh u
,
1
2 cos θ2 sinh u
.
(B.2)
In these equations u is defined by (5.8) in the main text. Solving the system
of equations (B.1) eventually gives
γ1 = γ2 = 0,
γ3 =
eiN1 θ ei
(∆N+1)θ
2
e−∆N u [p/(1 − p)]
1
∆N −2
2
4F {(1 − p) cos θ2 − [p(1 − p)] 2 e−u }
Plugging these values for γi =
R 2π
0
.
(B.3)
dφ′ bi (φ′ )h(φ′ ) into (5.6) in the main text
gives the expression for the generating function of the system’s probability
distribution in Laplace domain.
112
Appendix C
Solution of Fokker-Planck
equation from section 5.3
In order to solve (5.24) with boundary condition (5.25) from the main text, it
is convenient to convert to coordinates xs = x2 − x1 and xc = (x1 + x2 )/2 so
that xs is the separation distance between the particles and xc is the centroid.
This equation (5.24) to be written as
∂R
(xc , xs , t) = D
∂t
1 ∂2
∂2
+
2
2 ∂x2c
∂x2s
R(xc , xs , t) + 2v
∂R
(xc , xs , t),
∂xs
(C.1)
where R(xc , xs , t) = Q(x1 , x2 , t). The flux vector of equation (C.1) is
∂R
J = − 2D
(xc , xs , t) + 2vR(xc , xs , t)
∂xs
(C.2)
so the zero-flux boundary condition mentioned in the main text is n̂·J|xs =0 = 0
where n̂ is a unit normal to the line xs = 0 (Ambjörnsson, 2008). By writing
R(xc , xs , t) = Rc (xc , t)Rs (xs , t), (C.1) becomes
∂Rc
D ∂ 2 Rc
(xc , t),
(xc , t) =
∂t
2 ∂x2c
(C.3)
∂ 2 Rs
∂Rs
∂Rs
(xs , t) = 2D
(xs , t) + 2v
(xs , t),
2
∂t
∂xs
∂xs
(C.4)
and
with the boundary condition
113
∂Rs
D
(xs , t) + vRs (xs , t) ∂xs
= 0.
(C.5)
xs =0
The solution to (C.3) is a Gaussian and the solution to (C.4) with boundary
condition (C.5) can be found in e.g. Polyanin (2002) with the result
(xc −xc,0 )2
2Dt
e−
Rs (xc , t) = √
"
(xs −xs,0 +2vt)2
8Dt
2πDt
,
(C.6)
v
e 2D (xs,0 −vt−xs ) e−
√
√
Rc (xs , t) =H(xs )
+
8πDt
8πDt
#
xs + xs,0 − 2vt − vx
v
√
erfc
+
e D ,
2D
8Dt
e−
(xs +xs,0 )2
8Dt
(C.7)
where xs,0 = ∆x0 and xc,0 = (x1,0 + x2,0 )/2 are the initial conditions, and
H(x) is the Heaviside step function (H(x) = 0 if x < 0, H(x) = 1 if x ≥ 0).
The solution to (5.24) from the main text can now be written down as
Q(x1 , x2 , t) =H(x2 − x1 )
e−
(x1 −x1,0 +x2 −x2,0 )2
2Dt
√
8πDt
"
e−
(x2 −x2,0 −x1 +x1,0 +2vt)2
8Dt
√
2πDt
+
(x2 −x1 +x2,0 −x1,0 )2
v
8Dt
e 2D (x2,0 −x1,0 −vt−x2 +x1 ) e−
√
+
2πDt
#
x2 − x1 + x2,0 − x1,0 − 2vt − vx
v
√
erfc
e D
D
8Dt
(C.8)
In order to show that (C.8) is equivalent to (5.23) from the main text, the
following integral is calculated
−
t
[xs +2v(t−s)]2
8D(t−s)
[xs + 2v(t − s)]e
p
ds
I(xs , t) =
4D π(t − s)3
0
#
∆x0 − 2vs
v
√ erfc
√
.
2D
8Ds
Z
114
"
e−
(∆x0 −2vs)2
8Ds
√
πs
+
(C.9)
Since this is the sum of two convolutions in time, its Laplace transform can
be found by using the identity L[f ∗ g] = L[f ]L[g], where the asterix denotes
Rt
the convolution f ∗ g = 0 dsf (s)g(t − s). Since
−
[xs +2v(t−s)]2
−
8D(t−s)
[xs + 2v(t − s)]e
p
4D π(t − s)3
[xs +2v(t−s)]2
8D(t−s)
∂ e
p
=−
,
∂xs
π(t − s)
the Laplace transform of I(xs , t) can be written as
L[I(xs , t)] = −
"
−
∂ e
∂xs
v(xs −∆x0 )
2D
ǫ+
v2
2
e
q
q
v2
ǫ+ 2D
v2
2D
|xs |+∆x0
√
2D
v(|xs |+∆x0 )
−
2D
v e
√
2D
ǫ+
e
|xs |+∆x0
√
2D
ǫ+
q
v2
2
+
2
ǫ+ v2 − √v
−
2D
√v
2D
#
.
(C.10)
By repeatedly using the formula (5.21) from the main text, expression (C.10)
can be Laplace inverted to give
"
#
v
∂
|x
|
+
∆x
−
2vt
s
√ 0
e− 2D (|xs |+xs ) erfc
I(xs , t) = −
.
∂xs
8Dt
(C.11)
Performing the differentiation with respect to xs gives
v
v
(sgn(xs ) + 1) e− 2D (|xs |+xs ) erfc
I(xs , t) =
2D
v
(|xs |+∆x0 )2
8Dt
e 2D (∆x0 −xs −vt) e
√
sgn(xs )
2πDt
|xs | + ∆x0 − 2vt
√
8Dt
+
(C.12)
where sgn(x) is the sign of x (sgn(x) = −1 if x < 0 and sgn(x) = 1 if x ≥ 0).
After replacing the second term of (5.23) from the main text with I(xs , t)
one can show that the continuum limit of (5.7) is indeed the solution of the
Fokker-Planck equation (5.24) with the above mentioned zero-flux boundary
conditions.
115
116
Appendix D
Asymptotic continuous-time
expressions from section 5.3
Since the values of d(t) and ∆x2 (t) depend only on the initial condition ∆x0
and not the specific values of x1,0 and x2,0 , calculations are simplified by
assuming x1,0 = 0. The marginal probability distribution for the left-hand
(right-hand) particle in Fourier-Laplace domain is found by setting k2 = 0
(k1 = 0) in equation (5.20). Focussing on the left-hand particle gives the
following expression
k1
1
iei∆x0 2 k1
Q̃1 (k1 , ǫ) =
−
2
2
ǫ − ik1 v + Dk1
ǫ − ik1 v + D
2 k1
q
D
2e
q
∆x
√ 0
2D
ǫ+
√v −
2D
v2
2D
+
q
2
v
ǫ+ 2D
+D
k2
2 1
D 2
2 k1
−
√v
2D
.
(D.1)
This allows the mean position hx1 (ǫ)i of the left-hand particle to be caluclated
in Laplace domain, by differentiating (5.10) with respect to k1 , multiplying by
−i and setting k1 = 0
hx1 (ǫ)i =
√
∆x
√ 0
2D
q
v
ǫ+ 2D
−
√v
2D
√v −
2D
De
v
−
√ q
ǫ2
ǫ 2
ǫ+
v2
2D
2
.
(D.2)
Differentiating (D.1) twice with respect to k1 , multiplying by −1 and again
setting k1 = 0 gives the second moment of the distribution
117
q
v
v2
√ √∆x2D0 √2D
− ǫ+ 2D
2
2v
2D (∆x0 ǫ + 2v) De
hx21 (ǫ)i = 3 + 2 −
.
√ q
ǫ
ǫ
2
v
v
ǫ2 2
ǫ + 2D − √2D
(D.3)
By using the formula (5.21) from the main text, (D.2) and (D.3) can be inverted exactly to give the respective formulae in time domain. Performing the
same calculations for the right-hand particle allows the following expressions
for the mean separation and MSD to be found
d(t) = ∆x0 − v
Z
t
0
2
2vs − ∆x0
√
ds erfc
8Ds
2
Z
t
+
r
2D
π
Z
t
ds
e−
(∆x0 −2vs)2
8Ds
0
2vs − ∆x0
√
8Ds
√
s
, (D.4)
∆x (t) =2Dt − ∆x0 vt + v
ds (t − s)erfc
−
0
r Z
∆x0 + 2v(t − s) (∆x0 −2vs)2
D t
√
e 8Ds −
ds
2π 0
s
2

(∆x0 −2vs)2
r Z t
Z t
−
8Ds
D
2vs
−
∆x
e
v
0
 . (D.5)

√
√
−
ds erfc
ds
2 0
2π 0
s
8sD
Expressions (5.26) and (5.27) from the main text are obtained by applying the
2
R t As+B − As−B
Rt
√
1
As−B
As−B
s
= t erf √s
− √π 0 ds √s e
throughformula 0 dserf √s
out (D.4) and (D.5).
118
Appendix E
Matlab code for the
distribution of a territorial
animal
% Function to calculate the dimensionless marginal distribution of a
% particle undergoing a telegraphers equation inside a fluctuating
% square territory with arbitrary initial conditions.
%
% Parameters: bar_K - dimensionless K: KT/L^2
%
bar_v - dimensionless speed: vT/L
%
bar_gamma - dimensionless gamma: gamma*T
%
t_bar - dimensionless time to find marginal
%
sum_acc - accuracy of sum
%
R_acc - place to cut of tails in R(lambda,t)
function marginal = proverbial_square_marginal(bar_K, bar_v, ...
bar_gamma, t_bar, x_bar, y_bar, sum_acc, R_acc)
% Trapz lambda-vals; lambda is peaked around 1
lambda_vals = [0.001:0.001:0.009 0.01:0.01:0.09 0.1:0.1:0.4 ...
0.5:0.001:0.99 0.991:0.0001:1.009 1.01:0.001:1.5 1.6:0.1:10 11:1:100];
% Calculate bt_bar, ct_bar, R_bar and U_bar
bt_bar = ...
((4*bar_K)/bar_gamma)*(1-exp(-2*bar_gamma*(t_bar/log(1+t_bar)-1)));
ct_bar = 2*bar_K*(t_bar/log(1+t_bar)-1);
R_bar = (exp(-((lambda_vals-1).^2)./bt_bar)+ ...
exp(-((lambda_vals+1).^2)./bt_bar))./sqrt(pi.*bt_bar);
% Remove negligible values of R(lambda,t)
R_low_count = 1;
119
while R_bar(R_low_count) < R_acc
R_low_count = R_low_count + 1;
end
R_high_count = R_low_count;
while R_bar(R_high_count) >= R_acc
R_high_count = R_high_count + 1;
end
R_bar = R_bar(R_low_count:R_high_count);
lambda_vals = lambda_vals(R_low_count:R_high_count);
% Calculate first summand
U_bar_x = erf((lambda_vals-2.*x_bar)./(2.*sqrt(ct_bar)))+ ...
erf((lambda_vals+2.*x_bar)./(2.*sqrt(ct_bar)));
U_bar_y = erf((lambda_vals-2.*y_bar)./(2.*sqrt(ct_bar)))+ ...
erf((lambda_vals+2.*y_bar)./(2.*sqrt(ct_bar)));
marginal = ...
trapz(lambda_vals, U_bar_x.*U_bar_y.*R_bar./(4.*(lambda_vals.^2)));
% Loop though the other summands, stopping when desired accuracy is
% reached.
error_ny0 = sum_acc*2;
% x=0 cases
% Done y=0 so just need y>0
nx = 0;
ny = 0;
error = sum_acc*2;
while abs(error) > sum_acc
ny = ny + 1;
integrand = U_bar_x./(2.*lambda_vals);
integrand = integrand.*proverbial_square_marginal_xy(y_bar, ...
lambda_vals, t_bar, ct_bar, bar_v, ny,U_bar_y);
error = trapz(lambda_vals, integrand.*R_bar);
marginal = marginal + error;
end
% x>0 cases
while abs(error_ny0) > sum_acc
nx = nx + 1;
% Initial summand (y=0)
integrand = proverbial_square_marginal_xy(x_bar, lambda_vals, ...
t_bar, ct_bar, bar_v, nx,U_bar_x);
integrand = integrand.*U_bar_y./(2.*lambda_vals);
120
error_ny0 = trapz(lambda_vals, integrand.*R_bar);
marginal = marginal + error_ny0;
% Other summands (y>0)
ny = 0;
error = sum_acc*2;
while abs(error) > sum_acc
ny = ny + 1;
integrand = proverbial_square_marginal_xy(x_bar, ...
lambda_vals, t_bar, ct_bar, bar_v, nx,U_bar_x);
integrand = integrand.*proverbial_square_marginal_xy(y_bar, ...
lambda_vals, t_bar, ct_bar, bar_v, ny,U_bar_y);
error = trapz(lambda_vals, integrand.*R_bar);
marginal = marginal + error;
end
end
----------------------------------------------------------------------% Subfunction of proverbial_square_marginal()
function output = proverbial_square_marginal_xy(xy_bar, ...
lambda_vals, t_bar, ct_bar, bar_v,n,U_bar)
erf_acc = 10;
theta_n = sqrt((n.*pi.*bar_v./lambda_vals).^2-1/4);
Sn_val1_in = (2.*lambda_vals.*xy_bar-(lambda_vals.^2)+ ...
2.*complex(0,1).*n.*pi.*ct_bar)./(2.*lambda_vals.*sqrt(ct_bar));
Sn_val2_in = (2.*lambda_vals.*xy_bar+(lambda_vals.^2)+ ...
2.*complex(0,1).*n.*pi.*ct_bar)./(2.*lambda_vals.*sqrt(ct_bar));
Sn_val1 = cerf(Sn_val1_in,erf_acc,complex(0,1).*n.*pi.*xy_bar./ ...
lambda_vals-(n.^2).*(pi.^2).*ct_bar./(lambda_vals.^2));
Sn_val2 = cerf(Sn_val2_in,erf_acc,complex(0,1).*n.*pi.*xy_bar./ ...
lambda_vals-(n.^2).*(pi.^2).*ct_bar./(lambda_vals.^2));
Sn_val = imag(Sn_val1-Sn_val2);
Sn_val = (Sn_val./(n.*pi)).*sin(n.*pi./(2.*lambda_vals));
output = exp(-t_bar./2).*(cos(theta_n.*t_bar)+ ...
sin(theta_n.*t_bar)./(2.*theta_n));
output = output.*(((-1).^n).*Sn_val - (U_bar./(n.*pi)).* ...
sin(n.*pi.*xy_bar./lambda_vals).*sin(n.*pi./(2.*lambda_vals)));
----------------------------------------------------------------------% Calculate the complex error function: erf(z)=1-w(iz)exp(-z^2) if
% imag(iz)>0 or erf(z)=-[1-w(-iz)exp(-z^2)] otherwise.
121
%
% Parameters: z - the value to calculate cerf(z)
%
erf_acc - the accuracy of the cef() function
%
extra_exp - multiply the result by exp(extra_exp)
function output = cerf(z,erf_acc,extra_exp)
% Need two cases since cef(x) is only valid for imag(x)>=0
erfsign = sign(imag(complex(0,1).*z));
output = erfsign.*(exp(extra_exp)- ...
cef(erfsign.*complex(0,1).*z,erf_acc).*exp(-z.^2+extra_exp));
----------------------------------------------------------------------% Computes the function w(z) = exp(-z^2) erfc(-iz) using a rational
% series with N terms. It is assumed that Im(z) > 0 or Im(z) = 0.
%
%
Andre Weideman, 1995
function w = cef(z,N)
M = 2*N; M2 = 2*M; k = [-M+1:1:M-1]’;
L = sqrt(N/sqrt(2));
theta = k*pi/M; t = L*tan(theta/2);
f = exp(-t.^2).*(L^2+t.^2); f = [0; f];
a = real(fft(fftshift(f)))/M2;
a = flipud(a(2:N+1));
Z = (L+i*z)./(L-i*z); p = polyval(a,Z);
w = 2*p./(L-i*z).^2+(1/sqrt(pi))./(L-i*z);
%
%
%
%
%
%
%
%
M2 = no. of sampling points.
Optimal choice of L.
Define variables theta and t.
Function to be transformed.
Coefficients of transform.
Reorder coefficients.
Polynomial evaluation.
Evaluate w(z).
-----------------------------------------------------------------------
122
Appendix F
Movie captions
To help visualise some of the dynamic processes described in this thesis, the
attached CD-ROM contains a variety of movies. These are each explained
below.
F.1
Territory movement
File: territory movie.gif
Caption: A movie of the territorial dynamics of 25 animals with a dimensionless active scent time of TAS F = 2, 500 in a box of 100 × 100 sites with
periodic boundary conditions, where F is the rate of jumping between lattice
sites. The initial movie frame is recorded after a small transient obtained
from an initial condition with the animals periodically placed on the lattice
and without any scent profile. The snapshots of the simulations are taken at
dimensionless time intervals of 2, 500tF , where t is time.
F.2
Time dependence of the probability distribution
File: space use distribution.gif
Caption: The time-evolution of the probability distribution M2D (x, y, t|Θ)
(equation 3.40). The contours on the movie show M2D (x, y, t|Θ) for the values
of Θ that best-fit the pre-mange data. The dots on the movie show cumulative
locations of foxes through time from the pre-mange data set, normalised so
that the centres of their home ranges are all at (0, 0) and the distances from
the centre are divided by L = 435m.
123
F.3
Dynamics of territorial acquisition
File: territory acquisition.gif
Caption: The dynamics of territorial acquisition, showing 25 territorial random walkers on a 100 × 100 square lattice with periodic boundary conditions.
Each territory is denoted by a different colour. The white squares are intersti-
tial regions, where there is no active scent. Part way through the movie, the
simulated animal with the cyan territory is removed, shown by the colour of
the territory turning black. As the scent of this animal becomes inactive, the
black squares turn white, allowing the other animals to move in and acquire
the territory.
124
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