On chaos, transient chaos and ghosts in single population models

Transcription

Nonlinear Analysis: Real World Applications 13 (2012) 1647–1661
Contents lists available at SciVerse ScienceDirect
Nonlinear Analysis: Real World Applications
journal homepage: www.elsevier.com/locate/nonrwa
On chaos, transient chaos and ghosts in single population models with
Allee effects
Jorge Duarte a,b,∗ , Cristina Januário a , Nuno Martins b , Josep Sardanyés c,∗∗,1
a
ISEL - Engineering Superior Institute of Lisbon, Department of Mathematics, Lisboa, Portugal
b
Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Departamento de Matemática, Instituto Superior Técnico, Lisboa, Portugal
c
Instituto de Biología Molecular y Celular de Plantas, Consejo Superior de Investigaciones Científicas-Universitat Politècnica de València, València, Spain
article
info
Article history:
Received 15 August 2011
Accepted 30 November 2011
Keywords:
Allee effects
Chaos
Extinction transients
Scaling laws
Single species dynamics
Theoretical ecology
Topological entropy
abstract
Density-dependent effects, both positive or negative, can have an important impact on the
population dynamics of species by modifying their population per-capita growth rates. An
important type of such density-dependent factors is given by the so-called Allee effects,
widely studied in theoretical and field population biology. In this study, we analyze two
discrete single population models with overcompensating density-dependence and Allee
effects due to predator saturation and mating limitation using symbolic dynamics theory.
We focus on the scenarios of persistence and bistability, in which the species dynamics
can be chaotic. For the chaotic regimes, we compute the topological entropy as well as
the Lyapunov exponent under ecological key parameters and different initial conditions.
We also provide co-dimension two bifurcation diagrams for both systems computing the
periods of the orbits, also characterizing the period-ordering routes toward the boundary
crisis responsible for species extinction via transient chaos. Our results show that the
topological entropy increases as we approach to the parametric regions involving transient
chaos, being maximum when the full shift R(L)∞ occurs, and the system enters into the
essential extinction regime. Finally, we characterize analytically, using a complex variable
approach, and numerically the inverse square-root scaling law arising in the vicinity of a
saddle–node bifurcation responsible for the extinction scenario in the two studied models.
The results are discussed in the context of species fragility under differential Allee effects.
© 2011 Elsevier Ltd. All rights reserved.
1. Introduction
The presence of deterministic fluctuations in biological systems has been widely studied over the last decades, especially
in the field of theoretical ecology. Time-continuous predator–prey dynamical systems have shown periodic or quasiperiodic
dynamics [1,2], as well as chaos when three or more interacting species are considered [3–5]. Interestingly, some other
theoretical studies also showed potential chaos in this type of models considering ecologically meaningful parameters based
on empirical allometric scaling relationships [6,7]. Equivalent results showing chaotic dynamics were found in discrete
theoretical models for single population dynamics [8], metapopulations [9] or high-dimensional ecosystems [10]. More
recently, a real full trophic web was studied in laboratory experiments, with the dynamics of the populations suggesting the
presence of deterministic chaos [11]. Laboratory experiments with insect populations also hinted chaotic fluctuations to be
possible in population dynamics [12,13].
∗
∗∗
Corresponding author at: ISEL - Engineering Superior Institute of Lisbon, Department of Mathematics, Lisboa, Portugal.
Corresponding author.
E-mail addresses: jduarte@deq.isel.pt (J. Duarte), josep.sardanescayuela@gladstone.ucsf.edu (J. Sardanyés).
1 Current affiliation: Gladstone Institute of Virology and Immunology, University of California SF, San Francisco, CA, USA.
1468-1218/$ – see front matter © 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.nonrwa.2011.11.022
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J. Duarte et al. / Nonlinear Analysis: Real World Applications 13 (2012) 1647–1661
Species fluctuations can have a negative impact on populations survival under the so-called Allee effect, as there may exist
a critical population threshold which, if overcome, could drive species to extinction. Generically, the Allee effect describes
a positive correlation between any measure of species fitness and population numbers [14,15]. The Allee effect occurs
when positive density-dependence dominates at low population densities. Positive density-dependent factors have been
suggested to occur in populations with cooperative hunting, conspecific enhancement of reproduction, predator saturation
and increased availability of mates [16–18,15], among others. For the case of predator saturation, the Allee effect occurs
in species predated by a predator with a saturating functional response. Here, the risk of predation for a given individual
decreases as the population density increases. Examples of Allee effects due to predator saturation have been found in field
studies on Quercus robur, in which the per-capita rate of acorn loss due to invertebrate seed predators was greatest amongst
low acorn crops and lower on large acorn crops [19]. Moreover, Williams and collaborators [20] showed that some birds
could consume almost the whole population of adult cicadas in their summer decline, but their consumption was lower for
much larger population densities. In addition, some other studies about predation on bird eggs and nestlings [21] as well as
predation on fish species [22], showed that increasing prey population size minimizes vulnerability due to predation.
The Allee effect due to mating limitation may also be important in sexually reproducing populations, where finding mates
becomes difficult at low densities. This phenomenon has been shown to be relevant in the pollinization of plants by animal
vectors [23] as well as in the fertilization of gametes in benthic invertebrates [24,25]. For this latter case, the fertilization by
free spawning gametes for the species Strongylocentrotus franciscanus can become insufficient at low densities. However, a
fertilization of 82.2% was found for larger colonies.
The dynamics associated to the previously described Allee effects can be studied for single populations with nonoverlapping generations using discrete models, represented by the general difference equation
Nn+1 = Nn f (Nn ).
(1)
Here, Nn is the population density at generation n, and f (Nn ) corresponds to the per-capita growth rate of the population.
For the sake of clarity, we will briefly explain the dynamics of such models, studied in detail by Schreiber [26]. In these
models, very large population densities at time n can give place to very small population numbers at time n + 1. Moreover,
there is a population density at time n, namely C , that leads to the maximum population density at n + 1. Two different
dynamical regimes can be found for such systems: (i) persistence and (ii) extinction. Scenario (i) occurs if f (0) > 1 and
f (N ) > 0, ∀N > 0. Under such conditions, the population will always persist for all positive densities. Scenario (ii), in
which the population becomes extinct independently of the initial condition, occurs if f (N ) < 1, ∀N.
The previous models can undergo an Allee effect when the per-capita growth rate increases at low population numbers.
There exists a critical equilibrium density, named A in [26], such that if f (N ) < 1 for population densities below A, or
f (N ) > 1 for some population numbers greater than A, a so-called strong Allee effect takes place. If the population density
falls at some time below this critical threshold extinction takes place. It is known that under the strong Allee effect, two
other dynamical scenarios are possible (see [26]): (iii) bistability and (iv) essential extinction. In the bistability scenario, the
population will persist or will disappear depending on the initial densities. In scenario (iv) extinction occurs for almost every
initial population density (i.e., for a randomly chosen initial condition extinction occurs with probability one).
An interesting phenomenon associated to scenario (iv) is given by transient chaos, which asymptotically drives the
populations to extinction via a transitory chaotic dynamics [27,26]. Generally talking, transient chaos is tied to the concept
of chaotic saddles [28–30]. Chaotic saddles are bounded sets with fractal structure in both stable and unstable directions.
The fractality in the unstable direction implies the existence of an infinite number of gaps of all sizes along its unstable
manifold [31]. Such topological structures have been identified in predator–prey continuous dynamical systems [32,7,31].
The mechanisms generating transient chaos can be different. For instance, chaotic transients can arise due to the
collision of limit sets as occurs in the unstable–unstable pair bifurcations or in riddling bifurcations, responsible for
superpersistent chaotic transients. In the first type of bifurcation, an unstable periodic orbit of a chaotic attractor collides
with another unstable periodic orbit on the separatrix, giving place to the so-called chaotic crisis [33,34]. Crises have been
described in discrete [35,36] and continuous-time predator–prey models [7,31]. More recently, such a phenomenon has
been investigated in the context of cooperative hunting [37]. Riddling and riddled basins [30], which have been described
in experiments with coupled electronic circuits [38,39], occur when a fixed point loses its transverse stability as a control
parameter crosses its critical value, colliding with two repellers, which are located symmetrically to an invariant subspace.
This type of saddle–repeller pitchfork bifurcation causes the appearance of a tongue near the invariant subspace responsible
for superpersistent chaotic transients [40]. The chaotic transients found in the models studied in this work are associated to
the collision of the basin of attraction of the attractor corresponding to population persistence with the basin of attraction
of the origin [26]. After such a collision, named boundary crisis, the population becomes extinct via transient chaos.
In this study we will analyze with symbolic dynamics theory two discrete ecological models considering Allee effects
due to predator saturation and mating limitation (see [26] for more details of the studied models). By doing so, we can
characterize new measures such as the topological entropy for the chaotic dynamics and study their changes toward the
boundary crisis found in such models. As we will explain later on, we also provide a general mathematical framework to
calculate the time-to-extinction for any mathematical model represented by a quadratic one-dimensional map undergoing
a saddle–node bifurcation here illustrated with the two studied ecological models. The models investigated in our study can
be represented with the following general difference equation:
Nn+1 = Nn er (1−Nn /K ) β(Nn ),
(2)
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where er (1−Nn /K ) is a negative density-dependent factor and the term β(Nn ) is used to introduce a positive density-dependent
factor. Here r is the intrinsic growth rate and K the carrying capacity.
The manuscript is organized as follows. In Section 2 we compute the Lyapunov exponent as well as the topological
entropy for the model with predation saturation. These two measures are used to investigate the chaotic dynamics
associated to persistent and transient chaos. In Section 3 we apply the same measures for the model with mating limitation.
For both models we characterize co-dimension two bifurcation diagrams for key ecological parameters, also showing the
dependence of the asymptotic dynamics on the initial condition, characterized in one-dimensional parameter space. In
Section 4 we study, both analytically and numerically, the dynamical properties of extinction transients in the context of
delayed transitions for both models. Using a complex variable approach, we derive the inverse square-root scaling law tied
to saddle–node bifurcations giving place to the extinction scenario.
2. Model with predator saturation
In this section we study the Allee effects due to predator saturation. To do so, we use β(N ) = e−m/(1+sN ) (see Eq. (2)).
Here β(N ) corresponds to the probability of escaping predation by a predator with a saturating functional response, being m
the intensity of predation and s a parameter proportional to the handling time [41]. By non-dimensionalizing Eq. (2), taking
xn = Nn K −1 , we obtain the model under study, given by

xn+1 = xn exp r (1 − xn ) −
m
1 + sKxn

.
(3)
Focusing on the dynamics associated to scenarios (i) persistence and (iii) bistability, we first study stability properties of
the previous model by means of numerical evaluation of the Lyapunov exponent. The Lyapunov exponent, λ̂, for a point x0
is obtained from
λ̂(x0 ) = lim sup
n→+∞
n −1
1
n j =0
ln |f ′ (xj )|,
(4)
where xj = f j (x0 ). If the absolute value of f ′ (xj ) is greater than one, the Lyapunov exponent is positive, and then the system
is chaotic having sensitive dependence on the initial conditions. For our system we have
f ′ (x) = e
r (1−x)− 1+msKx


1+x
msK
(1 + sKx)2

−r
.
In Fig. 1 we display several bifurcation diagrams showing the dynamics, together with the corresponding Lyapunov
exponent for the chaotic semistability scenario. In all the studied parameters, the system undergoes the period-doubling
route to chaos, with a first flip bifurcation involving periodic dynamics that further evolves by the period-doubling cascade
leading to the Feigenbaum scenario. Under the selected parameter values, the increase in the handling time (i.e., increase
in sK ) involves the emergence of chaos, but at some critical threshold (indicated with the dashed red line) the population
collapses, entering into the essential extinction (iv) scenario. The positive Lyapunov exponent is shown to increase toward
the critical parameter value involving scenario (iv). Fig. 1(b) shows that an increase in the intensity of predation, m, stabilizes
the dynamics, although for large values of m the population becomes extinct due to the entry into the scenario of extinction
(ii). As shown in Fig. 2(d), where the unimodal map is below the diagonal, this is due to a saddle–node bifurcation. The
stabilization of the dynamics at increasing m is not a continuous process since, for some parameter values, the system falls
into the scenario of essential extinction (iv), where the population has overcome the Allee effect threshold and becomes
extinct via transient chaos (see Figs. 1(b) and 2(b)). Finally, the increase in the growth rate, r, displays a similar behavior as
the increase in the handling time, that is, after a saddle–node bifurcation the population reaches a nontrivial steady state
and after a flip bifurcation, the population achieve chaos via period-doubling cascade, and then finally collapses entering
into scenario (iv).
The investigation of one-dimensional basins of attraction may be useful to further understand the role of key parameters
together with the initial population numbers. Fig. 3 shows the asymptotic dynamics for several initial conditions (we use
0 < x0 ≤ 2) at increasing predator’s handling time, sK (Fig. 3(1.a)), and predation intensity, m (Fig. 3(1.b)). We here
differentiate between chaos (blue), order (i.e., not chaotic, shown in red) and extinction (gray) in population dynamics.
Chaotic dynamics were identified computing the Lyapunov exponent (see Fig. 3(2.a) and (2.b)). These analyses show that
for small values of sK the population is in the extinction (ii) scenario, but increasing handling time involves survival of
the populations which enter into the bistability scenario. Further increase in the handling time involves the appearance of
chaotic dynamics, and the initial population numbers driving the system to extinction is reduced. As previously mentioned,
very large values of sK cause the extinction for all possible values of the initial population, now in the essential extinction
scenario. The basins of attraction for predator’s intensity, m, are different. Here, for small values of m the populations persist
chaotically, and after an increase of m, the populations enter into the essential extinction scenario where transient chaos
can occur. Higher predation intensity drives the population to the bistability scenario, which is initially chaotic and then
is stabilized at growing m (see Fig. 3(1.b)). The increase of m in the bistability scenario involves an increase in the initial
1650
a
b
c
Fig. 1. Bifurcation diagrams for Eq. (3). In (a) r = 4.5 and m = 8. In (b) r = 4.5 and sK = 16. In (c) m = 8 and sK = 16. The diagrams are obtained using
random initial conditions x0 ∈ [0, 1], plotting x100 for each parameter value. The dashed red lines indicate the parameter values causing the boundary
crisis. The black arrows indicate the parameter regions where the trajectories reach the stable equilibrium x∗ = 0, for any arbitrary value of x0 . We also
show the Lyapunov exponent, λ̂, computed (using x0 = 0.4) for the following parameter ranges: (a) 4.85 ≤ sK ≤ 12.25, (b) 10.3 ≤ m ≤ 18 and (c)
3.25 ≤ r ≤ 4.25. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
a
b
c
d
Fig. 2. Cobweb maps representing the dynamics for the four different scenarios shown in Fig. 1(b), using r = 4.5 and sK = 16. In (a) the system falls in
the (i) persistence scenario, with m = 3 and x0 = c = 0.3144. In (b) we show the dynamics for the (iv) essential extinctions where transient chaos can
occur using m = 10.2913 and x0 = 0.925. In (c) we show the map for the (iii) chaotic semistability scenario with m = 12 and x0 = c = 0.4898. Finally, in
(d) we show the (ii) extinction scenario which occurs after a saddle–node bifurcation, here with m = 20.5 and x0 = c = 0.6075. The arrow indicates the
bottleneck region of the ghost causing the delayed transition.
population numbers with asymptotic extinctions. After a critical value of m ≈ 20, the saddle–node bifurcation occurs and
the system also enters into the extinction scenario (ii).
In the following lines we will characterize co-dimension two bifurcation diagrams computing the periods of the orbits
in the survival regimes using symbolic dynamics theory, either in the scenarios of (i) persistence and (iii) bistability. For
the chaotic semistability behavior we will also compute the topological entropy using this theory. For the sake of clarity,
we will explain in detail the steps followed to compute these measures using the theory of unimodal maps [42–44]. Let us
consider an iterated map f on the interval I = [a, b], which is a 2 -piecewise monotonic map with one turning point c. Thus
1651
.
^
λ
(iv) essential extinction
chaos
sK
order
(iii) bistability
(2.a)
(1.a)
0.693
0.00
^
x0
λ
(ii) extinction
–0.75
sK
x0
^
(2.b)
(1.b)
x0
(ii) extinction
order
λ
0.670
(iii) bistability
m
0.00
^
chaos
λ
(i) persistence
–0.75
x0
m
Fig. 3. (1) Basins of attraction for different initial conditions (x0 ) computed in one-dimensional parameter space using sK (with m = 8) (a); and m (with
sK = 16) (b), both with r = 4.5. Extinction domains are indicated in gray color. The initial conditions involving chaotic and ordered persistence are
indicated in blue and red, respectively. The dotted line indicates the parametric boundaries after which sustained chaos switches to transient chaos. The
dashed line separates the scenarios of (iii) bistability from (ii) extinction. (2) Computation of the Lyapunov exponent, λ̂, for magnified regions of the basins
of attraction shown in (1). The transparent planes intersect at the value λ̂ = 0, where bifurcations occur. (For interpretation of the references to colour in
this figure legend, the reader is referred to the web version of this article.)
I is subdivided into the following sets:
IL = [a, c [,
IC ∗ = {c },
IR = ]c , b],
in such way that the restriction of f to interval IL is strictly increasing and the restriction of f to interval IR is decreasing.
Each of such maximal intervals on which the function f is monotone is called a lap of f , and the number ℓ = ℓ (f ) of
distinct laps
of f . Starting with the critical point of f , c (relative extremum), we obtain the orbit
 is called the lap number

O(c ) = xi : xi = f i (c ), i ∈ N . With the purpose of studying the topological properties, we associate to the orbit O(c ) a
sequence of symbols, which leads to the itinerary (i(x))j = S = S1 S2 · · · Sj · · · , where Sj ∈ A = {L, C ∗ , R} and
Sj = L
Sj = C ∗
if f j (x) < c ,
if f j (x) = c ,
Sj = R if f j (x) > c .
The turning point c plays an important role, since the dynamics of the interval is characterized by the symbolic sequence
associated to the critical point orbit. When O(c ) is a k-periodic orbit, we obtain a sequence of symbols that can be
characterized by a block of length k, the kneading sequence S (k) = S1 S2 · · · Sk−1 C ∗ . The concept of kneading increments
and kneading-matrix was introduced by Milnor and Thurston [42]. These are power series that measure the discontinuity
evaluated at the turning points ci , i = 1, 2, . . . , m, of m-modal maps. For the case of unimodal maps we have one kneadingincrement defined by ν(t ) = θc + (t ) − θc − (t ), where θx (t ) is the invariant coordinate of the sequence S0 S1 S2 · · · Sj · · ·
associated to the itinerary of point x. The invariant coordinate is defined by θx (t ) =
for j > 0, τ0 = 1 for j = 0,

−1
ε(Si ) = 0
1
if Si = R
if Si = C ∗
if Si = L,
and θc ± (t ) = limx→c ± θx (t ).
∞
j =0
τj t j Sj , where τj =
J −1
i=0
ε(Si )
1652
Taking as an example the kneading sequence RLLLLLLRRLLC obtained for m = 4 and sK = 9.365 (with r = 4.5) with
period 12, we will illustrate the computation of the topological entropy. The symbolic sequences that correspond to the orbits
of the points c + and c − , are c + −→ R(RLLLLLLRRLLR)∞ and c − −→ L(RLLLLLLRRLLR)∞ . Note that the block RLLLLLLRRLLR
corresponds to the sequence RLLLLLLRRLLC where the symbol C is replaced by R because the parity of the block RLLLLLLRRLL
is odd. In fact, according to the kneading theory, the shared block of the orbits of the points c + and c − , RLLLLLLRRLLC , must
be even, that is, the symbol C is replaced by R in order to have an even number of R symbols. The invariant coordinates are
θc + (t ) = R − Rt + Lt 2 + Lt 3 + Lt 4 + Lt 5 + Lt 6 + Lt 7 + Rt 8 − Rt 9 + Lt 10 + Lt 11 + Rt 12 − Rt 13 + Lt 14 + · · ·


= R − t R − Lt − Lt 2 − Lt 3 − Lt 4 − Lt 5 − Lt 6 − Rt 7 + Rt 8 − Lt 9 − Lt 10 − Rt 11
− t 13 (R − Lt − · · · − Rt 11 ) − · · ·


= R − t R − Lt − Lt 2 − Lt 3 − Lt 4 − Lt 5 − Lt 6 − Rt 7 + Rt 8 − Lt 9 − Lt 10 − Rt 11 (1 + t 12 + t 24 + · · ·)

 

= R − t R − Lt − Lt 2 − Lt 3 − Lt 4 − Lt 5 − Lt 6 − Rt 7 + Rt 8 − Lt 9 − Lt 10 − Rt 11 / 1 − t 12
and
θc − (t ) = L + Rt − Lt 2 − Lt 3 − Lt 4 − Lt 5 − Lt 6 − Lt 7 − Rt 8 + Rt 9 − Lt 10 − Lt 11 − Rt 12 + Rt 13 − Lt 14 − · · ·


= L + t R − Lt − Lt 2 − Lt 3 − Lt 4 − Lt 5 − Lt 6 − Rt 7 + Rt 8 − Lt 9 − Lt 10 − Rt 11
+ t 13 (R − Lt − · · · − Rt 11 ) + · · ·


= L + t R − Lt − Lt 2 − Lt 3 − Lt 4 − Lt 5 − Lt 6 − Rt 7 + Rt 8 − Lt 9 − Lt 10 − Rt 11 (1 + t 12 + t 24 + · · ·)

 

= L + t R − Lt − Lt 2 − Lt 3 − Lt 4 − Lt 5 − Lt 6 − Rt 7 + Rt 8 − Lt 9 − Lt 10 − Rt 11 / 1 − t 12 .
Separating the terms associated to the symbols L and R, we obtain
ν(t ) = N11 (t )L + N12 (t )R.
In our case, the kneading increment is given by
ν(t ) = θc + (t ) − θc − (t )

 

= R − t R − Lt − Lt 2 − Lt 3 − Lt 4 − Lt 5 − Lt 6 − Rt 7 + Rt 8 − Lt 9 / 1 − t 12


 

− L + t R − Lt − Lt 2 − Lt 3 − Lt 4 − Lt 5 − Lt 6 − Rt 7 + Rt 8 − Lt 9 / 1 − t 12




−2t + 2t 8 − 2t 9 + 2t 12
2t 2 + 2t 3 + 2t 4 + 2t 5 + 2t 6 + 2t 7 + 2t 10 + 2t 11
L+ 1+
R.
= −1 +
1 − t 12
1 − t 12
The general form of the kneading-matrix, N (t ), can be expressed as follows
N (t ) = N11 (t )
N12 (t ) .


For the system under study we have

 −1 +
2t 2 + 2t 3 + 2t 4 + 2t 5 + 2t 6 + 2t 7 + 2t 10 + 2t 11
1 − t 12
−2t + 2t 8 − 2t 9 + 2t 12
1+
1 − t 12
The corresponding kneading determinant, D(t ), is
[N (t )]T = 

D(t ) =
D1 (t )
1 − ε(L)t
=−
D2 (t )
1 − ε(R)t
=
D1 (t )
1−t
=−
D2 (t )
1+t
T

 .

,
where D1 (t ) = N12 (t ) and D2 (t ) = N11 (t ). Again, for our model we obtain
D(t ) =
1 − 2t + 2t 8 − 2t 9 + t 12


(1 − t ) 1 − t 12
.
The topological entropy of f , denoted by htop (f ), is given by htop (f ) = log s, where s is the growth number of a unimodal
interval map f , and is given by s = 1/t ∗ , being t ∗ the root of D(t ), which has the lowest modulus. Therefore t ∗ =
0.502140 . . . , and the topological entropy is given by
htop (f ) = log

1
t∗

= 0.688874 . . . .
Some of the kneading sequences and the topological entropy for our system, obtained as described above, are shown in the
following table for small periods (n ≤ 5):
a
b
20
3
20
5
3
5
5
1653
5
4
15
15
4
2
2
sK
10
5
1
5
sK
10
1
4
4
5
5
(ii) extinction
(ii) extinction
0
0
m
m
5
c
5
3
d
20
5
5
4
ss
2
sK
(iv
)e
)e
10
2
en
4
(iv
sK
4
le
15
tia
sse
nti
xt
in
nc
ex
ti
5
al
15
5
cti
tio
on
n
20
3
10
1
1
5
5
(ii) extinction
0
(ii) extinction
4
0
m
0
5
10
15
20
m
Fig. 4. Periods (in solid black lines) of the dynamics in the parameter space (m, sK ) using: (a) r = 3.8269, (b) r = 4, (c) r = 4.5 and (d) r = 4.8. We
also indicate the regions with asymptotic extinctions given by scenarios (ii) extinction and (iv) essential extinctions. The periods are found in the parametric
regions delimited by (i) persistence and (iii) bistability. The number of the periods are indicated on each period line inside a black circle. For all the diagrams,
the kneading sequences (at decreasing m) are as follows: period one C , period two (RC )∞ , period four (RLRC )∞ , period five (RLRRC )∞ , period three (RLC )∞ ,
period five (RLLRC )∞ , period four (RLLC )∞ and period five (RLLLC )∞ .
Kneading
sequences
C
RC
RLRC
RLRRC
RLC
RLLRC
RLLC
RLLLC
Topological
entropy
0
0
0
0.414013. . .
0.481212. . .
0.543535. . .
0.609378. . .
0.656256. . .
As we mentioned before, we also use the previous steps to first characterize co-dimension two bifurcation diagrams, which
allow us to distinguish the scenarios where species persist, showing the periods of the orbits, as well as to identify the
parametric regions involving extinction of the populations. We specifically study the dynamics in the parameter space
(m, sK ), using four different and increasing values of the intrinsic growth rate of the population, r (see Fig. 4). We show that
the region of extinction tied to scenario (ii), which is delimited by the period-1 line, is reduced at increasing r. Moreover,
increasing r enlarges the region of essential extinctions (iv), where transient chaos can occur. The enlargement of the region
for scenario (iv) involves that the region where the periods appear is stretched and thus the change of period is found in
more narrow regions of the parameter values.
1654
a
0.6
htop
h top
0.4
0.2
0.
0
5
10
15
20
SsK
k
b
0.6
0.4
htop
h top
0.2
0.
0
5
10
15
m
Fig. 5. Topological entropy, htop , computed at increasing sK (a) and m (b) using x0 = c and r = 4.5. In (a) we use ( from left to right): m = 4, m = 8
and m = 12. In (b) we show, with black points, the values of htop corresponding to sK = 4 (left) and sK = 8 (right). The arrows indicate the values of
sk where the full shift R(L)∞ takes place, and where transient chaos can appear. For this case, the topological entropy remains at the maximum value at
increasing sK values (results not shown). In red, we show the values of htop considering the phases of (i) persistence, (iv) essential extinctions and (iii)
chaotic semistability, using sK = 12 (see Fig. 1(b)). Note that in scenario (iv), the topological entropy is maximum, also corresponding to the full shift
R(L)∞ , where the asymptotic dynamics is the extinction of the population. (For interpretation of the references to colour in this figure legend, the reader
is referred to the web version of this article.)
The topological entropy, htop , is computed using as control parameters sK (Fig. 5(a)) and m (Fig. 5(b)). In agreement with
Figs. 1(a) and 3(a), the topological entropy is positive once the dynamics falls into the chaotic semistability (scenario (iii)),
and, as we show, the chaotic region starts at higher values of sK as m increases, that is, as predation intensity increases,
chaos appears for longer handling times. This effect can be interpreted in another way: an increase in predation intensity
stabilizes the dynamics. For the three values of m studied in Fig. 5(a), we show that the topological entropy increases at
growing sK , being maximum once the population dynamics enters into the essential extinction regime, which takes place
when the full shift (RL)∞ occurs and the topological entropy is maximum.
The values of the topological entropy at increasing predation intensity, m, are similar. For the example with sK = 12,
indicated in red in Fig. 5(b), we show that for very small values of m (corresponding to the persistence scenario (i)), the
topological entropy is positive because the dynamics is chaotic (equivalently to the bifurcation diagram shown in Fig. 1(b)).
Once parameter m reaches the critical value ( for this case m ≈ 4) the topological entropy achieves the full shift (RL)∞ ,
being maximum, and then decreases again once the system enters again into the bistability region. A further increase of m
diminishes the topological entropy as the system undergoes the inverse period-doubling bifurcations.
3. Model with mating limitation
In this section we further analyze the model with Allee effects due to mating limitation studied in [26]. For this case
β(Nn ) = α N /(α N + 1) (see Eq. (2)). Here β(Nn ) is the probability of finding a mate and α is the searching efficiency for an
individual. We also non-dimensionalize the system by setting xn = Nn /K , thus obtaining the model that we will explore,
given by
xn+1 = xn exp (r (1 − xn ))
α Kxn
1 + α Kxn
.
(5)
We also computed the Lyapunov exponent using Eq. (4), now with
f ′ (x) =
xα Ker (1−x)
1 + xα K

2−
xα K
1 + xα K

− rx .
Similarly to the previous model studied in Section 2, the system undergoes the period-doubling cascade at increasing
both the individual searching efficiency, α K , and the intrinsic growth rate, r (see Fig. 6). For both cases, at low values of
1655
a
x
b
x
Fig. 6. Same as in Fig. 1 for Eq. (5) with (a) r = 4.5 and (b) α K = 2. For the Lyapunov exponent diagrams we explore the ranges: (a) 0.15 ≤ α K ≤ 1.42
and (b) 2.5 ≤ r ≤ 4.25, using the initial condition x0 = 0.4.
a
b
c
Fig. 7. Same as in Fig. 2 now showing three cases of the dynamics of the model with mating limitation. In (a) we display the cobweb map for the (ii)
extinction scenario, with r = 2, α K = 1.03 and x0 = c = 0.7776 (here the arrow also indicates the slow passage of the iterations through the bottleneck
region of the ghost). In (b) the system is in the (iii) chaotic semistability scenario with r = 4.5, α K = 1 and x0 = c = 0.3829. Finally, we show in (c) the
(iv) essential extinction scenario with transient chaos, using r = 4.5 and α K = 1.2 and x0 = 1.4.
the parameters, the system is at the (ii) extinction scenario, and the increase of the parameters involves the saddle–node
bifurcation (see Fig. 7(a)), where the population can persist in the bistability scenario (iii), via fixed point, periodic or
quasi-periodic and chaotic dynamics (see Fig. 7(b)). After the critical parameter values, the system enters into the essential
extinction scenario (here also indicated with the dashed red lines), where transient chaos occurs (see Fig. 7(c)).
As we already did for the previous model, we characterize one-dimensional basins of attraction at increasing the
searching efficiency (see Fig. 8). For this case, small values of α K involve extinction independently of the initial condition,
thus corresponding to scenario (ii) essential extinction. As the searching efficiency increases, and because of the appearance
of the fixed point allowing persistence due to the saddle–node bifurcation, the dynamics falls into the bistability scenario
(iii). The increase of the searching efficiency in this region has two effects: the basin of attraction involving extinction is
reduced and the dynamics in the survival phase becomes chaotic. Once a critical threshold in α K is overcome (α K ≈ 14.1
in the analysis shown in Fig. 8), the populations enter into the essential extinction regime, and populations become extinct
via transient chaos (as shown in Fig. 7(c)).
Co-dimension two bifurcation diagrams are also investigated in the parameter space (r , α K ) (see Fig. 9). We show
that the periods of the orbits found for intermediate values of the searching efficiency, and as the intrinsic growth rate
increases, follow the next sequence: period-1, (C )∞ ; period-2, (RC )∞ ; period-4, (RLRC )∞ ; period-5, (RLRRC )∞ ; period-3,
(RLC )∞ ; period-5, (RLLRC )∞ ; period-4, (RLLC )∞ ; and period-5, (RLLLC )∞ . Note that the entrance to the essential extinction
corresponds to the chaotic dynamics with period 5. In Fig. 9 we show the values of the topological entropy, htop , computed
1656
.
^
λ
2.0
0.6635
1.5
0.5
αK
(iii) bistability
chaos
1.0
0.5
order
0.0
0.0
0.5
1.0
1.5
2.0
0.5
1.5
1.0
x0
(ii) extinction
0.0
0.0
0.00
–0.5
^
λ
1.0
1.5
0.5
2.0
αK
–0.75
0.0
2.0
x0
Fig. 8. Same as in Fig. 3 for α K using r = 4.5.
5
3
4
5
a
b
5
0.6
4
5
(iv) essential
extinction
4
3
2
αK
hto p
h top
2
0.4
0.2
1
1
0.
0
0
(ii) extinction
1
2
3
4
5
1
6
2
3
α Kk
r
Fig. 9. (a) Period boundaries for the model with mating limitation given by Eq. (5) using as control parameters r and α K . The line in between period
1 and period 2 corresponds to the particular dynamics f (c ) = c. (b) Topological entropy, htop , at increasing α K , with x0 = c and ( from left to right):
r = 4, r = 4.4 and r = 4.8. Note that here the three curves also achieve the (iv) essential extinction scenario (see Fig. 8) reaching the maximum value of
topological entropy, corresponding to the full shift R(L)∞ (indicated with the arrows), which is also found at increasing values of α K (results not shown).
using the searching efficiency as control parameters for three values of the growth rate. These results are in agreement with
the previous figures, which showed that the increase of the searching efficiency involves the period-doubling bifurcation
scenario and the entrance into the chaotic semistability regime, which has positive topological entropy. As we found in
the previous model, the topological entropy increases as we approach to the critical value of the parameter involving the
collision of the basins of attraction of the persistence attractor and the origin. Once the full shift (RL)∞ , where the topological
entropy is maximum, occurs, the population goes to extinction via transient chaos.
4. Delayed extinctions: dynamics near a saddle–node bifurcation
In this section we study in detail the extinction scenario (ii) for the system with predator saturation and for the
system with mating limitation. More precisely, we analytically derive the inverse square-root scaling law arising near the
saddle–node bifurcation by using complex variable techniques. The same approach was recently applied to study the scaling
properties for saddle–node bifurcations in a general one-dimensional quadratic map [45]. For the sake of clarity, we will
summarize the calculations in the following lines.
For a discrete dynamical system, the evolution of an iterate xn can be represented by the formula
xn+1 = f (xn ).
As pointed out in [45], the equation can also be viewed as a difference equation
xn+1 − xn = f (xn ) − xn .
Hence, defining g (x) ≡ f (x) − x, gives
xn+1 − xn = g (xn ) × 1,
1657
which can be read ‘‘as n changes by 1 unit, x changes by g (x)’’. In this section we are going to consider firstly the model with
predator saturation to study the scaling properties for the saddle–node bifurcation, analyzing

xn+1 = f (xn , m) = xn exp r (1 − xn ) −

m
1 + sK xn
.
(6)
Therefore, defining
g (x, m) = f (x, m) − x,
(7)
we obtain xn+1 − xn = g (xn , m), with
g ( xn , m ) = xn


exp r (1 − xn ) −
m

1 + sKxn

−1 .
The equilibrium points occur when g (xn , m) = 0. Besides the trivial solution x∗n = 0, we have two equilibrium points,
∗
xn =
sK − 1

±
r 2 (1 + 2sK + (sK )2 ) − 4rsKm
,
2sK
2rsK
which are central in the study of the saddle–node bifurcation. We have the following general scenarios:
1. For r 2 (1 + 2sK + (sK )2 ) − 4rsKm > 0, that is, m < mc , with mc = r 2 1 + 2sK + (sK )2 /(4rsK ), there are two real
equilibrium points, a stable and an unstable one.
2. For m = mc the saddle–node bifurcation occurs. We denote this bifurcation value of the parameter by mc .
3. For m > mc , there are two complex equilibrium points.


When m is extremely close to mc (m > mc ), many iterations of the map are required to pass through the bifurcation
point. The reason for this local delay is the presence of a bottlenecking manifold, which is also named ghost [46]. The scaling
law measures the number of iterates required to pass through xc = (sK − 1)/(2sK ), as a function of the difference of the
parameter m to its bifurcation value mc .
The saddle–node bifurcation for the map displayed in Fig. 2(d) takes place at x = xc when m = mc . The function
(7) is analytic with respect
to x in the neighborhood of xc . Sufficient conditions for such a bifurcation, when (xc , mc ) =


sK −1
2sK
r 2 1+2sK +(sK )2 )
, ( 4rsK
, are
g (xc , mc ) = 0,
Dm g (xc , mc ) ̸= 0,
(8)
and
Dx g (xc , mc ) = 0,
D2x g (xc , mc ) ̸= 0.
(9)
Indeed,
g ( xn , m ) = xn


exp r (1 − xn ) −
m

1 + sKxn

−1
⇒ g (xc , mc ) = 0;
Dm g (xn , m) = −

xn
exp r (1 − xn ) −
m

1 + sKxn
1 + sKxn
⇒ Dm g (xc , mc ) = −0.05514 . . . ̸= 0;

 


msK
m
Dx g (xn , m) = exp r (1 − xn ) −
x n −r +
+
1
−1
1 + sKxn
(1 + sKxn )2
⇒ Dx g (xc , mc ) = 0;
D2x g (xc , mc ) = −7.94117 . . . ̸= 0.
In the following lines we consider a = Dm g (xc , mc ) and b = D2x g (xc , mc )/2!. As previously mentioned, the equilibrium
points are the solutions of g (xn , m) = 0. Condition (8) allows us to apply the implicit function theorem and obtain that,
locally, the zeros of g close to (xc , mc ) are the points of the graph of an analytic function m = h(xn ), such that h(xc ) = mc .
D g (x ,m)
Differentiating implicitly g (xn , h(xn )) = 0, we obtain Dx g (xn , m) + Dm g (xn , m)Dh(xn ) = 0 and Dh(xn ) = − D x g (xn ,m) .
m
D g (x ,m )
n
For (xn , m) = (xc , mc ), we have Dh(xc ) = − D x g (xc ,mc ) . Since Dx g (xc , mc ) = 0, we obtain Dh(xc ) = 0. Therefore, we
m
c
c
must proceed by computing the second derivative for the purposes of constructing the function h. From Dx [Dx g (xn , m) +
D 2 g ( x ,m )
Dm g (xn , m)Dh(xn )] = 0, and since Dh(xc ) = 0, we have D2x h(xc ) = − D x g (xc ,mc ) . Using a Taylor expansion to construct
m
c
c
m = h(x), we obtain
m = h(x) = h(xc ) +
m = mc −
b
a
D2x h(xc )
2!
(x − xc )2 + · · · ,
(x − xc )2 + O(x − xc ).
and thus,
1658
With the purpose of having the zeros in terms of m, we have to invert the previous expression. Except for m = mc , we get
the following solutions
x = xc ±

a
− (m − mc ) + O(m − mc ).
b
For m < mc , we have two real solutions
1
 a
2
xj (m) = xc + − (m − mc ) eiπ j + O(m − mc ),
b
√
−1. For m > mc , we obtain two complex solutions
a
 21  2j+1 
i
π
xj (m) = xc +
( m − mc ) e 2
+ O(m − mc ),
with j = 0, 1 and i =
b
also with j = 0, 1.
As we already mentioned, our goal is to estimate the number of iterates needed to go from x0 to xn in terms of m(>mc ).
Let us consider the equation presented at the beginning of this section, now expressed as
xn+1 − xn = g (xn , m) × 1,
(10)
meaning that as n changes by 1 unit, x changes g (xn , m). Thus, the number of iterates required to go from x0 to xn can be
represented by
N =
n

xi+1 − xi
i=0
g (xi , m)
.
(11)
Let us consider again expression (10) and make 1x = xn+1 − xn . As a consequence of the application of the implicit theorem
function, we notice that g (xn , m) = 0 ⇒ 1x → 0, thus, 1x gives place to dx. Therefore, in order to compute an approximate
value of the sum (11), we consider the path integral

dx
g (x, m)
γ
= 2π i
p−1


Res
1
g
j=0

, xj (m) ,
(12)
where xj , with 0 ≤ j ≤ p − 1, 
are the poles
 in the upper-half plane inside the path γ . In our case, we only consider x0 (m).
Therefore, γ g (xdx,m) = 2π i Res
, xj (m) , with j = 0 (see [47]). The computation of the residue is given by

1
1
, xj (m) = lim (x − xj (m))
.
Res
x→xj (m)
g
g (x, m)
1
g


Since g (xj (m), m) = 0, we can write

Res
1
g
x − xj (m)

, xj (m) = lim
g (x, m) − g (xj (m), m)
x→xj (m)
,
which leads to

Res
1
g

, xj (m) =
1
Dx g (xj (m), m)
.
Taking j = 0, the integral (12) is given by

dx
γ
g (x, m)
=
2π i
Dx g (x0 (m), m)
√
+ K + O( m − mc ),
m > mc ,
where K is a constant independent of m. The number ofiterates, N, requiredto pass through the bifurcation point for the
m
model with predator saturation, given by xn+1 = xn exp r (1 − xn ) − 1+sK
Nm =
2π i
Dx g (x0 (m), m)
xn
, is approximated by
,
(13)
where

Dx g (x0 (m), m) = exp r (1 − x0 (m)) −
and x0 (m) = xc +
a
b
(m − mc )
 21
i.
m
1 + sKx0 (m)


x0 (m) −r +
msK
(1 + sKx0 (m))2


+ 1 − 1;
a
1659
b
t ext
t ext
–1/2
–1/2
φm
φ αK
Fig. 10. Delayed transitions after the saddle–node bifurcations found in the system with predator saturation (a) and mating limitation (b). In both plots
we show the time to extinction, text , at increasing the bifurcation parameters beyond the critical bifurcation values. In (a) we use φm = m − mc , where
mc = 20.3203125, r = 4.5, sK = 16 and x0 = c = 0.605293. In (b) we use φα K = α Kc − α K where α Kc = 1.06976, r = 2 and x0 = c = 0.77359.
The blue dots show the extinction times computed numerically from (a) Eq. (3); and (b) Eq. (5). The solid lines correspond to the analytical solutions, (13)
and (14) characterized in Section 4 for each model. We notice that delayed transitions occur in all discrete models given by unimodal maps undergoing a
saddle–node bifurcation. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Now, we consider the system with mating limitation
xn+1 = f (xn , α K ) = xn exp (r (1 − xn ))
α Kxn
1 + α Kxn
,
and follow precisely the procedure described above. We start by defining
g (x, α K ) = xn


α Kxn
exp (r (1 − xn ))
−1 .
1 + α Kxn
The equilibrium points occur when g (xn , α K ) = 0 and the saddle–node bifurcation of the one-dimensional map of Fig. 7
(a) takes place at x = xc = 0.2157056 . . . , when α K = α Kc = 0.1400549, for r = 4.5. When α K < α Kc , there are two
complex equilibrium points.
For α K extremely close to α Kc (α K < α Kc ), many iterations of the map are required to pass through the bifurcation point.
In this case, the number of iterates, N, required to pass through the bifurcation point for the model with mating limitation,
is approximated by
Nα K =
2π i
Dx g (x0 (α K ), α K )
,
(14)
where
Dx g (x0 (α K ), α K ) = −1 −
exp(r (1 − x0 (α K )))x20 (α K )(α K )2
(1 + x0 (α K )α K )2
2 exp(r (1 − x0 (α K )))x0 (α K )α K
exp(r (1 − x0 (α K )))rx20 (α K )α K
+
−
,
1 + x0 (α K )α K
1 + x0 (α K )α K

1
and x0 (α K ) = xc + ba (α K − α Kc ) 2 i, with a = Dα K g (xc , α Kc ) = 1.49499 . . . and b = D2x g (xc , α Kc )/2! = −4.63196 . . . .
Fig. 10(a) displays the extinction time, text (number of iterates until the fixed point x∗ = 0 is reached), as the intensity of
predation parameter grows above the limit point bifurcation. We display the extinction times obtained numerically from Eq.
(3) (blue dots) and analytically (solid black line) as described above. Note that the analytical results are in perfect agreement
with the numerical ones. The equivalent results are displayed for the model with mating limitation given by Eq. (5) (see
Fig. 10(b)).
5. Conclusions
The investigation of Allee effects in population ecology is a current subject of research. Field studies have shown the
importance of population densities in the overall population dynamics under different ecological interactions [19–25]. The
study of Allee effects in complex ecosystems has been also carried out by investigating ecological dynamical systems. For
instance, two classic examples of Allee effects, given by predator saturation and mating limitation, have been thoroughly
1660
studied by using one-dimensional discrete models [48,26]. Some works have also extended the previous one-dimensional
systems to two-species competition models with predator–prey dynamics and Allee effects [49,50], and several conditions
for species extinction and permanence have been mathematically determined [51]. Other studies analyzing Allee effects in
continuous-time dynamical systems have identified periodic solutions [52–54] and limit cycles governing the coexistence
dynamics in predator–prey systems [49]. The study of the spatio-temporal dynamics of predator–prey systems with Allee
effects have been also carried out in the context of species survival and diffusion-induced chaos [55,56].
In the present article we have studied one-dimensional maps describing single-species dynamics with predator
saturation and mating limitation Allee effects. We have focused on the chaotic dynamics governing species survival in,
also studying the parametric relations involving the essential extinction scenario where transient chaos can occur. The
theory of symbolic dynamics allowed us to compute the topological entropy and co-dimension two bifurcation diagrams
by identifying the periods of the orbits. Both models show common properties. In particular, we found the same periodordering toward the critical parameter values responsible for transient chaos. We have also found that the appearance of
transient chaos corresponds to the maximum topological entropy, with dynamics (RL)∞ , where the full shift takes place and
the population becomes asymptotically extinct.
From an ecological point of view, the first model reveals that longer predator handling times unstabilize the dynamics of
the populations or even are responsible for extinctions via transient chaos, which are independent of the initial population
values. In the bistability scenario, where species can either survive or extinguish, the Allee effect at increasing handling times
might not be so important because as the dynamics switches from the ordered to the chaotic regime, the basin of attraction to
the extinction dynamics is reduced. That is, only a very small initial population will become extinct. The inverse phenomenon
is found at increasing predation intensity. For this case, the bistability scenario is mainly characterized by an initial chaotic
regime, in which small initial populations will become extinct, but as the dynamics changes to the non-chaotic regime, the
basin of attraction for small population enlarges, and might experience a stronger Allee effect. Interestingly, the regions
with smaller basins of attraction involving extinction are found when the dynamics in the bistable scenario are chaotic. The
same phenomenon is also found in the model with mating limitation, where the largest basin of attraction for population
extinctions in the bistability scenario is found in the non-chaotic regime. These results indicate that the extinction of the
population is less probable at low initial population densities for the chaotic regime. In this context, Allen and co-workers
found that chaos could decrease species extinction in metapopulation dynamical systems under random disturbances, for
which the extinction probability increased under periodic or quasi-periodic regimes [9]. Our work suggests that, under the
chaotic regime, the initial population values driving the population to extinction might be less than the ones for the ordered
regime.
Finally, we have also investigated delayed extinction transients due to saddle–node bifurcations found in both studied
models. Delayed transitions tied to saddle–node bifurcations have been studied in theoretical models for catalytic networks
(e.g., hypercycles) (see [57]) and in models of charge density waves [58]. This delaying phenomenon was also empirically
identified in an electronic circuit modeling Duffing’s oscillator [59]. Here, we have analytically derived the inverse squareroot scaling law near the bifurcation for the time-to-extinction by using a complex variable approach. The expressions
derived analytically for such a time delay are in complete agreement with the numerical results carried out for the two
studied ecological models.
Acknowledgments
This work was partially funded by the Fundação para a Ciência e a Tecnologia through the Program POCI 2010/FEDER
and by the Human Frontier Science Program Organization (Grant RGP12/2008). JS wants to thank the Kavli Institute for
Theoretical Physics (University of California Santa Barbara) where part of this work was developed (National Science
Foundation Grant No. NSF PHY05-51164).
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On chaos, transient chaos and ghosts in single population models

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