Rhino

Transcription

Rhino

Optimal Strategy for the Conservation of the Rhinoceros
Population in South Africa
Prof. Michael Sears, Prof. Tim Myers, Prof. Chris Breward, Ashleigh
Hutchinson
Kimesha Naicker, Despina Newman, Maria-Helena Wate, Ayobami
Akinyelu, Sylvester Mothapo, Nelson Phora, Tanki Motsepa, Collen
Matuwane
January 16, 2015
Prof. Michael Sears, Prof. Tim Myers, Prof. Chris Breward,
Rhino
Ashleigh
Conservation
Hutchinson[0.3cm]Kimesha Naicker,January
Despina16,
Newman,
2015 Maria-Hele
1 / 24
Introduction
Our objective:
To conserve the rhino population as well as decrease poaching to a
manageable level.
Rhino
Ashleigh
Conservation
Despina16,
Newman,
2015 Maria-Hele
2 / 24
Introduction
We attempted to answer the following questions:
- Can we characterize the rhino poaching situation in South Africa
mathematically?
- What effects do translocation of rhinos and legalisation of selling
rhino horns have on the rhino population?
- How can we design a pricing model that would reflect the rhino horn
market?
Rhino
Ashleigh
Conservation
Despina16,
Newman,
2015 Maria-Hele
3 / 24
Governing Equations
The population equations:
Logistic growth term
Rate of change of wild rhinos
z}|{
dR1
dt
z
}|
{
R1
= α1 R1 1 −
k1
Poaching effect term
−
z }| {
χ1 R 1
−
z }| {
η2 R 1
Rhino translocation
−
Trophy hunting
(1)
z }| {
η1 R 1
Natural wild rhino death
−
z }| {
ρ1 R1
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Conservation
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2015 Maria-Hele
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Governing Equations
Here,
Poaching effort
z }| {
γ1 q3 P3
χ1 =
ν η R + q1 (P1 − C1 ) + T + C + ν1 q2 (P1 − C2 )
|2 2 1
{z
}
(2)
Anti-poaching effort
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Ashleigh
Conservation
Despina16,
Newman,
2015 Maria-Hele
5 / 24
Governing Equations
Logistic growth term
Rate of change of farm rhinos
z}|{
dR2
dt
z
}|
{
R2
= α2 R 2 1 −
k2
Poaching effect term
−
z }| {
χ2 R 2
(3)
Rhino translocation
+
z }| {
η1 R 1
Natural farm rhino death
−
z }| {
ρ2 R2
Rhino
Ashleigh
Conservation
Despina16,
Newman,
2015 Maria-Hele
6 / 24
Governing Equations
Here,
Poaching effort
z }| {
γ2 q3 P3
χ2 =
(1 − ν2 )η2 R1 + (1 − ν1 )q2 (P1 − C2 )
|
{z
}
(4)
Anti-poaching effort
Rhino
Ashleigh
Conservation
Despina16,
Newman,
2015 Maria-Hele
7 / 24
Governing Equations
Rhino horn stockpile equations:
Rate of change of wild rhino horn stockpile
z}|{
dS1
dt
Horn obtained from wild rhino deaths
=
z }| {
λ1 ρ1 R1
(5)
Horn removed from stock pile for legal sale
−
z}|{
q1
(6)
Rhino
Ashleigh
Conservation
Despina16,
Newman,
2015 Maria-Hele
8 / 24
Governing Equations
Rate of change of farmed rhino horn stockpile
z}|{
dS2
dt
Horn obtained from farm rhino deaths
=
z }| {
λ2 ρ2 R2
(7)
Horn obtained from horn harvesting
+
z}|{
µR2
Horn removed from stock pile for legal sale
−
z}|{
q2
Rhino
Ashleigh
Conservation
Despina16,
Newman,
2015 Maria-Hele
9 / 24
Governing Equations
Rate of change of illegal horn stockpile
z}|{
dS3
dt
Horn obtained from poaching
=
z
}|
{
λ3 (χ1 R1 + χ2 R2 )
(8)
Horn removed from stock pile for illegal sale
−
z}|{
q3
Rhino
Ashleigh
Conservation
Hutchinson[0.3cm]Kimesha Naicker,
January
Despina
16,Newman,
2015
Maria-Hele
10 / 24
Governing Equations
We assume a three-player game where all players are acting optimally.
Our price model:
P = P0 − (β1 q1 + β2 q2 + β3 q3 ),
(9)
where P0 is the ”crunch price”.
The resulting cash flow for the i th player:
wi = (P − Ci )qi .
(10)
Each player employs the optimal strategy to maximize their cash flow,
resulting in optimal qi values.
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Ashleigh
Conservation
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2015
Maria-Hele
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Governing Equations
These are:
−3C1 + C2 + C3 + P0
,
4β1
C1 − 3C2 + C3 + P0
q2 =
,
4β2
C1 + C2 − 3C3 + P0
q3 =
.
4β3
q1 =
(11)
(12)
(13)
With C1 = C2 = $0 and C3 = $10000, we get a price of P = $65000.
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Ashleigh
Conservation
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16,Newman,
2015
Maria-Hele
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Graphs
Population sizes
14 000
12 000
10 000
8000
6000
4000
2
4
6
8
10
Time
Figure : Population sizes of rhinos over time without legalisation. The blue line
represents wild rhinos, the orange line represents farmed rhinos.
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Ashleigh
Conservation
January
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16,Newman,
2015
Maria-Hele
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Graphs
Population sizes
20 000
15 000
10 000
5000
5
10
15
20
Time
Figure : Population sizes of rhinos over time with legalisation. The blue line
represents wild rhinos, the orange line represents farmed rhinos.
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Ashleigh
Conservation
January
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16,Newman,
2015
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Graphs
Kg of horn
140 000
120 000
100 000
80 000
60 000
40 000
20 000
5
10
15
20
Time
Figure : Stock pile sizes of rhino horns over time. The blue line represents wild
rhino horn stock, the orange line represents farmed rhino horn stock, the green
line represents illegal stock.
Rhino
Ashleigh
Conservation
January
Despina
16,Newman,
2015
Maria-Hele
15 / 24
Graphs
Number of rhinos after 20 years
20 000
15 000
10 000
5000
0.0
0.2
0.4
0.6
0.8
1.0
ν1
Figure : Graph showing the effect of varying cross-subsidy due to selling of
farmed rhino horn on the population after 20 years. Blue is wild rhino population,
green is farmed rhino population.
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Ashleigh
Conservation
January
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16,Newman,
2015
Maria-Hele
16 / 24
Horn growth model
Assume the horn growth of a mature rhino is governed by
dh
= k1 − k2 f (h),
da
(14)
where h is horn length, a is age and f (h) represents the rate of natural
horn erosion. We get h = g (a).
√
A possible function for the erosion may be f (h) = πR R 2 + h2 .
The first harvest occurs at age a = am and thus h(am ) = 0. If we harvest
N times over a period of L years, the height of the horn at harvest is given
by
L
+ am .
(15)
hharvest = g
N
We can obtain V (hharvest ) volume of horn. Thus in L years, we get
V (hharvest ) · N.
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Ashleigh
Conservation
January
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16,Newman,
2015
Maria-Hele
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Age-structured model
We let the age of the rhinos be an independent variable
Ri = Ri (a, t).
(16)
Our model:
∂R1 ∂R1
+
= −ρ1 (a)R1 − η1 (a)R1 − η2 (a)R1 − γ1 χ1 R1 ,
∂t
∂a
∂R2 ∂R2
+
= −ρ2 (a)R2 + η1 (a)R1 − γ2 χ2 R2 ,
∂t
∂a
dS1
= λ1 ρ1 (a)R1 − q1 ,
dt
dS2
= λ2 ρ2 (a)R2 + µ(a)R2 − q2 ,
dt
dS3
= λ1 (γ1 R1 + γ2 R2 ) − q3 .
dt
(17)
(18)
(19)
(20)
(21)
A similar economic model to the homogeneous age model presented prior
will be utilized.
Rhino
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Conservation
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16,Newman,
2015
Maria-Hele
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In this model,
Death rates will increase as age increases
Hunting is normally allowed for older (male) rhinos
Translocation is normally done for pregnant rhinos and older rhinos
The amount of horn removed from rhinos on the farm increases with
the rhino age until they are mature
Births of rhino enter the model through the boundary conditions on
a=0
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Conservation
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16,Newman,
2015
Maria-Hele
19 / 24
Figure : Births of rhino enter the model through the boundary conditions on a = 0
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Ashleigh
Conservation
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16,Newman,
2015
Maria-Hele
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Two-sex model
The aim for this model is to show how the population evolves in a two-sex
population.
The model is formulated like a predator-prey model. The prey population
is sub-divided into two different populations, male and female. The
equations for the model are given below:
df
= −θf + R,
dt
dm
= −θm + R,
dt
(22)
(23)
where f (t) and m(t) refers to the number of male and female rhinos in
the population at a given point in time, t, and θ corresponds to the
respective mortality rates. Additionally, R, is the birth rate.
Rhino
Ashleigh
Conservation
January
Despina
16,Newman,
2015
Maria-Hele
21 / 24
Two-sex model
Fredrickson defined R as:
R=
γf γm f · m
,
γf f + γm m
(24)
where γf and γm are some positive constants.
R must satisfy the following four conditions:
R(0, m) = R(f , 0) = 0,
(25)
R(kf , km) = kR(f , m),
(26)
R(m, f ) ≥ 0,
(27)
R(m + u, f + v ) > R(m, f ), for u, v > 0.
(28)
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Ashleigh
Conservation
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2015
Maria-Hele
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Conclusion
Our conclusions:
Without intervention, the wild rhino population will decline under
poaching pressure.
With controlled legalization, not only does the wild population
increase, but a sustainable industry is suggested.
The model is generally insensitive to the values of the parameters
except for γ1 and γ2 .
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Ashleigh
Conservation
January
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16,Newman,
2015
Maria-Hele
23 / 24
Further work
Improvements to the model can be provided by building in the age
structure of the population and by using sex based models
Analysis of horn growth and harvest timing can assist with optimal
strategies for rhino farming management
The pricing model needs further refinement and investigation
The functions γ1 χ1 and γ2 χ2 are important in the results of the
model. Further investigation of these will be important.
Rhino
Ashleigh
Conservation
January
Despina
16,Newman,
2015
Maria-Hele
24 / 24

Rhino

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