Optimal Strategy for the Conservation of the Rhinoceros
Population in South Africa
Ashleigh Hutchinson
Kimesha Naicker, Xichavo Vukeya, Sibabalwe Mdingi, Despina
Newman, Maria-Helena Wate, Ayobami Akinyelu, Sylvester Mothapo,
Nelson Phora
January 10, 2015
Ashleigh Hutchinson[0.3cm]Kimesha Naicker, Xichavo Vukeya,
RhinoSibabalwe
Conservation
Mdingi, Despina Newman, Maria-Helena
January 10,
Wate,
2015Ayobami
1 /Akiny
17
Introduction
Our goal: To find an optimal strategy to reduce poaching and conserve
the rhino population in South Africa.
Ashleigh Hutchinson[0.3cm]Kimesha Naicker, Xichavo Vukeya,
RhinoSibabalwe
Conservation
Mdingi, Despina Newman, Maria-Helena
January 10,
Wate,
2015Ayobami
2 /Akiny
17
Introduction
We attempted to solve the following questions:
Can we characterize the rhino poaching situation in South Africa
mathematically?
- How would translocation affect the rhino populations?
Could rhino horn legalization eliminate poaching?
- How would a combined strategy of both legalization and tranlocation
affect populations and impact the black market?
Ashleigh Hutchinson[0.3cm]Kimesha Naicker, Xichavo Vukeya,
RhinoSibabalwe
Conservation
Mdingi, Despina Newman, Maria-Helena
January 10,
Wate,
2015Ayobami
3 /Akiny
17
Black Market Model
Our model characterizing the current poaching situation in South Africa:
R1
dR1
= α1 R1 1 −
− β 1 R 1 Q − H1 R 1 ,
(1)
dt
k1
dR2
R2
= α2 R2 1 −
− β2 R2 Q + γH1 R1 − σR2 ,
(2)
dt
k2
dQ
δP
= τQ
−1 ,
(3)
dt
P∗
dP
= ν(Pmax − P) − µQ(β1 R1 + β2 R2 ),
(4)
dt
where R1 is the number of rhinos in public reserves, R2 is the number of
rhinos in private reserves, Q is the amount of poachers at a given time and
P is the profit per kilogram of a rhino horn.
Ashleigh Hutchinson[0.3cm]Kimesha Naicker, Xichavo Vukeya,
RhinoSibabalwe
Conservation
Mdingi, Despina Newman, Maria-Helena
January 10,
Wate,
2015Ayobami
4 /Akiny
17
Stability Analysis
Our first equilibrium point was (0, 0, 0, P ∗ ).
This corresponds to the situation where rhino populations are extinct and
thus there are no poachers either.
The eigenvalues are
λ1 = α1 − H1 ,
λ3 = τ (δ − 1) ,
λ2 = α2 − σ,
λ4 = −ν.
We note that there is the relationship which we assume to be
δ = β1 − β2 + .
We want to avoid this case. For most sets of the parameters, this is a
repeller.
Ashleigh Hutchinson[0.3cm]Kimesha Naicker, Xichavo Vukeya,
RhinoSibabalwe
Conservation
Mdingi, Despina Newman, Maria-Helena
January 10,
Wate,
2015Ayobami
5 /Akiny
17
Stability Analysis
Our second equilibrium point was 0, k2 1 −
σ
α2
, 0, P ∗ .
This corresponds to the situation where poachers are eliminated, however
only private parks will have rhinos. While this is not ideal, it is preferable
to total extinction.
The eigenvalues are
λ1 = α1 − H1 ,
λ3 = τ (δ − 1) ,
λ2 = σ − α2 ,
λ4 = −ν.
We can control the translocation rate to force this point to be an
attractor, which could be a possible solution.
Ashleigh Hutchinson[0.3cm]Kimesha Naicker, Xichavo Vukeya,
RhinoSibabalwe
Conservation
Mdingi, Despina Newman, Maria-Helena
January 10,
Wate,
2015Ayobami
6 /Akiny
17
Stability Analysis
Our final equilibrium point was k1 1 −
H1
α1
, R2∗ , 0, P ∗ .
This is an ideal situation as poachers are eliminated while still maintaining
populations in both private and public parks.
The eigenvalues are
λ1 = H1 − α1 ,
λ3 = τ (δ − 1) ,
λ2 = A,
λ4 = −ν.
For most sets of parameters, this point will be a repeller which is reflective
of the current situation.
Ashleigh Hutchinson[0.3cm]Kimesha Naicker, Xichavo Vukeya,
RhinoSibabalwe
Conservation
Mdingi, Despina Newman, Maria-Helena
January 10,
Wate,
2015Ayobami
7 /Akiny
17
Graphs
Population sizes
Population sizes
50 000
100
40 000
80
30 000
60
40
20 000
20
10 000
2
4
6
8
10
Time
(a) Short term scale.
5
10
15
20
Time
(b) Long term scale.
Figure: Blue = public rhino population,
orange = private rhino population,
green = poacher numbers
Ashleigh Hutchinson[0.3cm]Kimesha Naicker, Xichavo Vukeya,
RhinoSibabalwe
Conservation
Mdingi, Despina Newman, Maria-Helena
January 10,
Wate,
2015Ayobami
8 /Akiny
17
Legal and black market trade model
Our model showing legal trade against the black market:
dR1
R1
= α1 R1 1 −
− β 1 R 1 Q − H1 R 1 ,
dt
k1
dR2
R2
= α2 R2 1 −
− β2 R2 Q + γH1 R1 − σR2 ,
dt
k2
δPmax
dQ
=Q
P2 − 1 ,
dt
P∗
dP1
= ξ2 P2 − θS = 0,
dt
dP2
= ξ(P ∗ − P2 ) − µ(β1 R1 Q + β2 R2 Q),
dt
(5)
(6)
(7)
(8)
(9)
where R1 is the number of rhinos in public reserves, R2 is the number of
rhinos in private reserves, Q is the amount of poachers at a given time, P1
is the profit of legalized horn per kilogram and P2 is the profit per
kilogram of a rhino horn on the black market.
Ashleigh Hutchinson[0.3cm]Kimesha Naicker, Xichavo Vukeya,
RhinoSibabalwe
Conservation
Mdingi, Despina Newman, Maria-Helena
January 10,
Wate,
2015Ayobami
9 /Akiny
17
Stability Analysis
Our first equilibrium point was (0, 0, 0, P ∗ ).
This corresponds to the situation where rhino populations are extinct and
thus there are no poachers either.
The eigenvalues are
λ1 = α1 − H1 ,
λ3 = δPmax − 1,
λ2 = α2 − σ,
λ4 = −ξ.
We want to avoid this case. For most realistic sets of the parameters, this
is a repeller.
Ashleigh Hutchinson[0.3cm]Kimesha Naicker, Xichavo Vukeya,
RhinoSibabalwe
Conservation
Mdingi, Despina Newman, Maria-Helena
January 10,Wate,
2015 Ayobami
10 /Akiny
17
Stability Analysis
Our second equilibrium point was 0, k2 1 −
σ
α2
, 0, P ∗ .
Once again, this corresponds to the situation where poachers are
eliminated, however only private parks will have rhinos. While this is not
ideal, it is preferable to total extinction.
The eigenvalues are
λ1 = α1 − H1 ,
λ3 = δPmax − 1,
λ2 = σ − α2 ,
λ4 = −ξ.
We can control the translocation rate to force this point to be an
attractor, which could be a possible solution.
Ashleigh Hutchinson[0.3cm]Kimesha Naicker, Xichavo Vukeya,
RhinoSibabalwe
Conservation
Mdingi, Despina Newman, Maria-Helena
January 10,Wate,
2015 Ayobami
11 /Akiny
17
Stability Analysis
Our final equilibrium point was k1 1 −
H1
α1
, R2∗ , 0, P ∗ .
This is an ideal situation as poachers are eliminated while still maintaining
populations in both private and public parks.
The eigenvalues are
λ 1 = H1 − α 1 ,
λ3 = δPmax − 1,
λ2 = A,
λ4 = −ξ.
For most sets of parameters, this point will be a repeller which is reflective
of the current situation.
Ashleigh Hutchinson[0.3cm]Kimesha Naicker, Xichavo Vukeya,
RhinoSibabalwe
Conservation
Mdingi, Despina Newman, Maria-Helena
January 10,Wate,
2015 Ayobami
12 /Akiny
17
Graphs
Population sizes
Population sizes
100 000
80 000
150
60 000
100
40 000
50
20 000
2
4
6
8
10
Time
(a) Short term scale.
5
10
15
20
25
30
Time
(b) Long term scale.
Figure: Blue = public rhino population,
orange = private rhino population,
green = poacher numbers
Ashleigh Hutchinson[0.3cm]Kimesha Naicker, Xichavo Vukeya,
RhinoSibabalwe
Conservation
Mdingi, Despina Newman, Maria-Helena
January 10,Wate,
2015 Ayobami
13 /Akiny
17
Graphs
Profit ($)
13 000
12 000
11 000
10 000
0
2
4
6
8
10
Time
Blue = Profit of rhino horn per kg on the black market, orange = Profit of
rhino horn per kg legally
Ashleigh Hutchinson[0.3cm]Kimesha Naicker, Xichavo Vukeya,
RhinoSibabalwe
Conservation
Mdingi, Despina Newman, Maria-Helena
January 10,Wate,
2015 Ayobami
14 /Akiny
17
Conclusions and possible strategies
By introducing legalization, we can drive the horn price down to the
point where it is no longer viable to poach
Translocation to private parks is a good strategy to decrease poaching
Legalization should only occur if a constant supply of rhino horn can
be maintained ie. the amount of horn at a given time released into
the market must be carefully controlled.
The type of strategy we choose (ie. model 1 or model 2) depends on
the security present in private parks
Ashleigh Hutchinson[0.3cm]Kimesha Naicker, Xichavo Vukeya,
RhinoSibabalwe
Conservation
Mdingi, Despina Newman, Maria-Helena
January 10,Wate,
2015 Ayobami
15 /Akiny
17