52 A. K. Supriatna and H. P. Possingham
where p
X
is the price of the harvested stock X and is assumed to be constant, while c
X
i
is the unit cost of harvesting and is assumed to be a non-increasing function of X
i
and may depend on the location of the stock. To obtain the optimal harvest for a two-patch predator–prey population we define a value function
J
T
N
10
, N
20
, P
10
, P
20
= max
0≤S
Xi0
≤ X
i 0
T
X
k=0
ρ
k 2
X
i =1 P
X
X =N
5
X
i
X
i k
, S
X
i k
6 which is the sum of the discounted net revenue resulting from harvesting both
populations in both locations up to period t = T . This function is maximized by choosing appropriate optimal escapements S
∗ X
i k
. Equation 6 is used recursively to obtain the value function at time T + 1, that is
J
T +1
N
10
, N
20
, P
10
, P
20
= max
0≤S
Xi0
≤ X
i 0
ρ J
T
N
11
, N
21
, P
11
, P
21
+
2
X
i =1 P
X
X =N
5
X
i
X
i 0
, S
X
i 0
. 7
Thus the optimal escapements, S
∗ N
i 0
and S
∗ P
i 0
, for a two-patch predator–prey sys- tem can be found by iterating this equation back from time T .
First, consider the net revenue in equation 6 for time horizon T = 0. The resulting net revenue, J
N
10
, N
20
, P
10
, P
20
, represents immediate net revenue taken from the next harvest without considering the future value of the harvest,
hence the maximum value is exactly the same as the maximum value of P V
∞
in 4. We consider two cases.
C
ASE
1. If the unit cost of harvesting is constant, let c
X
i
X
i
= c
X
i
, then p
X
− c
X
i
in 5 is constant. Hence, the integral in 5, and thus P V
∞
in 4, is maximized by S
∗ X
i ∞
satisfying S
∗ X
i ∞
= X
i
if p
X
≤ c
X
i
if p
X
c
X
i
. 8
Therefore, if the unit cost of harvesting is constant and lower than the unit price of harvested stock then it is optimal to drive the stock to extinction see
also Fig. 2 for a relatively large discount rate. On the other hand, if the unit cost of harvesting is constant and greater than or equal to the unit price of harvested
stock then we should not harvest the stock at all.
C
ASE
2. If the unit cost of harvesting is not constant then P V
∞
in 4 is maxi- mized by S
∗ N
i ∞
and S
∗ P
i ∞
satisfying
∂5
Xi
X
i 0
, S
Xi0
∂ S
Xi0
S
Xi0
= S
∗ Xi∞
= 0. Differentiate the in-
tegral in 5 with respect to S
∗ X
i 0
to obtain p
N
− c
N
S
∗ N
i ∞
= 0 and p
P
− c
P
S
∗ P
i ∞
= 0. The last two equations say that optimal escapements occur if the marginal rev-
enue equals the marginal value of cost. This condition is known as ‘bionomic equilibrium’ Gordon, 1954.
Harvesting Predator–Prey Metapopulation 53
a Profit is maximum at 14505,9010 for discount rate 10
x0 x
x50 x
x 1105
1100 1095
1090
100 300
5000 10000
15000 20000
5000 10000
15000 20000
b Profit is maximum at 145050,145050 for discount rate 10
x0 x
x50 x100
x 8000
7500 7000
300 50000
100000 150000
200000 50000
100000 150000
200000
Figure 2. Contour plot for the profit in 3 as a function of a predator escapements and b prey escapements, calculated in millions unit with discount rate 10. Escapements
S
P
1
= 14,505 and S
P
2
= 9010 are found as the predator optimal escapements a and
escapements S
N
1
= S
N
2
= 145,050 are found as the prey optimal escapements b. The
symbol ‘×’ indicates the position of equilibrium escapements for various discount rates, e.g. ×0 indicates the position with no discount rate.
Next, to obtain the net revenue for time horizon T = 1 we substitute these immediate escapements into equation 7. As in the case for the time horizon
T = 0, to maximize the revenue, we use the necessary conditions for optimality by differentiating the resulting equation with respect to escapements for the time
horizon T = 1, that is S
N
i 0
and S
P
i 0
. This procedure yields p
N
− c
N
i
S
∗ N
i 0
ρ =
a
i
− α
i
S
∗ P
i 0
+ p
i i
F
′ i
S
∗ N
i 0
p
N
− c
N
i
N
i 1
+ p
i j
F
′ i
S
∗ N
i 0
p
N
− c
N
j
N
j 1
+β
i
S
∗ P
i 0
p
P
− c
P
i
P
i 1
, 9
54 A. K. Supriatna and H. P. Possingham
p
P
− c
P
i
S
∗ P
i 0
ρ =
b
i
+ β
i
S
∗ N
i 0
+ q
i i
G
′ i
S
∗ P
i 0
p
P
− c
P
i
P
i 1
+ q
i j
G
′ i
S
∗ P
i 0
p
P
− c
P
j
P
j 1
−α
i
S
∗ N
i 0
p
N
− c
N
i
N
i 1
. 10
These equations are the general form of the optimal harvesting equation for a two- patch predator–prey population system. The escapements S
∗ X
i 0
found by solving these equations are the optimum escapements of the prey and the predator on
each patch that maximize revenue provided the Hessian matrix J
′′ 1
S
N 10
, S
N 20
, S
P10
, S
P20
satisfies [ J
′′ 1
S
∗ X
S
X
− S
∗ X
] · [S
X
− S
∗ X
] 0 where S
X
= S
N 10
, S
N 20
, S
P10
, S
P20
and S
∗ X
= S
∗ N 10
, S
∗ N 20
, S
∗ P10
, S
∗ P20
. It can be shown that these optimal escapements of predator–prey metapopulation are independent of the time horizon
considered.
3. R
