The model in this paper has a similar structure and assumptions to the model in Supriatna and Possingham 1997a, 1998. These papers assume that the juveniles of the
predator as a result of food conversion from the captured prey are sedentary. In this paper, I modify the model in Supriatna and Possingham 1997a, 1998 to allow some proportion of
these juveniles to migrate between patches. Using dynamic programming theory I found optimal harvesting strategies to harvest the predator-prey metapopulation. The results show that
some properties of the optimal escapements for a single-species metapopulation are preserved in the presence of predators, such as the strategies on how to harvest a relative sourcesink and
exporterimporter sub-population.
2. The Model
Consider a predator-prey metapopulation that coexists in two different patches; patch one and patch two. The movement of individuals between the local populations is a result of
dispersal by juveniles. Adults are assumed to be sedentary, and they do not migrate from one patch to another patch. Assume that the dynamics of the prey metapopulation is given by
Prey in patch i now = the number of surviving adult prey from last period
+ prey recruitment from patch i + prey recruitment from patch j
- the number of prey killed by the predator in patch i, where
i =1,2 and
j =1,2 . If we assume that predation affects the predator recruitment then
Predator in patch i now = the number of surviving adult predator from last period
+ predator recruitment from patch i
+ predator recruitment from patch j. Let the number of prey predator in patch
i where
i =1,2 at the beginning of period
k be denoted by
N
ik
and P
ik
and the survival rate of adult prey predator in patch i
be denoted by a
i
and b
i
, respectively. If the proportion of juvenile prey and predator from patch i
that successfully migrate to patch
j are
p
ij
and q
ij
, respectively, then the above equation can be written as
N
ik+1
= a
i
N
ik
+ p
ii
F
i
N
ik
+ p
ji
F
j
N
jk
+ α
i
N
ik
P
ik
, 1
P
ik+1
= b
i
P
ik
+ q
ii
[G
i
P
ik
+ β
i
N
ik
P
ik
] + q
ji
[G
j
P
jk
+ β
j
N
jk
P
jk
], 2
where the functions F
i
N
ik
and G
i
P
ik
are the recruit production functions of the prey and predator in patch
i at time period
k ,
α
i
negative and β
i
positive. If
S
Nik
=N
ik
-H
Nik
S
Pik
=P
ik
-H
Pik
is the escapement of the prey predator in patch i
at the end of that period, and
H
Nik
H
Pik
is the harvest taken from the prey predator, then we obtain the dynamic of exploited population
N
ik+1
= a
i
S
Nik
+ p
ii
F
i
S
Nik
+ p
ji
F
j
S
Njk
+ α
i
S
Nik
S
Pik
, 3 P
ik+1
= b
i
S
Pik
+ q
ii
[G
i
S
Pik
+ β
i
S
Nik
S
Pik
] + q
ji
[G
j
S
Pjk
+ β
j
S
Njk
S
Pjk
]. 4
Furthermore, we assume that in the absence of predator-prey interaction, there is an environmental carrying capacity in the recruitment of each species, and that recruitment is
linearly dependent on population size, whenever the population size is far below the carrying capacity. With these assumptions, we can choose a logistic function as a recruit production
function, that is F
i
N
ik
=r
i
N
ik
1-N
ik
K
i
and G
i
P
ik
=s
i
P
ik
1-P
ik
L
i
, where r
i
s
i
denotes the intrinsic growth of the prey predator and
K
i
L
i
denotes the carrying capacity of the prey predator, respectively.
To find optimal escapements for the metapopulation, I use dynamic programming as in Supriatna and Possingham 1998 to maximise the net present value
,
2 1
Xik ik
i P
N X
Xi T
k k
S X
PV
∑ ∑ ∑
= =
=
Π =
ρ
5 subject to the equations 3 and 4, and ignoring all costs of harvesting, to obtain optimal
escapements:
, 2
2 1
i i
i i
i i
i i
Ni
B C
L s
q q
m A
S ∆
+ +
=
6
, 2
2 1
i i
i i
i i
i i
Pi
A C
K r
p p
B S
∆ +
+ =
7 where:
ρ =11+
δ with
δ≥ denoting a periodic discount rate,
∆
i
=C
i 2
-mp
i1
+p
i2
[2r
i
K
i
] q
i1
+q
i2
[2s
i
L
i
] is not zero,
ξ ξ
d c
p S
X
Xi Xik
SXik X
Xik ik
Xi
, −
= Π
∫
is the net revenue from the harvest
H
Xik
of the local population X
i
in period k
, A
i
=1 ρ
-[p
i1
+p
i2
]r
i
-a
i
, B
i
=m1 ρ
-[q
i1
+q
i2
]s
i
-b
i
, C
i
= α
i
+mq
i1
+q
i2
β
i
, s
im
=mq
i1
+q
i2
s
i
, r
im
=p
i1
+p
i2
r
i
, and m
is the relative price of the predator to the prey.
Appendix A shows the derivation of the optimal escapements 6 and 7 and Appendix B shows that these escapements are independent of the time horizon considered. It can also be
shown that if A
i
and B
i
are negative and C
i
is non-positive with C
i
max
{2B
i
K
i
,2mA
i
L
i
} then
∆
i
0 and all resulting escapements, S
Ni
and S
Pi
, are positive. For the remaining part of this
paper I will assume that A
i
and B
i
are negative, so that extinction is not optimal.
3. Results and Conclusion
