Application Of Dynamic Programming In Harvesting A Predator-prey Metapopulation.
Bulletin of Indonesian Sciences Technology and Economics, Volume 5 Number 2 (1998)
Application of Dynamic Programming
Bulletin of Indonesian Sciences Technology and Economics, Volume 5 Number 2 (1998)
2
example, the abalone are eaten by crabs, lobsters, octopi and many species of fish (Kojima,
1990). In this paper I develop a model to describe a predator-prey metapopulation in which the
in Harvesting a Predator-Prey Metapopulation
Asep K. Supriatna
Department of Mathematics, University of Padjadjaran, Indonesia
predator-prey interaction takes place in the adult life stage of the prey.
Predation on adult life stage is not uncommon in nature. Zaret (1980) divides predators
in freshwater communities into two types: ‘gape-limited predators’ and ‘size-dependent
predators’. The first type of predator eats prey by swallowing it whole. Hence the prey needs to
Abstract
be smaller than the predator's mouth diameter. The probability of prey with body size larger
In this paper I use dynamic programming theory to obtain optimal harvesting strategies for a two-
than the predator's gape being eaten by the predator is zero. The second type of predator eats
patch predator-prey metapopulation. I found that if predator economic efficiency is relatively high then we
prey by piercing, crushing or sucking it, and hence can eat prey with a relatively larger body
should protect a relative source prey sub-population in two different ways. Directly, with a higher
escapement of the relative source prey sub-population, and indirectly, with a lower escapement of the
size than the predator's mouth diameter. In general, the second type of predators also can be
predator living in the same patch with the relative source prey sub-population. Other rules are also found
found both in freshwater and marine communities. For example, sea lumprey (Petromyzon
as generalisations of rules to harvest a single-species metapopulation.
marinus) that prey on many species of fish, such as lake trout, salmon, rainbow trout, whitefish,
Abstrak
Di dalam paper ini, penulis menggunakan pemograman dinamik untuk mendapatkan strategi yang
optimal dalam eksploitasi sumber alam yang mempunyai struktur metapopulasi dan mempunyai relasi
burbot, walleye and catfish, is an example in freshwater communities, and octopus that prey on
rock lobster is an example in marine communities.
Some predators of both types have preferential feeding habits. For example, several
biologi ‘predator-prey’. Teori di dalam paper ini menyebutkan bahwa dalam keadaan tertentu, proteksi
populasi prey yang relatif ‘source’ bisa dilakukan dengan dua cara. Pertama dengan sisa tangkapan yang
lebih banyak dibandingkan sisa tangkapan untuk prey yang relatif ‘sink’ dan dengan eksploitasi predator
species of Coregonus and many planktivorous fish feed on the largest individuals of their prey
(De Bernardi and Giussani, 1975; Vanni, 1987). The maximum body size of the prey that is
yang lebih besar dibandingkan dengan eksploitasi predator di tempat lainnya. beberapa aturan umum juga
diperoleh sebagai perluasan dari teori eksploitasi untuk species tunggal.
captured by the gape-limited predators is limited by the diameter of the predator's mouth, while
the maximum body size of the prey that is captured by the size-dependent predators is only
Keywords: Dynamic Programming, Natural Resource Modelling, Harvesting Theory, Predator-Prey Metapopulation
limited by the predator capability in capturing and handling the prey (Zaret, 1980). Although it
1. Introduction
Many marine organisms which have commercial value are known to have a
metapopulation structure, for example abalone, Haliotis rubra. The adults are sedentary
occupying suitable space in reefs that are separated by some distance from other suitable reefs.
These sub-populations are connected by the dispersal of their larvae (Prince et al., 1987). In
addition, many of these marine metapopulations are part of predator-prey interactions. For
is not regarded as feeding habits, large crabs can prey on large abalone up to 200 mm (Shepherd
and Breen, 1992), and large octopi eat large mussels by drilling the shells (McQuaid, 1994).
Body size in many aquatic organisms is often related to age of maturity; a larger individual
often means an older individual. Hence the predator-prey interaction can be regarded as
interaction in the adult life stage of the prey.
Bulletin of Indonesian Sciences Technology and Economics, Volume 5 Number 2 (1998)
3
Bulletin of Indonesian Sciences Technology and Economics, Volume 5 Number 2 (1998)
4
+ predator recruitment from patch j.
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
Let the number of prey (predator) in patch i (where i =1,2) at the beginning of period k
modify the model in Supriatna and Possingham (1997a, 1998) to allow some proportion of
be denoted by Nik and Pik and the survival rate of adult prey (predator) in patch i be denoted by
these juveniles to migrate between patches. Using dynamic programming theory I found
ai and bi, respectively. If the proportion of juvenile prey and predator from patch i that
optimal harvesting strategies to harvest the predator-prey metapopulation. The results show that
successfully migrate to patch j are pij and qij , respectively, then the above equation can be
some properties of the optimal escapements for a single-species metapopulation are preserved
written as
in the presence of predators, such as the strategies on how to harvest a relative source/sink and
exporter/importer sub-population.
Ni(k+1) = aiNik + piiFi(Nik) + pjiFj(Njk) + αiNikPik,
(1)
Pi(k+1) = biPik + qii [Gi(Pik)+β iNikPik] + qji [Gj(Pjk)+β jNjkPjk],
(2)
where the functions Fi(Nik) and Gi(Pik) are the recruit production functions of the prey and
predator in patch i at time period k, αi negative and β i positive.
2. The Model
Consider a predator-prey metapopulation that coexists in two different patches; patch
If SNik=Nik-HNik (SPik=Pik-HPik) is the escapement of the prey (predator) in patch i at the
one and patch two. The movement of individuals between the local populations is a result of
end of that period, and HNik (HPik) is the harvest taken from the prey (predator), then we obtain
dispersal by juveniles. Adults are assumed to be sedentary, and they do not migrate from one
the dynamic of exploited population
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
Ni(k+1) = aiSNik + piiFi(SNik) + pjiFj(SNjk) + αiSNikSPik,
(3)
Pi(k+1) = biSPik + qii [Gi(SPik)+β iSNikSPik] + qji [Gj(SPjk)+β jSNjkSPjk].
(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
+ prey recruitment from patch j
linearly dependent on population size, whenever the population size is far below the carrying
- the number of prey killed by the predator in patch i,
capacity. With these assumptions, we can choose a logistic function as a recruit production
function, that is Fi(Nik)=riNik(1-Nik/Ki) and Gi(Pik)=siPik(1-Pik/Li), where ri (si) denotes the
where i =1,2 and j =1,2. If we assume that predation affects the predator recruitment then
intrinsic growth of the prey (predator) and Ki (Li) denotes the carrying capacity of the prey
(predator), respectively.
Predator in patch i now
= the number of surviving adult predator from last period
+ predator recruitment from patch i
To find optimal escapements for the metapopulation, I use dynamic programming as in
Supriatna and Possingham (1998) to maximise the net present value
Bulletin of Indonesian Sciences Technology and Economics, Volume 5 Number 2 (1998)
T
PV = ∑ ρ
k =0
k 2
P
∑ ∑Π
Xi
( X ik , S Xik )
5
(5)
i =1 X = N
Bulletin of Indonesian Sciences Technology and Economics, Volume 5 Number 2 (1998)
6
S1. If prey sub-population one is a relative source sub-population, i.e. r1(p11+p12) >
r2(p21+p22), and both predator sub-populations are identical with predator efficiency Ci
subject to the equations (3) and (4), and ignoring all costs of harvesting, to obtain optimal
satisfying max {2Bi/Ki,2mAi/Li}|α2|, and both predator sub-populations are identical with predator
efficiency Ci satisfying max {(-rmBi)/(AiKi),(-smAi)/(BiLi)}β 2/|α2|, and both prey sub-populations are identical with predator efficiency Ci
satisfying max {(-rmBi)/(AiKi),(-smAi)/(BiLi)}max {2Bi/Ki,2mAi/Li} then
∆iS*N2 and S*P1>S*P2. It can be shown that if Ai
limited dispersal of haliotid larvae (genus Haliotis; Mollusca: Gastropoda). J. Exp. Mar.
and Bi are negative and Ci is non-positive with Ci>max {2Bi/Ki,2mAi/Li} then ∆ip21+p22, then let
causes. In Abalone of the World: Biology, Fisheries and Culture, Shepherd et al. (Eds.), pp.
R1=R2=R, S1=S2=S, and s1m=s2m=sm. Let us define ∆SN=(S*N1-S*N2)∆1∆2. Using equations (6)
276-304, Fishing News Books, Oxford.
6. Supriatna, A. K. and H. P. Possingham (1997a). The exploitation of marine metapopulation:
and (7) we obtain
∆ SN
4s 2
2 R
2S
2C
= − m (r2 m − r1m )
−C −
C −
s m (r1m − r2 m )
K
L
KL
L
2
2 B 4 s m R
s m C C −
=
(r2 m − r1m ).
−
K KL
L
a modelling perspective. Mar. Res. Indonesia (submitted).
7. Supriatna, A. K. and H. P. Possingham (1997b). Harvesting a two-patch predator-prey
metapopulation. A paper presented in 1997 World Conference on Natural Resource
Modelling, Hobart, Tasmania.
Since 2B/K
Application of Dynamic Programming
Bulletin of Indonesian Sciences Technology and Economics, Volume 5 Number 2 (1998)
2
example, the abalone are eaten by crabs, lobsters, octopi and many species of fish (Kojima,
1990). In this paper I develop a model to describe a predator-prey metapopulation in which the
in Harvesting a Predator-Prey Metapopulation
Asep K. Supriatna
Department of Mathematics, University of Padjadjaran, Indonesia
predator-prey interaction takes place in the adult life stage of the prey.
Predation on adult life stage is not uncommon in nature. Zaret (1980) divides predators
in freshwater communities into two types: ‘gape-limited predators’ and ‘size-dependent
predators’. The first type of predator eats prey by swallowing it whole. Hence the prey needs to
Abstract
be smaller than the predator's mouth diameter. The probability of prey with body size larger
In this paper I use dynamic programming theory to obtain optimal harvesting strategies for a two-
than the predator's gape being eaten by the predator is zero. The second type of predator eats
patch predator-prey metapopulation. I found that if predator economic efficiency is relatively high then we
prey by piercing, crushing or sucking it, and hence can eat prey with a relatively larger body
should protect a relative source prey sub-population in two different ways. Directly, with a higher
escapement of the relative source prey sub-population, and indirectly, with a lower escapement of the
size than the predator's mouth diameter. In general, the second type of predators also can be
predator living in the same patch with the relative source prey sub-population. Other rules are also found
found both in freshwater and marine communities. For example, sea lumprey (Petromyzon
as generalisations of rules to harvest a single-species metapopulation.
marinus) that prey on many species of fish, such as lake trout, salmon, rainbow trout, whitefish,
Abstrak
Di dalam paper ini, penulis menggunakan pemograman dinamik untuk mendapatkan strategi yang
optimal dalam eksploitasi sumber alam yang mempunyai struktur metapopulasi dan mempunyai relasi
burbot, walleye and catfish, is an example in freshwater communities, and octopus that prey on
rock lobster is an example in marine communities.
Some predators of both types have preferential feeding habits. For example, several
biologi ‘predator-prey’. Teori di dalam paper ini menyebutkan bahwa dalam keadaan tertentu, proteksi
populasi prey yang relatif ‘source’ bisa dilakukan dengan dua cara. Pertama dengan sisa tangkapan yang
lebih banyak dibandingkan sisa tangkapan untuk prey yang relatif ‘sink’ dan dengan eksploitasi predator
species of Coregonus and many planktivorous fish feed on the largest individuals of their prey
(De Bernardi and Giussani, 1975; Vanni, 1987). The maximum body size of the prey that is
yang lebih besar dibandingkan dengan eksploitasi predator di tempat lainnya. beberapa aturan umum juga
diperoleh sebagai perluasan dari teori eksploitasi untuk species tunggal.
captured by the gape-limited predators is limited by the diameter of the predator's mouth, while
the maximum body size of the prey that is captured by the size-dependent predators is only
Keywords: Dynamic Programming, Natural Resource Modelling, Harvesting Theory, Predator-Prey Metapopulation
limited by the predator capability in capturing and handling the prey (Zaret, 1980). Although it
1. Introduction
Many marine organisms which have commercial value are known to have a
metapopulation structure, for example abalone, Haliotis rubra. The adults are sedentary
occupying suitable space in reefs that are separated by some distance from other suitable reefs.
These sub-populations are connected by the dispersal of their larvae (Prince et al., 1987). In
addition, many of these marine metapopulations are part of predator-prey interactions. For
is not regarded as feeding habits, large crabs can prey on large abalone up to 200 mm (Shepherd
and Breen, 1992), and large octopi eat large mussels by drilling the shells (McQuaid, 1994).
Body size in many aquatic organisms is often related to age of maturity; a larger individual
often means an older individual. Hence the predator-prey interaction can be regarded as
interaction in the adult life stage of the prey.
Bulletin of Indonesian Sciences Technology and Economics, Volume 5 Number 2 (1998)
3
Bulletin of Indonesian Sciences Technology and Economics, Volume 5 Number 2 (1998)
4
+ predator recruitment from patch j.
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
Let the number of prey (predator) in patch i (where i =1,2) at the beginning of period k
modify the model in Supriatna and Possingham (1997a, 1998) to allow some proportion of
be denoted by Nik and Pik and the survival rate of adult prey (predator) in patch i be denoted by
these juveniles to migrate between patches. Using dynamic programming theory I found
ai and bi, respectively. If the proportion of juvenile prey and predator from patch i that
optimal harvesting strategies to harvest the predator-prey metapopulation. The results show that
successfully migrate to patch j are pij and qij , respectively, then the above equation can be
some properties of the optimal escapements for a single-species metapopulation are preserved
written as
in the presence of predators, such as the strategies on how to harvest a relative source/sink and
exporter/importer sub-population.
Ni(k+1) = aiNik + piiFi(Nik) + pjiFj(Njk) + αiNikPik,
(1)
Pi(k+1) = biPik + qii [Gi(Pik)+β iNikPik] + qji [Gj(Pjk)+β jNjkPjk],
(2)
where the functions Fi(Nik) and Gi(Pik) are the recruit production functions of the prey and
predator in patch i at time period k, αi negative and β i positive.
2. The Model
Consider a predator-prey metapopulation that coexists in two different patches; patch
If SNik=Nik-HNik (SPik=Pik-HPik) is the escapement of the prey (predator) in patch i at the
one and patch two. The movement of individuals between the local populations is a result of
end of that period, and HNik (HPik) is the harvest taken from the prey (predator), then we obtain
dispersal by juveniles. Adults are assumed to be sedentary, and they do not migrate from one
the dynamic of exploited population
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
Ni(k+1) = aiSNik + piiFi(SNik) + pjiFj(SNjk) + αiSNikSPik,
(3)
Pi(k+1) = biSPik + qii [Gi(SPik)+β iSNikSPik] + qji [Gj(SPjk)+β jSNjkSPjk].
(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
+ prey recruitment from patch j
linearly dependent on population size, whenever the population size is far below the carrying
- the number of prey killed by the predator in patch i,
capacity. With these assumptions, we can choose a logistic function as a recruit production
function, that is Fi(Nik)=riNik(1-Nik/Ki) and Gi(Pik)=siPik(1-Pik/Li), where ri (si) denotes the
where i =1,2 and j =1,2. If we assume that predation affects the predator recruitment then
intrinsic growth of the prey (predator) and Ki (Li) denotes the carrying capacity of the prey
(predator), respectively.
Predator in patch i now
= the number of surviving adult predator from last period
+ predator recruitment from patch i
To find optimal escapements for the metapopulation, I use dynamic programming as in
Supriatna and Possingham (1998) to maximise the net present value
Bulletin of Indonesian Sciences Technology and Economics, Volume 5 Number 2 (1998)
T
PV = ∑ ρ
k =0
k 2
P
∑ ∑Π
Xi
( X ik , S Xik )
5
(5)
i =1 X = N
Bulletin of Indonesian Sciences Technology and Economics, Volume 5 Number 2 (1998)
6
S1. If prey sub-population one is a relative source sub-population, i.e. r1(p11+p12) >
r2(p21+p22), and both predator sub-populations are identical with predator efficiency Ci
subject to the equations (3) and (4), and ignoring all costs of harvesting, to obtain optimal
satisfying max {2Bi/Ki,2mAi/Li}|α2|, and both predator sub-populations are identical with predator
efficiency Ci satisfying max {(-rmBi)/(AiKi),(-smAi)/(BiLi)}β 2/|α2|, and both prey sub-populations are identical with predator efficiency Ci
satisfying max {(-rmBi)/(AiKi),(-smAi)/(BiLi)}max {2Bi/Ki,2mAi/Li} then
∆iS*N2 and S*P1>S*P2. It can be shown that if Ai
limited dispersal of haliotid larvae (genus Haliotis; Mollusca: Gastropoda). J. Exp. Mar.
and Bi are negative and Ci is non-positive with Ci>max {2Bi/Ki,2mAi/Li} then ∆ip21+p22, then let
causes. In Abalone of the World: Biology, Fisheries and Culture, Shepherd et al. (Eds.), pp.
R1=R2=R, S1=S2=S, and s1m=s2m=sm. Let us define ∆SN=(S*N1-S*N2)∆1∆2. Using equations (6)
276-304, Fishing News Books, Oxford.
6. Supriatna, A. K. and H. P. Possingham (1997a). The exploitation of marine metapopulation:
and (7) we obtain
∆ SN
4s 2
2 R
2S
2C
= − m (r2 m − r1m )
−C −
C −
s m (r1m − r2 m )
K
L
KL
L
2
2 B 4 s m R
s m C C −
=
(r2 m − r1m ).
−
K KL
L
a modelling perspective. Mar. Res. Indonesia (submitted).
7. Supriatna, A. K. and H. P. Possingham (1997b). Harvesting a two-patch predator-prey
metapopulation. A paper presented in 1997 World Conference on Natural Resource
Modelling, Hobart, Tasmania.
Since 2B/K