Application of Dynamic Programming

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)

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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)

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Bulletin of Indonesian Sciences Technology and Economics, Volume 5 Number 2 (1998)

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+ 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

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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