Application of Dynamic Programming in Harvesting a Predator-Prey Metapopulation
Asep K. Supriatna
Department of Mathematics, University of Padjadjaran, Indonesia
Abstract
In this paper I use dynamic programming theory to obtain optimal harvesting strategies for a two- patch predator-prey metapopulation. I found that if predator economic efficiency is relatively high then we
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
predator living in the same patch with the relative source prey sub-population. Other rules are also found as generalisations of rules to harvest a single-species metapopulation.
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
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 yang lebih besar dibandingkan dengan eksploitasi predator di tempat lainnya. beberapa aturan umum juga
diperoleh sebagai perluasan dari teori eksploitasi untuk species tunggal.
Keywords: Dynamic Programming, Natural Resource Modelling, Harvesting Theory, Predator-Prey Metapopulation
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
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
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
be smaller than the predators mouth diameter. The probability of prey with body size larger than the predators gape being eaten by the predator is zero. The second type of predator eats
prey by piercing, crushing or sucking it, and hence can eat prey with a relatively larger body size than the predators mouth diameter. In general, the second type of predators also can be
found both in freshwater and marine communities. For example, sea lumprey Petromyzon marinus that prey on many species of fish, such as lake trout, salmon, rainbow trout, whitefish,
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 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 captured by the gape-limited predators is limited by the diameter of the predators mouth, while
the maximum body size of the prey that is captured by the size-dependent predators is only limited by the predator capability in capturing and handling the prey Zaret, 1980. Although it
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.
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
