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Next: A partial differential equation Up: A Deterministic Size-Structured Metapopulation Previous: Introduction

The model

In the metapopulation model we consider an infinite number of equal patches. The metapopulation is represented by the fraction of patches unsuitable for habitation tex2html_wrap_inline1962 , and a measure tex2html_wrap_inline1964 on habitable patches with local population density tex2html_wrap_inline1966 . Since we assume that loss or creation of new patches does not occur, the metapopulation dynamics is constrained by conservation, i.e.


Individuals within a patch are assumed to be well mixed, and demographic events are not separated in time, i.e. birth, death, and migration of an individual can happen at any time of year. We consider local populations where the number of individuals is so high that demographic stochasticity can be neglected, and a deterministic model for local population growth can be used. We propose, following Gyllenberg & Hanski [4] that immigration into patches is such a frequent event that it influences the local dynamics. The rate of change in the local density x is


where g(x) describes the birth and death process, tex2html_wrap_inline1972 is the rate of emigration, and a the per capita rate of immigration of a disperser from the pool of dispersers (D).

The dispersers are assumed to be homogeneously mixed over space. Therefore we can restrict ourselves to modeling the patches, do not regard any space different from patch, and concentrate on the mean density of dispersing individuals per patch formerly defined as D. The rate of change in D is equal to


where l is the death rate of a dispersing individual.

A local population suffers from disasters. Disasters happen at a rate tex2html_wrap_inline1984 . This rate depends on the population density, allowing for a dependence on biotic factors such as diseases. A disaster results in a decline in population size. The probability that a population of size y falls to a population of size x;SPMgt;0 is tex2html_wrap_inline1990 and the probability of a total disaster is tex2html_wrap_inline1992 . Note that since a patch cannot vanish we must have


On the metapopulation level we will provide two technically distinct formulations of the same model. We first present a partial differential equation, which is conceptually close to the model of Levins [8]. The second more general model, a Stieltjes renewal equation is beneficial since it circumvents the elaborate job of justifying the use of the pde [2]. Furthermore it is easier to obtain stability conditions for the equilibria.

next up previous
Next: A partial differential equation Up: A Deterministic Size-Structured Metapopulation Previous: Introduction

John Val
Wed Feb 26 07:30:07 EST 1997