In the metapopulation model we consider an infinite number of equal patches. The metapopulation is represented by the fraction of patches unsuitable for habitation , and a measure on habitable patches with local population density . 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,
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 . 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 and the probability
of a total disaster is . 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.

- A partial differential equation at the metapopulation level.
- A Stieltjes renewal equation at the metapopulation level.

Wed Feb 26 07:30:07 EST 1997