Apart from ordinary slaughter, humanity has a severe influence on the persistence of species due to its intensive use of space and natural resources, thus influencing the quality and amount of a species habitat. Human influence has been beneficial for some species (e.g. [14]), but has also led to the local and even global extinction of many. Since extinction follows a route via low population densities, human activity must be examined as to prevent species from becoming rare.
Metapopulation models provide a tool for the examination of the influence of habitat fragmentation on species persistence. Metapopulation models range from strategic models (e.g. [8],[5],[4],[6]), with the aim of studying qualitative behavior, to tactical models (e.g. [12]) with the aim of providing quantitative predictions of the effect of a specific human activity. Mostly, strategic models are built on heuristic interpretations of a certain level of organization. A better method would be to derive the parameter-sparse strategic models from models containing more biological detail. Verboom et al. [13] provides an example of such an approach for badger populations.
In this paper, continuous-time metapopulation dynamics is analyzed for species which can reach high population densities per patch, and for which migration is such a frequent event that it influences local patch dynamics. Organisms we have in mind include insects, small birds, and small mammals. The model is built at the level of the dynamics of the local patch, and keeps track of the state of all the patches in the metapopulation. Each patch is surrounded by an amount of inhabitable area. We assume an infinite number of equally reachable patches, and a deterministic model on the metapopulation level can be used. On the patch level stochastic processes allow for jumps in the number of individuals. Our modeling approach is similar to that of Gyllenberg & Hanski [4] and Hastings [6]. Gyllenberg & Hanski assume that patches having suffered a total disaster are not colonized immediately and colonization is made dependent on the state of the metapopulation. This assumption is contradicting to their assumption that migration has an influence on local patch dynamics. In our model we only allow for empty patches if they become inhabitable after a total disaster. There are other processes, most likely not influenced by the state of the metapopulation, which cause habitat recovery. Hastings assumes that the state of the metapopulation has only effect on the rate of starting up empty patches. Once initiated patches are only affected by local events. Migration does not contribute to local growth, and therefore does not allow for multiple steady states based on rescue effects.
This modeling approach is made in order to obtain more insight in patch dynamics from underlying local behavior, and insight in the dynamics of the total population of a spatially distributed population. The model is compared to winking-patch models ([8],[7],[3]) and single-population models.
For winking patch models it is assumed that local dynamics are fast compared to colonization and extinction and the population density in occupied patches is set at the carrying capacity. These models can not be used to address questions involving the effect of local growth and migration on metapopulation dynamics. However the winking patch models should be a result of a simplification of underlying processes. For the biological system considered in this paper our model does not simplify to the Levins [8] model exactly, but results carrying an equal message are obtained.
Single population models, e.g. logistic growth, follow from our model if we simplify to dynamics for the mean population size. However when put forward on heuristic arguments they lack insight in spatial events. We show that single population models are very useful for the determination of possible existence of species in a fragmented habitat. That is, one does not need to go to the elaborate analysis presented here if one is only interested in the stability of the trivial steady state. It then suffice to study local patch dynamics adding an additional mortality due to migration.