Persistence and Extinction of Populations in Finite Domains

Abstract

We now turn our attention to steady states of reaction–transport systems. We focus first on steady states that arise in RD models on finite domains. Such models are important from an ecological point of view, since they describe population dynamics in island habitats. The main problem consists in determining the critical patch size, i.e., the smallest patch that can minimally sustain a population. As expected intuitively, the critical patch size depends on a number of factors, such as the population dynamics in the patch, on the nature of the boundaries, the patch geometry, and the reproduction kinetics of the population. The first critical patch model was studied by Kierstead and Slobodkin [228] and Skellam [414] and is now called the KISS problem. A significant amount of work has focused on systems with partially hostile boundaries, where individuals can cross the boundary at some times but not at others, or systems where individuals readily cross the boundary but the region outside the patch is partially hostile, or a combination of the above. In this chapter we deal with completely hostile boundaries and calculate the critical patch size for different geometries, reproduction processes, and dynamics.