Random population dispersal in a linear hostile environment.

We consider the Fisher equation and its generalization for an asocial population in a linear, hostile environment. The method of center manifold analysis is used to obtain the time-dependent solution of the former, nonlinear equation. The correct critical habitat size is obtained; in addition, the result for the steady state central density compares favorably with the exact result for relatively large population sizes (up to one half of the carrying capacity). For a model of asocial growth we obtain the expanded criteria for survival. This includes the habitat size, the population size at which positive growth begins, and also the minimum initial central density.