Permanence and Extinction in Logistic and Lotka Volterra Systems with Diffusion1

In this paper we consider the effect of diffusion on the permanence and extinction of single and multiple endangered species that live in changing patch environments. Differing from former studies, our discussion includes the important situation in conservation biology in which species live in a weak patchy environment, in the sense that species will become extinct in some of the isolated patches. For the single population model, we show that identical species can persist for some diffusion rates and can also vanish for another set of restrictions on the diffusion rates, although the single endangered species will vanish in some isolated patches without contributions from other patches. Furthermore, we consider the existence, uniqueness, and global stability of the positive periodic solution. For prey predator systems we can make both the prey and the predator species permanent by choosing the diffusion rates appropriately, even if the prey species has a negative intrinsic growth rate in some patches. Moreover, we introduce an exotic competitive species y into the habitat occupied by the native species x. Competitive permanence and competitive exclusion both are considered. The implications of these results are significant for the conservation of endangered species.