Persistence and Extinction in Two Species Reaction-Diffusion Systems with Delays

A fundamental problem in population dynamics is to study uniform persistence of the ecosystems, that is, to study the long term survival of interacting species. Abstract persistence theory, started in [BFW, BW], has been well-developed for both continuous and discrete semi-dynamical systems (see, e.g., [FRT, HW, FS, HS1, Th2, YR]) and has been applied to various types of equations including reaction diffusion equations (see, e.g., [CC1, CCH, FLG, HS2, LG, Zh1, Zh2, ZH]) and functional differential equations (see, e.g., [FR]). For more details and references, we refer to a survey paper [HS2]. In realistic ecosystem models, diffusion and time delay should be taken into account. As argued in [Br] and pointed out by the referee, since individuals in the populations are moving, they may not have been at the same location in space at previous times, and the terms involving delay must be nonlocal in space. We refer to [GB1, GB2] for two-species comArticle ID jdeq.1998.3599, available online at http: www.idealibrary.com on