Cyclic competition of three species in the time periodic and diffusive case

We consider a semilinear reaction diffusion system with time periodic coefficients. The system models the competitive interaction of three species, which inhabit a bounded domain. We make assumptions that may be loosely stated (cyclically for i ? {1,2,3}) as: Species i outcompetes species i + 1 in the absence of species i + 2. Under those assumptions we prove the existence of a time periodic solution, which is strictly positive in each of its three components. Moreover, we obtain a new result on the nonexistence of positive time-periodic solutions and the extinction of one species for the related two-species subsystem. Our discussion includes a situation, known as cyclic competition in the autonomous ODE-case, in a more general frame-work that includes temporal and spatial heterogeneity. © Springer-Verlag 1996.