Competing species near a degenerate limit

We consider a competitive reaction-diffusion model of two species in a bounded domain which are identical in all aspects except for their birth rates, which differ by a function g. Under a fairly weak hypothesis, the semitrivial solutions always exist. Our analysis provides a description of the stability of these solutions as a function of the diffusion rate μ and the difference between the birth rates g. In the case in which the magnitude of g is small we provide a fairly complete characterization of the stability in terms of the zeros of a single function. In particular, we are able to show that for any fixed number n, one can choose the difference function g from an open set of possibilities in such a way that the stability of the semitrivial solutions changes at least n times as the diffusion n is varied over (0, ∞). This result allows us to make conclusions concerning the existence of coexistence states. Furthermore, we show that under these hypotheses, coexistence states are unique if they exist. The biological implication is that there is a delicate balance between resource utilization and dispersal rates which can have a dramatic impact with regards to extinction. Furthermore, we show that there is no optimal form of resource utilization. To be more precise, given a fixed diffusion rate and a particular spatially dependent utilization of resources which are expressed in terms of the birth rate, there always exists a birth rate, which on average is the same but differs pointwise, which allows the corresponding species to invade.