Positive Steady-State Solutions of a Competing Reaction-Diffusion System

This paper is concerned with positive steady-state solutions of a coupled reaction-diffusion system which models the coexistence problem of two competing species in ecology. The main purpose of the paper is to determine the set Λ of natural growth rate (r1, r2) of the two competing species so that the coupled system possesses positive solutions. It is shown that Λ is a connected unbounded region in R2+, whose boundary consists of two monotone nondecreasing curves r1=H2(r2) and r2=H1(r1). For every (r1, r2) inside Λ the coupled system has positive solutions and for (r1, r2) outside Λ there exists no positive solution. The functions H1(r1, ) and H2(r2) are constructed in terms of the limit of the corresponding time-dependent solution with a specific initial function.