Multiple coexistence states for a prey–predator system with cross-diffusion

Abstract

We study the multiple existence of positive solutions for the following strongly coupled elliptic system: Δ[(1+αv)u]+u(a−u−cv)=0 in Ω, Δ[(1+βu)v]+v(b+du−v)=0 in Ω, u=v=0 on ∂Ω, where α, β, a, b, c, d are positive constants and Ω is a bounded domain in R N . This is the steady-state problem associated with a prey–predator model with cross-diffusion effects and u (resp. v) denotes the population density of preys (resp. predators). In particular, the presence of β represents the tendency of predators to move away from a large group of preys. Assuming that α is small and that β is large, we show that the system admits a branch of positive solutions, which is S or ⊃ shaped with respect to a bifurcation parameter. So that the system has two or three positive solutions for suitable range of parameters. Our method of analysis uses the idea developed by Du-Lou (J. Differential Equations 144 (1998) 390) and is based on the bifurcation theory and the Lyapunov–Schmidt procedure.