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

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.