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: Delta[(1 + alphaupsilon)u] + u(a - u - cupsilon) = 0 in Omega, Delta[(1 + betau)upsilon] + upsilon(b + du - upsilon) = 0 in Omega, u = v = 0 on Omega, where alpha, beta, a, b, c, d are positive constants and Omega is a bounded domain in R-N. This is the steadystate problem associated with a prey-predator model with cross-diffusion effects and u (resp. upsilon) denotes the population density of preys (resp. predators). In particular, the presence of beta represents the tendency of predators to move away from a large group of preys. Assuming that a is small and that beta is large, we show that the system admits a branch of positive solutions, which is S or D 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. (C) 2003 Elsevier Inc. All rights reserved.