We study the multiple existence of positive solutions for the following strongly coupled elliptic system:where a; b; a; b; c; d are positive constants and O 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. vÞ denotes the population density of preys (resp. predators). In particular, the presence of b represents the tendency of predators to move away from a large group of preys. Assuming that a is small and that b 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. r

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This paper is concerned with a cross-diffusion system arising in a prey-predator population model. The main purpose is to discuss the stability analysis for coexistence steady-state solutions obtained by Kuto and Yamada (J. Differential Equations, to appear). We will give some criteria on the stability of these coexistence steady states. Furthermore, we show that the Hopf bifurcation phenomenon occurs on the steady-state solution branch under some conditions. r

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