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Multiple coexistence states for a prey-predator system with cross-diffusion. (English) Zbl 1205.35116

This paper concerns the nonnegative steady-states of the following parabolic system

u t =Δ[(d 1 +ρ 12 v)u]+u(a 1 -b 1 u-c 1 v)inΩ×(0,+),v t =Δ[(d 2 +ρ 21 u)v]+v(a 2 +b 2 u-c 2 v)inΩ×(0,+),u=v=0,onΩ×(0,+),u(x,0)=u 0 (x)0,v(x,0)=v 0 (x)0,xΩ,

where Ω N (N1) is a bounded domain, ρ 12 , ρ 21 0, a 1 ,α i ,b i ,c i (i=1,2) are positive constants and a 2 .

This is a Lotka-Volterra prey-predator model with cross-diffusion effects. It is shown tha under certain assumptions (on the parameters) the system admits a branch of positive steady-states, which is S or I shaped with respect to a bifurcation parameter. The analysis is based on the bifurcation theory and the Lyapunov-Schmidt procedure.

MSC:
35J65Nonlinear boundary value problems for linear elliptic equations
35B32Bifurcation (PDE)
92D25Population dynamics (general)