Multiple existence and stability of steady-states for a prey-predator system with cross-diffusion
Volume 66 / 2004
Banach Center Publications 66 (2004), 199-210
MSC: Primary 35J65; Secondary 92D25.
DOI: 10.4064/bc66-0-13
Abstract
This article discusses a prey-predator system with cross-diffusion. We obtain multiple positive steady-state solutions of this system. More precisely, we prove that the set of positive steady-states possibly contains an S or $\supset$-shaped branch with respect to a bifurcation parameter in the large cross-diffusion case. Next we give some criteria on the stability of these positive steady-states. Furthermore, we find the Hopf bifurcation point on the steady-state solution branch in a certain case. Our method of analysis uses the idea developed by Du and Lou \cite{DL} and is based on the bifurcation theory and the Lyapunov-Schmidt reduction technique.
