On the fixed point index and multiple steady-state solutions of reaction-diffusion systems

Abstract

This paper is concerned with the fixed point index of a compact operator and its application to the study of multiple steady-state solutions of nonlinear reaction-diffusion systems. The method is simplified under the condition that the Banach space $X$ can be decomposed as $X=Y\oplus S_\varphi$, which is frequently satisfied by various reaction-diffusion models. A new method for proving the existence of positive steady-state solutions is developed by using this simplified method to semiflows. The result is applied to a three-species ecological model for which some sufficient conditions for the existence of positive steady-state solutions are obtained.