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On the fixed point index and multiple steady-state solutions of reaction- diffusion systems. (English) Zbl 0815.35017
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.
Reviewer: J.F.Toland (Bath)

##### MSC:
 35J60 Nonlinear elliptic equations 35K57 Reaction-diffusion equations 92D25 Population dynamics (general)