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Qualitative analysis of a modified Leslie-Gower model with Allee effect in predator. (Chinese. English summary) Zbl 1449.35274
Summary: The stability and existence of positive solutions of a modified Leslie-Gower model with Allee effect in predator are considered, subject to homogeneous Neumann boundary condition. Firstly, the priori estimates of positive solutions are obtained by using the maximum principle and the Harnack inequality, then the asymptotic stability of positive constant solution is acquired by means of stability theory. Secondly, the non-existence of the non-constant positive solutions is verified through the integral property and Poincaré inequality. Finally, based on the methods of Leray-Schauder degree theory, sufficient conditions for the existence of non-constant positive solutions are derived. The fact shows that the two species can be co-existed under some conditions.
35K57 Reaction-diffusion equations
35B09 Positive solutions to PDEs
35B35 Stability in context of PDEs
92D25 Population dynamics (general)
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