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Coexistence states of a nonlinear Lotka-Volterra type predator-prey model with cross-diffusion. (English) Zbl 1238.35162
Summary: A Lotka-Volterra type predator-prey elliptic system with nonlinear cross-diffusion is considered, representing the tendency of predators to get closer to prey under Robin boundary condition. The necessary and sufficient conditions for the existence of coexistence states are given in terms of principal eigenvalues, involving semi-trivial solutions using the degree theory. This paper also discusses the ecological meaning and effect of decreasing cross-diffusion induced on predators by prey in a specific model.
MSC:
35Q92PDEs in connection with biology and other natural sciences
92D25Population dynamics (general)