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Dynamical behaviors of a diffusive predator-prey system with Beddington-DeAngelis functional response. (English) Zbl 1305.34072

Summary: We present a diffusive predator-prey system with Beddington-DeAngelis functional response, where the prey species can disperse between the two patches, and there is competition between the two predators. Sufficient conditions for the permanence and extinction of system are established based on the upper and lower solutions method and comparison theory. Furthermore, the global asymptotic stability of positive solutions is obtained by constructing a suitable Lyapunov function. By using the continuation theorem in coincidence degree theory, we show the periodicity of positive solutions. Finally, we illustrate global asymptotic stability of the model by a simulation.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
34D20 Stability of solutions to ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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