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| In this talk, we study the asymptotic property of a diffusive delayed predator-prey model with Beddington-DeAngelis type functional response under homogeneous Neumann boundary conditions, where the discrete time delay covers the period from the birth of immature preys to their maturity. We establish the threshold dynamics of their permanence and the extinction of the predator. We also give sufficient conditions for the global attractiveness of the semi-trivial and coexistence equilibria. Furthermore, we provide sufficient conditions for the local asymptotical stability at these equilibria under some assumptions.
Biologically, the results imply that the variation of prey stage structure can affect the permanence of the system and drive the predator into extinction by having a large handling time or a low capture rate. Our results show that the predator coexists with prey if the system has sufficiently large mutual interference of predators or low death rate of it.
The methods used in this paper are comparison argument and upper-lower solution method to examine the existence of unique coexistence solution. Furthermore, the linearized stability theory and Lyapunov function theory for the asymptotical stability of equilibria are employed. |
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