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Global dynamics of a predator-prey model with stage structure for the predator. (English) Zbl 1120.92045
Summary: The global properties of a predator-prey model with nonlinear functional response and stage structure for the predator are studied using Lyapunov functions and LaSalle’s invariance principle. It is found that, under hypotheses which ensure the uniform persistence of the system and the existence of a unique positive steady state, a feasible a priori lower bound condition on the abundance of the prey population ensures the global asymptotic stability of the positive steady state. A condition which leads to the extinction of the predators is indicated. We also obtain results on the existence and stability of periodic solutions. In particular, when a condition in the Routh-Hurwitz theorem fails to hold and the unique positive steady state $E^*$ becomes unstable, the coexistence of prey and predator populations is ensured for initial populations not on the one-dimensional stable manifold of $E^*$, albeit with fluctuating population sizes.

MSC:
92D40Ecology
37N25Dynamical systems in biology
34D20Stability of ODE
34D23Global stability of ODE
93D20Asymptotic stability of control systems
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