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An impulsive predator-prey system with Beddington-DeAngelis functional response and time delay. (English) Zbl 1155.92045

Summary: This paper deals with an impulsive predator-prey model with Beddington-DeAngelis functional response and time delay, in which the evolution of the predators takes them through two stages, juvenile and mature. It is assumed that only mature predators are able to hunt for prey and reproduce and the time delay is understood as being the time spent by the juvenile predators from birth to maturity. It is first seen that the dynamics of the model can be completely determined through the use of a reduced system consisting of the equations for prey and mature predators, respectively. Using the discrete dynamical system determined by the stroboscopic map, one first determines the mature predator-free periodic solution of the reduced system. By means of comparison techniques, one then deduces sufficient criteria for the global stability of the mature predator-free periodic solution and for the permanence of the reduced system, which yield similar properties for the initial system. As a result, it is observed that time delays and pulses have a crucial effect upon the dynamics of our model.

MSC:

92D40 Ecology
34K45 Functional-differential equations with impulses
37N25 Dynamical systems in biology
34K20 Stability theory of functional-differential equations
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