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Seasonal effects on a Beddington-DeAngelis type predator-prey system with impulsive perturbations. (English) Zbl 1187.34058

The authors consider the following periodical impulsive differential equations which represent a predator-prey system

x ' (t)=rx(t)1 - x(t) k-ax(t)y(t) by(t)+x(t)+c+λx(t)sin(ωt),y ' (t)=-dy(t)+eax(t)y(t) by(t)+x(t)+c,tnτ,x(t + )=(1-p 1 )x(t),t=nτ,y(t + )=(1-p 2 )y(t)+q,(x(0 + ),y(0 + ))=(x 0 ,y 0 ),(1)

where τ is the period of impulsive immigration or stock of the predator, 0p 1 , p 2 <1 and x(t), y(t) represent the population densities of prey and predator, respectively.

The authors perform a numerical analysis of (1) for the case without of impulses, i.e. p 1 =p 2 =0. Sufficient conditions for the local asymptotic stability of (1) are derived.

For the periodic solution (0,y * (t)) sufficient conditions for its local asymptotic stability are found. A numerical analysis of seasonal effect and impulsive perturbations is performed.

34C60Qualitative investigation and simulation of models (ODE)
34A37Differential equations with impulses
92D25Population dynamics (general)
34C25Periodic solutions of ODE