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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
$\begin{gathered} x'(t)= rx(t)\Biggl(1- {x(t)\over k}\Biggr)- {ax(t) y(t)\over by(t)+ x(t)+c}+\lambda x(t)\sin(\omega t),\\ y'(t)= -dy(t)+ {eax(t) y(t)\over by(t)+ x(t)+c},\quad t\neq n\tau,\\ x(t^+)= (1- p_1) x(t),\;t= n\tau,\;y(t^+)= (1- p_2) y(t)+ q,\;(x(0^+), y(0^+))= (x_0, y_0),\end{gathered}\tag{1}$
where $$\tau$$ is the period of impulsive immigration or stock of the predator, $$0\leq p_1$$, $$p_2< 1$$ 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.

##### MSC:
 34C60 Qualitative investigation and simulation of ordinary differential equation models 34A37 Ordinary differential equations with impulses 92D25 Population dynamics (general) 34C25 Periodic solutions to ordinary differential equations
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