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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 $$\gathered x'(t)= rx(t)\Biggl(1- {x(t)\over k}\Biggr)- {ax(t) y(t)\over by(t)+ x(t)+c}+\lambda x(t)\sin(\omega t),\\ y'(t)= -dy(t)+ {eax(t) y(t)\over by(t)+ x(t)+c},\quad t\ne n\tau,\\ x(t^+)= (1- p_1) x(t),\ t= n\tau,\ y(t^+)= (1- p_2) y(t)+ q,\ (x(0^+), y(0^+))= (x_0, y_0),\endgathered\tag1$$ where $\tau$ is the period of impulsive immigration or stock of the predator, $0\le p_1$, $p_2< 1$ 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:
34C60Qualitative investigation and simulation of models (ODE)
34A37Differential equations with impulses
92D25Population dynamics (general)
34C25Periodic solutions of ODE
WorldCat.org
Full Text: DOI EuDML
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