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Existence and stability of periodic solution of a Lotka-Volterra predator-prey model with state dependent impulsive effects. (English) Zbl 1162.34007
An impulsive system of Lotka-Volterra-predator-prey type, according to biological and chemical control strategy for pest is constructed
\[ \begin{aligned} & \left.\begin{aligned} & \frac{dx}{dt}=x(t)[b_1-a_{11} x(t)-a_{12} y(t)]\\ & \frac{dy}{dt}=y(t)[-b_2+a_{21}x(t)]\end{aligned}\right\}\;x\neq h_1h_2,\\ & \left.\begin{aligned} & \Delta x(t)=0\\ & \Delta y(t)=y(t^+)-y(t)=\alpha\end{aligned}\right\}\quad x=h_1\\ & \left.\begin{aligned} & \Delta x(t)=x(t^+)-x(t)=-px(t)\\ & \Delta y(t)=y(t^+)-y(t)=-qy(t)\end{aligned}\right\}\quad x=h_2\end{aligned}\tag{1} \]
Sufficient conditions for the existence of
– stable semi-trivial solution of (1)
– order-1 periodic solution of (1)
– positive locally orbitally stable solution of (1)
– positive order-1 periodic solution
are founded.

MSC:
34A37 Ordinary differential equations with impulses
34C25 Periodic solutions to ordinary differential equations
92D25 Population dynamics (general)
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