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Zbl 1162.34007
Nie, Linfei; Peng, Jigen; Teng, Zhidong; Hu, Lin
Existence and stability of periodic solution of a Lotka-Volterra predator-prey model with state dependent impulsive effects.
(English)
[J] J. Comput. Appl. Math. 224, No. 2, 544-555 (2009). ISSN 0377-0427

An impulsive system of Lotka-Volterra-predator-prey type, according to biological and chemical control strategy for pest is constructed \aligned & \left.\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)]\endaligned\right\}\ x\ne h_1h_2,\\ & \left.\aligned & \Delta x(t)=0\\ & \Delta y(t)=y(t^+)-y(t)=\alpha\endaligned\right\}\quad x=h_1\\ & \left.\aligned & \Delta x(t)=x(t^+)-x(t)=-px(t)\\ & \Delta y(t)=y(t^+)-y(t)=-qy(t)\endaligned\right\}\quad x=h_2\endaligned\tag1 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 2000:
*34A37 Differential equations with impulses
34C25 Periodic solutions of ODE
92D25 Population dynamics

Keywords: impulsive differential equations; Runge-Kutta method for stochastic differential equations; Lotka-Volterra predator-prey system

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