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Dynamic complexities of a Holling II two-prey one-predator system with impulsive effect. (English) Zbl 1136.34046
Summary: We investigate the dynamic behavior of a Holling II two-prey one-predator system with impulsive effect concerning biological control and chemical control strategies-periodic releasing natural enemies and spraying pesticide (or harvesting pests) at fixed time. By using the Floquet theory of linear periodic impulsive equation and small-amplitude perturbation we show that there exists a globally asymptotically stable two-prey eradication periodic solution when the impulsive period is less than some critical value. Further, we prove that the system is permanent if the impulsive period is larger than some critical value, and meanwhile the conditions for the extinction of one of the two prey and permanence of the remaining two species are given. Finally, numerical simulation shows that there exists a stable positive periodic solution with a maximum value no larger than a given level. Thus, we can use the stability of the positive periodic solution and its period to control insect pests at acceptably low levels.
See also G. Peng, F. Wang and J. Hui, Int. J. Pure Appl. Math. 34, No. 2, 177–190 (2007; Zbl 1129.34035).

##### MSC:
 34C60 Qualitative investigation and simulation of ordinary differential equation models 34C25 Periodic solutions to ordinary differential equations 92D25 Population dynamics (general) 34D05 Asymptotic properties of solutions to ordinary differential equations 34A37 Ordinary differential equations with impulses
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