Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
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DYNAMIC ANALYSIS OF A PERIODICALLY FORCED HOLLING-TYPE II TWO-PREY ONE-PREDATOR SYSTEM WITH IMPULSIVE CONTROL STRATEGIES

  • Kim, Hye-Kyung (DEPT. OF MATHEMATICS EDUCATION, CATHOLIC UNIVERSITY OF DAEGU) ;
  • Baek, Hun-Ki (DEPT. OF MATHEMATICS, KYUNGPOOK NATIONAL UNIVERSITY)
  • Received : 2010.09.30
  • Accepted : 2010.11.16
  • Published : 2010.12.25
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Abstract

In this paper, we establish a two-competitive-prey and one-predator Holling type II system by introducing a proportional periodic impulsive harvesting for all species and a constant periodic releasing, or immigrating, for the predator at different fixed time. We show the boundedness of the system and find conditions for the local and global stabilities of two-prey-free periodic solutions by using Floquet theory for the impulsive differential equation, small amplitude perturbation skills and comparison techniques. Also, we prove that the system is permanent under some conditions and give sufficient conditions under which one of the two preys is extinct and the remaining two species are permanent. In addition, we take account of the system with seasonality as a periodic forcing term in the intrinsic growth rate of prey population and then find conditions for the stability of the two-prey-free periodic solutions and for the permanence of this system. We discuss the complex dynamical aspects of these systems via bifurcation diagrams.

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