Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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EXISTENCE OF PERIODIC SOLUTION AND PERSISTENCE FOR A DELAYED PREDATOR-PREY SYSTEM WITH DIFFUSION AND IMPULSE

  • Shao, Yuanfu (Department of Mathematics, Guilin University of Technology) ;
  • Tang, Guoqiang (Department of Mathematics, Guilin University of Technology)
  • Received : 2011.06.30
  • Accepted : 2011.10.10
  • Published : 2012.05.30
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Abstract

By using Mawhin continuation theorem and comparison theorem, the existence of periodic solution and persistence for a predator-prey system with diffusion and impulses are investigated in this paper. An example and simulation are given to show the effectiveness of the main results.

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References

  1. R. Xu and Z. Ma, The effect of dispersal on the permanence of a predator-prey system with time delay, Nonl. Anal. RWA 9 (2008), 354-369. https://doi.org/10.1016/j.nonrwa.2006.11.004
  2. Q. Liu and S. Dong, Periodic solutions for a delayed predator-prey system with dispersal and impulses, Elec. J. Diff. Equat. 31 (2005), 1-14.
  3. Y.F. Shao. Analysis of a delayed predator-prey system with impulsive diffusion between two patches, Math. Comp. Model., 52 (2010), 120-127. https://doi.org/10.1016/j.mcm.2010.01.021
  4. X. Meng and L. Chen, Periodic solution and almost periodic solution for a nonautonomous Lotka-Volterra dispersal system with infinite delay, J. Math. Anal. Appl. 339 (2008), 125-145 https://doi.org/10.1016/j.jmaa.2007.05.084
  5. L. Dong, L. Chen and P. Shi, Periodic solutions for a two-species nonautonomous competition system with diffusion and impulses, Chaos Solit. Fract. 32 (2007), 1916-1926. https://doi.org/10.1016/j.chaos.2006.01.003
  6. F. Chen, On a nonlinear nonautonomous predator-prey model with diffusion and distributed delay, J. Comp. Appl. Math. 180 (2005), 33-49. https://doi.org/10.1016/j.cam.2004.10.001
  7. Y.F. Shao and B.X. Dai, The dynamics of an impulsive delay predator-prey model with stage structure and Beddington-type functional response, Nonlinear Anal. RWA. 11 (2010), 3567-3576. https://doi.org/10.1016/j.nonrwa.2010.01.004
  8. Y. Takeuchi, Diffusion-mediated persistence in two-species competition Lotka-Volterra model, Math. Biosci. 95 (1989), 65-83. https://doi.org/10.1016/0025-5564(89)90052-7
  9. Y. Takeuchi, Conflict between the need to forage and the need to avoid competition: persistence of two-species model, Math. Biosci. 99 (1990), 181-194. https://doi.org/10.1016/0025-5564(90)90003-H
  10. R. Gains and J. Mawhin, Coincidence degree and nonlinear differential equations, Springer-Verlag, Berlin, 1977.
  11. J.Hale, Theory of Functional Differential Equations, Springer, Heidelberg, 1977.
  12. D.D. Bainov and P.S. Simeonov, Impulsive Differential Equations:periodic solutions and Applications, Longman scientific, Technical, New York, 1993.
  13. G.G. Ding, Introduction to Banach Space, Scientific (in Chinese), Science Publication, Beijing, 1999.
  14. R. Xu, M. Chaplain and F. Davidson, Periodic solution of a Lotka-Voltterra predator-prey model with dispersion and time delay, Appl. Math. Comput. 148 (2004), 537-560.
  15. X. Song and L. Chen, Optimal harvesting and stability for a two-species competitive system with stage structure, Math. Biosci. 170 (2001), 173-186. https://doi.org/10.1016/S0025-5564(00)00068-7
  16. J. Yan and A. Zhao, Oscillation and stability of linear impulsive delay differential equation, J. Math. Anal. Appl. 227 (1998), 187-194. https://doi.org/10.1006/jmaa.1998.6093