Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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PERIODIC SOLUTIONS FOR DISCRETE ONE-PREDATOR TWO-PREY SYSTEM WITH THE MODIFIED LESLIE-GOWER FUNCTIONAL RESPONSE

  • Shi, Xiangyun (College of Mathematics and Science, Xinyang Normal University) ;
  • Zhou, Xueyong (College of Mathematics and Science, Xinyang Normal University) ;
  • Song, Xinyu (College of Mathematics and Science, Xinyang Normal University)
  • Published : 2009.05.31
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Abstract

In this paper, we study a discrete Leslie-Gower one-predator two-prey model. By using the method of coincidence degree and some techniques, we obtain the existence of at least one positive periodic solution of the system. By linalization of the model at positive periodic solution and construction of Lyapunov function, sufficient conditions are obtained to ensure the global stability of the positive periodic solution. Numerical simulations are carried out to explain the analytical findings.

Keywords

References

  1. A.F. Nindjin, M.A, Aziz-Alaoui and M. Cadivel, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with time delay, Nonlinear Anal. Real World Appl. 7 (2006), 1104-1118. https://doi.org/10.1016/j.nonrwa.2005.10.003
  2. R.K. Upadhyay and S.R.K. Iyengar, Effect of seasonality on the dynamics of 2 and 3 species prey-predator system, MNonlinear Anal.: Real World Appl. 6 (2005), 509-530. https://doi.org/10.1016/j.nonrwa.2004.11.001
  3. Y. Li and D. Xiao, Bifurcations of a predator-prey system of Holling and Leslie types,Chaos Solitons Fractals 34 (2007), 606-620. https://doi.org/10.1016/j.chaos.2006.03.068
  4. S. Gakkhar and B. Singh, Dynamics of modified Leslie-Gower-type prey-predator model with seasonally varying parameters, haos Solitons Fractals 27 (2006), 1239-1255. https://doi.org/10.1016/j.chaos.2005.04.097
  5. P.H. Leslie, Some further notes on the use of matrices in population mathematics, Biometrica 35(1948), 213-245.
  6. H.F. Huo and W.T. Li, Stable periodic solution of the discrete periodic Leslie-Gower predator-prey model, Math. Comput. Modelling 40 (2004), 261-269. https://doi.org/10.1016/j.mcm.2004.02.026
  7. Hai-Feng Huo and Wan-Tong Li, Existence and global stability of periodic solutions of a discrete ratio-dependent food chain model with delay, Applied Mathematics and Computation 162(3) (2005), 1333-1349. https://doi.org/10.1016/j.amc.2004.03.013
  8. Hai-Feng Huo and Wan-Tong Li, Existence and global stability of periodic solutions of a discrete predator-prey system with delays, Applied Mathematics and Computation 153(2)(2004), 337-351. https://doi.org/10.1016/S0096-3003(03)00635-0
  9. R.E. Gaines and J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer, Berlin, 1977.
  10. M. Fan and K. Wang, Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system, Math. Comput. Modelling 35(9-10)(2002), 951-961. https://doi.org/10.1016/S0895-7177(02)00062-6
  11. R.P. Agarwal, Difference Equations and Inequalities: Theory, Method and Applications, Monographs and textbooks in Pure and Applied Mathematics, No. 228, Marcel Dekker, New York, 2000.