Honam Mathematical Journal (호남수학학술지)

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QUALITATIVE ANALYSIS OF A LOTKA-VOLTERRA TYPE IMPULSIVE PREDATOR-PREY SYSTEM WITH SEASONAL EFFECTS

  • Baek, Hun-Ki (Department of Mathematics, Kyungpook National University)
  • Received : 2008.07.02
  • Accepted : 2008.08.25
  • Published : 2008.09.25
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Abstract

We investigate a periodically forced Lotka-Volterra type predator-prey system with impulsive perturbations - seasonal effects on the prey, periodic releasing of natural enemies(predator) and spraying pesticide at the same fixed times. We show that the solutions of the system are bounded using the comparison theorems and find conditions for the stability of a stable prey-free solution and for the permanence of the system.

Keywords

References

  1. R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics:Ratio-dependence, J. Theor. Biol., 139(1989), 311-326. https://doi.org/10.1016/S0022-5193(89)80211-5
  2. D. K. Arrowsmith and C. M. Place, Dynamical systems: Differential equations, maps and chaotic behaviour, Chapman and Hall(1992).
  3. M. Bardi, Predator-prey models in periodically fluctuatin environment, J. Math. BioI., 12(1981), 127-140.
  4. D.D. Bainov and P.S. Simeonov, Impulsive Differential Equations:Periodic Solutions and Applications, vol. 66, of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Science & Technical, Harlo, UK, 1993.
  5. F. Brauer and C. Castillo-Chcvez, Mathematical models in population biology and epidemiology, Texts in applied mathematics 40, Springer-Verlag, New York(2001).
  6. L, Chen and Z. Jing, The existence and uniqueness of predator-prey differential equations, Chin. Sci, Bull., 9(1984),521-523.
  7. Z. Cheng, Y. Lin and J. Cao, Dynamical behaviors of a partial-dependent predator-prey system, Chaos, Solitons and Fractals, 28(2006), 67-75. https://doi.org/10.1016/j.chaos.2005.05.002
  8. J. B. Collings, The effects of the functional response ont he bifurcation behavior of a mite predator-prey interaction model, J. Math. Biol., 36 (1997), 149-168. https://doi.org/10.1007/s002850050095
  9. J. M. Cushing, Periodic time-dependent predator-prey systems, SIAM J. Appl. Math., 32(1977), 82-95. https://doi.org/10.1137/0132006
  10. J. M. Cushing, Periodic Kolmogorov systems, SIAM J. Math. Anal., 13(1987), 811-827.
  11. H.I. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel Dekker, New York (1980)
  12. S.-B. Hsu and T.-W. Huang, Global stability for a class of predator-prey systems, SIAM J. Appi. Math., 55(3)(1995), 763-783. https://doi.org/10.1137/S0036139993253201
  13. G. Jiang, Q. Lu and L. Qian, Complex dynamics of a Holling type II prey-predator system with state feedback coontrol, Chaos, Solitons and Fractals, 31(2007), 448-461. https://doi.org/10.1016/j.chaos.2005.09.077
  14. A. Lakmeche and O. Arino, Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment, Dynamics of Continuous, Discrete and Impulsive Systems, 1(2000), 265-287.
  15. V Lakshmikantham, D. Bainov, P.Simeonov, Theory of Impulsive Differential Equations, World Scientific Publisher, Singapore, 1989.
  16. X. Liu and L. Chen, Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator, Chaos, Solitons and Fractals, 16(2003), 311-320. https://doi.org/10.1016/S0960-0779(02)00408-3
  17. B. Liu, Z. Teng and L. Chen, Analsis of a predator-prey model with Holling II functional response concerning impulsive control strategy, J. of Compo and Appl. Math., 193(1)(2006), 347-362 https://doi.org/10.1016/j.cam.2005.06.023
  18. B. Liu, Z. Teng and W. Liu, Dynamic behaviors of the periodic Lotka-Volterra competing system with impulsive perturbations, Chaos, Solitons and Fractals, 31(2007),356-370. https://doi.org/10.1016/j.chaos.2005.09.059
  19. B. Liu, Y. Zhang and L. Chen, Dynamic complexities in a Lotka-Volterra predator-prey model concerning impulsive control strategy, Int. J. of Bifur. and Chaos, 15(2)(2005), 517-531. https://doi.org/10.1142/S0218127405012338
  20. B. Liu, Y. Zhang and L. Chen, The dynamical behaviors of a Lotka-Volterra predator-prey model concerning integrated pest mangement, Nonlinear Analysis:Real World Applications, 6(2005), 227-243. https://doi.org/10.1016/j.nonrwa.2004.08.001
  21. Z. Lu, X. Chi and L. Chen, Impulsive control strategies in biological control and pesticide, Theoretical Population Biology, 64(2003), 39-47. https://doi.org/10.1016/S0040-5809(03)00048-0
  22. R. May, Limit cycles in predator prey communities, Science, 177(1972), 900-902. https://doi.org/10.1126/science.177.4052.900
  23. G. C. W. Sabin and D. Summers, Chaos in a periodically forced predator-prey ecosystem model, Math. Bioscience, 113(1993), 91-113. https://doi.org/10.1016/0025-5564(93)90010-8
  24. E. Saez and E. Gonzalez-Olivares, Dynamics of a predator-prey model, SIAM J. Appl. Math., 59(5)(1999), 1867-1878. https://doi.org/10.1137/S0036139997318457
  25. G. T. Skalski and J. F. Gilliam, Functional response with predator interference:Viable alternatives to the holling type II model, Ecology, 82(11(2001), 3083-3092. https://doi.org/10.1890/0012-9658(2001)082[3083:FRWPIV]2.0.CO;2
  26. J. Sugie, R. Kohno and R. Miyazaki, On a predator-prey system of Holling type, Proceedings of the A.M.S., 125(7)(1997), 2041-2050. https://doi.org/10.1090/S0002-9939-97-03901-4
  27. S. Tang, Y. Xiao, L. Chen and R.A. Cheke, Integrated pest management models and their dynamical behaviour, Bulletin of Math. Biol., 67(2005), 115-135. https://doi.org/10.1016/j.bulm.2004.06.005
  28. H. Wang and W. Wang, The dynamical complexity of a Ivlev-type preypredator system with impulsive effect, Chaos, Solitons and Fractals(2007), doi:10.1016/j.chaos.2007.02.008.
  29. S. Zhang, L. Dong and L. Chen, The study of predator-prey system with defensive ability of prey and impulsive perturbations on the predator, Chaos, Solitons and Fractals, 23(2005), 631-643. https://doi.org/10.1016/j.chaos.2004.05.044
  30. S. Zhang, D. Tan and L. Chen, Chaos in periodically forced Holling type II predator-prey system with impulsive perturbations, Chaos, Solitons and Fractals, 28(2006), 367-376. https://doi.org/10.1016/j.chaos.2005.05.037
  31. S. Zhang, O. Tan and L. Chen, Chaos in periodically forced Holling type IV predator-prey system with impulsive perturbations, Chaos, Solitons and Fractals, 27(2006), 980-990. https://doi.org/10.1016/j.chaos.2005.04.065