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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

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.

References

  1. H. Baek, Dynamic complexites of a three-species Beddington-DeAngelis system with impulsive control strategy, Acta Appl. Math., 110(1)(2010), 23-38. https://doi.org/10.1007/s10440-008-9378-0
  2. H. Baek, A food chain system with Holling-type IV functional response and impulsive perturbations, Computers and Mathematics with Applications, 60(2010), 1152-1163. https://doi.org/10.1016/j.camwa.2010.05.039
  3. H. Baek, Dynamics of an impulsive food chain system with a Lotka-Volterra functional response , J. of the Korean Society for Industrial and Applied Mathematics, 12(3)(2008), 139-151.
  4. D.D. Bainov and P.S. Simeonov, Impulsive Differential Equations:asymptotic properties of the solutions, Singapore:World Scientific, 1993.
  5. J. B. Collings, The effects of the functional response on the bifurcation behavior of a mite predator-prey interaction model, J. Math. Biol., 36(1997), 149-168. https://doi.org/10.1007/s002850050095
  6. B. A. Croft, Arthropod biological control agents and pesicides. Wiley, New York (1990).
  7. J. M. Cushing, Periodic time-dependent predator-prey systems, SIAM J. Appl. Math. 32(1977), 82-95. https://doi.org/10.1137/0132006
  8. A. Donofrio, Stability properties of pulse vaccination strategy in SEIR epidemic model, Math. Biosci, 179(2002), 57-72. https://doi.org/10.1016/S0025-5564(02)00095-0
  9. A. El-Gohary and A. S. Al-Ruzaiza, Chaos and adaptive control in two prey, one predator system with nonlinear feedback, Chaos, Solitions and Fractals ,34(2007), 443-453. https://doi.org/10.1016/j.chaos.2006.03.101
  10. S. Gakkhar and R. K. Naji, Chaos in seasonally perturbed ratio-dependent prey-predator system, Chaos, Solitons and Fractals, 15(2003), 107-118. https://doi.org/10.1016/S0960-0779(02)00114-5
  11. S. Gakkhar and B. Singh, The dynamics of a food web consisting of two preys and a havesting predator, Chaos, Solitions and Fractals, 34(2007), 1346-1356. https://doi.org/10.1016/j.chaos.2006.04.067
  12. P. Georgescu and G. Morosanu, Impulsive perturbations of a three-trophic prey-dependent food chain system, Mathematical and Computer Modeling(2008), doi:10.1016/j.mcm.2007.12.006.
  13. M. P. Hassell, The dynamics of competition and predation. p.68. Arnod, London (1976).
  14. C. S. Holling, The functional response of predator to prey density and its role in mimicy and population regulatio. Mem. Entomol. Soc. Can., 45(1965), 1-60.
  15. S.-B. Hsu and T.-W. Huang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55(3)(1995), 763-783. https://doi.org/10.1137/S0036139993253201
  16. V Lakshmikantham, D. Bainov, P.Simeonov, Theory of Impulsive Differential Equations, World Scientific Publisher, Singapore, 1989.
  17. 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, 7(2000), 265-287.
  18. 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
  19. B. Liu, Z. Teng and L. Chen, Analsis of a predator-prey model with Holling II functional response concerning impulsive control strategy, J. of Comp. and Appl. Math., 193(1)(2006), 347-362 https://doi.org/10.1016/j.cam.2005.06.023
  20. 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
  21. J. C. Panetta, A mathematical model of periodically pulsed chemotherapy: tumor recurrence and metastasis in a competitive environment, Bull. Math. Biol., 58(1996), 425-447. https://doi.org/10.1007/BF02460591
  22. M. Rafikov , J. M. Balthazar and H.F. von Bremen, Mathematical modeling and control of population systems: Applications in biological pest control, Appl. Math. Comput., 200(2008), 557-573. https://doi.org/10.1016/j.amc.2007.11.036
  23. S. Rinaldi ,S. Muratori S and YA. Kuznetsov Multiple attractors, catastrophes and chaos in seasonally perturbed predator-prey communities. Bull Math. Biol., 55(1993),15-35. https://doi.org/10.1007/BF02460293
  24. M. G. Roberts and R. R. Kao, The dynamics of an infectious disease in a population with birth purses, Math. Biosci., 149(1998), 23-36. https://doi.org/10.1016/S0025-5564(97)10016-5
  25. S. Ruan, D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math, 61(2001), 1445-1472. https://doi.org/10.1137/S0036139999361896
  26. 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
  27. 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
  28. G.T.Skalski and J.F.Gilliam, Funtional responses with predator interference: viable alternatives to the Holling type II mode, Ecology, 82(2001), 3083-3092. https://doi.org/10.1890/0012-9658(2001)082[3083:FRWPIV]2.0.CO;2
  29. B. Shulgin, L. Stone and Z. Agur, Pulse vaccination strategy in the SIR epidemic model, Bull. Math. Biol., 60(1998), 1-26. https://doi.org/10.1006/bulm.1997.0010
  30. B. Shulgin, L. Stone and Z. Agur, Pulse vaccination strategy in the SIR epidemic model, Bull. Math. Biol., 60(1998), 1-26. https://doi.org/10.1006/bulm.1997.0010
  31. X. Song and Y. Li, Dynamic complexities of a Holling II two-prey one-predator system with impulsive effect, Chaos, Solitions and Fractals ,33(2007), 463-478. https://doi.org/10.1016/j.chaos.2006.01.019
  32. S.Y. Tang and L. Chen, Density-dependent birth rate, birth pulse and their population dynamic consequences, J. Math. Biol., 44(2002), 185-199. https://doi.org/10.1007/s002850100121
  33. 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
  34. W.B. Wang , J.H. Shen and J.J. Nieto, Permanence periodic solution of predator prey system with Holling type functional response and impulses, Discrete Dynamics in Nature and Society, 2007, Article ID 81756, 15 pages.
  35. W. Wang, H. Wang and Z. Li, The dynamic complexity of a three-species Beddington-type food chain with impulsive control strategy, Chaos, Solitons and Fractals, 32(2007), 1772-1785. https://doi.org/10.1016/j.chaos.2005.12.025
  36. W. Wang, H. Wang and Z. Li, Chaotic behavior of a three-species Beddington-type system with impulsive perturbations, Chaos Solitons and Fractals, 37(2008), 438-443. https://doi.org/10.1016/j.chaos.2006.09.013
  37. H. Zhang, L. Chen and J.J. Nieto, A delayed epidemic model with stage-structure and pulses for pest management strategy, Nonlinear Anal.:Real World Appl., 9(2008),1714-1726. https://doi.org/10.1016/j.nonrwa.2007.05.004
  38. S. Zhang and L. Chen, Chaos in three species food chain system with impulsive perturbations, Chaos Solitons and Fractals, 24(2005), 73-83.
  39. S. Zhang and L. Chen, A Holling II functional response food chain model with impulsive perturbations, Chaos Solitons and Fractals, 24(2005), 1269-1278. https://doi.org/10.1016/j.chaos.2004.09.051
  40. S. Zhang and L. Chen, A study of predator-prey models with the Beddington-DeAngelis functional response and impulsive effect, Chaos, Solitons and Fractals, 27(2006), 237-248. https://doi.org/10.1016/j.chaos.2005.03.039
  41. S. Zhang, F.Wang and L. Chen, A food chain model with impulsive perturbations and Holling IV functional response, Chaos, Solitons and Fractals, 26(2005), 855-866. https://doi.org/10.1016/j.chaos.2005.01.053
  42. S. Zhang, D. Tan and L. Chen, Dynamic complexities of a food chain model with impulsive perturbations and Beddington-DeAngelis functional response, Chaos Solitons and Fractals, 27(2006), 768-777. https://doi.org/10.1016/j.chaos.2005.04.047
  43. S. Zhang, D. Tan and L. Chen, Chaotic behavior of a periocically forced predator-prey system with Beddington-DeAngelis functional response and impulsive perturbations, Advances in complex Systems, 9(3)(2006), 209-222. https://doi.org/10.1142/S0219525906000811
  44. 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
  45. Y. Zhang, B. Liu and L. Chen, Extinction and permanence of a two-prey one-predator system with impulsive effect, Mathematical Medicine and Biology, 20(2003), 309-325. https://doi.org/10.1093/imammb/20.4.309
  46. Y. Zhang, Z. Xiu and L. Chen, Dynamic complexity of a two-prey one-predator system with impulsive effect, Chaos Solitons and Fractals, 26(2005), 131-139. https://doi.org/10.1016/j.chaos.2004.12.037