Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

Permanence of a Three-species Food Chain System with Impulsive Perturbations

  • Baek, Hunki (Department of Mathematics, Kyungpook National University) ;
  • Lee, Hung-Hwan (Department of Mathematics, Kyungpook National University)
  • Received : 2008.05.02
  • Published : 2008.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

We investigate a three-species food chain system with Lotka-Volterra functional response and impulsive perturbations. In [23], Zhang and Chen have studied the system. They have given conditions for extinction of lowest-level prey and top predator and considered the local stability of lower-level prey and top predator eradication periodic solution. However, they did not give a condition for permanence, which is one of important facts in population dynamics. In this paper, we establish the condition for permanence of the three-species food chain system with impulsive perturbations. In addition, we give some numerical examples.

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.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.
  3. 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
  4. C. Cosner, D.L. DeAngelis, Effects of spatial grouping on the functional response of predators, Theoretical Popuation Biology, 56(1999), 65-75. https://doi.org/10.1006/tpbi.1999.1414
  5. H. I. Freedman and R. M. Mathsen, Persistence in predator-prey systems with ratio-dependent predator influence, Bulletin of Math. Biology, 55(4)(1993), 817-827. https://doi.org/10.1007/BF02460674
  6. M. P. Hassell and G. C. Varley , New inductive population model for insect parasites and its bearing on biological control, Nature, 223(1969), 1133-1136. https://doi.org/10.1038/2231133a0
  7. 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
  8. S.-B. Hsu, T.-W. Hwang and Y. Kuang, Global dynamics of a predator-prey model with Hassell-Varley type functional response, Submittion for publication.
  9. K. Kitamura, K. Kashiwagi, K.-i. Tainaka, T. Hayashi, J. Yoshimura, T. Kawai and T. Kajiwara, Asymmetrical effect of migration on a prey-predator model, Physics Letters A, 357(2006), 213-217. https://doi.org/10.1016/j.physleta.2006.04.067
  10. V Lakshmikantham, D. Bainov, P.Simeonov, Theory of Impulsive Differential Equations, World Scientific Publisher, Singapore, 1989.
  11. 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
  12. B. Liu, Y. J. Zhang, L. S. Chen and L. H. Sun, The dynamics of a prey-dependent consumption model concerning integrated pest management, Acta Mathematica Sinica, English Series, 21(3)(2005), 541-554. https://doi.org/10.1007/s10114-004-0476-2
  13. 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
  14. S. Ruan and D. Xiao, Golbal analysis in a predator-prey sytem with non-monotonic functional response, SIAM J. Appl. Math., 61(4)(2001), 1445-1472. https://doi.org/10.1137/S0036139999361896
  15. 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
  16. 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
  17. W. Wang, X. Wang and Y. Lin, Complicated dynamics of a predator-prey system with Watt-type functional response and impulsive control strategy, Chaos, Solitons and Fractals(2006), doi:10.1016/j.chaos.2006.10.032.
  18. Xiaoqin Wang, Weiming Wang, Yezhi Lin and Xiolin Lin, The dynamical complexity of an impulsive Watt-type prey-predator system, Chaos, Solitons and Fractals(2007), doi:10.1016/j.chaos.2007.08.019.
  19. H. Wang and W. Wang, The dynamical complexity of a Ivlev-type prey-predator system with impulsive effect, Chaos, Solitons and Fractals(2007), doi:10.1016/j.chaos.2007.02.008.
  20. 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
  21. 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
  22. Z. Xiang and X. Song, The dynamical behaviors of a food chain model with im- pulsive effect anf Ivlev functional response, Chaos, Solitons and Fractals(2007), doi:10.1016/j.chaos.2007.06.124.
  23. S. Zhang and L. Chen, Chaos in three species food chain system with impulsive perturbations, Chaos Solitons and Fractals, 24(2005), 73-83. https://doi.org/10.1016/j.chaos.2004.07.014
  24. 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
  25. 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
  26. 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
  27. S. Zhang, D. 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
  28. 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

Cited by

  1. Dynamic Complexities of a Three-Species Beddington-DeAngelis System with Impulsive Control Strategy vol.110, pp.1, 2010, https://doi.org/10.1007/s10440-008-9378-0
  2. Complex Dynamic Behaviors of an Impulsively Controlled Predator-prey System with Watt-type Functional Response vol.56, pp.3, 2016, https://doi.org/10.5666/KMJ.2016.56.3.831
  3. Stability and bifurcation analysis on a three-species food chain system with two delays vol.16, pp.9, 2011, https://doi.org/10.1016/j.cnsns.2010.12.042
  4. Impulsive Effect on Tri-Trophic Food Chain Model with Mixed Functional Responses under Seasonal Perturbations 2016, https://doi.org/10.1007/s12591-016-0328-4
  5. Hopf bifurcations in a three-species food chain system with multiple delays vol.15, pp.1, 2017, https://doi.org/10.1515/math-2017-0039