Journal of the Korean Society for Industrial and Applied Mathematics

  • 제23권2호
  • /
  • Pages.139-155
  • /
  • 2019
  • /
  • 1226-9433(pISSN)
  • /
  • 1229-0645(eISSN)
한국산업응용수학회 (Korean Society for Industrial and Applied Mathematics)
권호 표지
DOI QR코드

DOI QR Code

DYNAMICS OF A MODIFIED HOLLING-TANNER PREDATOR-PREY MODEL WITH DIFFUSION

  • SAMBATH, M. (DEPARTMENT OF MATHEMATICS, PERIYAR UNIVERSITY) ;
  • BALACHANDRAN, K. (DEPARTMENT OF MATHEMATICS, BHARATHIAR UNIVERSITY) ;
  • JUNG, IL HYO (DEPARTMENT OF MATHEMATICS, PUSAN NATIONAL UNIVERSITY)
  • 투고 : 2018.12.29
  • 심사 : 2019.06.18
  • 발행 : 2019.06.25
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, we study the asymptotic behavior and Hopf bifurcation of the modified Holling-Tanner models for the predator-prey interactions in the absence of diffusion. Further the direction of Hopf bifurcation and stability of bifurcating periodic solutions are investigated. Diffusion driven instability of the positive equilibrium solutions and Turing instability region regarding the parameters are established. Finally we illustrate the theoretical results with some numerical examples.

키워드

참고문헌

  1. J.T. Tanner, The stability and the intrinsic growth rates of prey and predator populations, Ecol, 56 (1975) 855-867. https://doi.org/10.2307/1936296
  2. D.J. Wollkind, J.B. Collings and J.A. Logan, Metastability in a temperature-dependent model system for predator-prey mite Outbreak interactions on fruit fies, Bull. Math. Biol, 50 (1988) 379-409. https://doi.org/10.1016/S0092-8240(88)90005-5
  3. E. Saez and E. Gonzalez-Olivares, Dynamics of a predator-prey model, SIAM J. Appl. Math, 59 (1999) 1867-1878. https://doi.org/10.1137/S0036139997318457
  4. M.P. Hassell, The Dynamics of Arthropod Predator-Prey Systems, Princeton University Press, Princeton, NJ, 1978.
  5. C.S. Holling, The functional response of invertebrate predators to prey density, Mem. Ent. Soc. Can, 45 (1965) 3-60.
  6. R.M. May, Stability and Complexity in Model Eco Systems, Second ed., Princeton Univ. Press, 1974.
  7. J.D. Murray, Mathematical Biology-I: An Introduction, Springer-Verlag, New York, 2002.
  8. J.R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol, 44 (1975) 331-340. https://doi.org/10.2307/3866
  9. D.L. DeAngelis, R.A. Goldstein and R.V. ONeill, A model for trophic interactions, Ecol, 56 (1975) 881-892. https://doi.org/10.2307/1936298
  10. R. Artidi and L.R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence, J. Theoret. Biol, 139 (1989) 311-326. https://doi.org/10.1016/S0022-5193(89)80211-5
  11. P.A. Braza, The bifurcation structure of the Holling-Tanner model for predator-prey interactions using twotiming, SIAM J. Appl. Math, 63 (2003) 889-904. https://doi.org/10.1137/S0036139901393494
  12. S. Chen and J. Shi, Global stability in a diffusive Holling-Tanner predator-prey model, Appl. Math. Lett, 25 (2012) 614-618. https://doi.org/10.1016/j.aml.2011.09.070
  13. Gul Zaman, Yong Han Kang and Il Hyo Jung, Stability analysis and optimal vaccination of an SIR epidemic model, Biosystems, 93 (2008) 240-249. https://doi.org/10.1016/j.biosystems.2008.05.004
  14. S.B. Hsu and T.W. Hwang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math, 55 (1995) 763-783. https://doi.org/10.1137/S0036139993253201
  15. S.B. Hsu and T.W. Huang, Hopf bifurcation analysis for a predator-prey system of Holling and Leslie Type, Taiwan. J. Math, 3 (1999) 35-53. https://doi.org/10.11650/twjm/1500407053
  16. R. Peng and M. Wang, Global stability of the equilibrium of a diffusive Holling-Tanner prey-predator model, Appl. Math. Lett, 20 (2007) 664-670. https://doi.org/10.1016/j.aml.2006.08.020
  17. R. Peng and M. Wang, Stationary patterns of the Holling-Tanner prey-predator model with diffusion and cross-diffusion, Appl. Math. Comput, 196 (2008) 570-577. https://doi.org/10.1016/j.amc.2007.06.019
  18. M. Sambath and K. Balachandran, Pattern formation for a ratio-dependent predator-prey model with cross diffusion, J. Korean Soc. Ind. Appl. Math, 16 (2012) 249-256. https://doi.org/10.12941/jksiam.2012.16.4.249
  19. M. Sambath, S. Gnanavel and K. Balachandran, Stability Hopf Bifurcation of a diffusive predator-prey model with predator saturation and competition, Applicable Analysis, 92 (2013) 2439-2456. https://doi.org/10.1080/00036811.2012.742185
  20. M. Sambath and K. Balachandran, Bifurcations in a diffusive predator-prey model with predator saturation and competition response, Mathematical Models and Methods in Applied Sciences, 38 (2015) 785-798. https://doi.org/10.1002/mma.3106
  21. M. Sambath and K. Balachandran, Influence of diffusion on bio-chemical reaction of the morphogenesis process, Journal of Applied Nonlinear Dynamics, 4 (2015) 181-195. https://doi.org/10.5890/JAND.2015.06.007
  22. M. Sambath, K Balachandran and M Suvinthra, Stability and Hopf bifurcation of a diffusive predator-prey model with hyperbolic mortality, Complexity, 21 (2016) 34-43. https://doi.org/10.1002/cplx.21708
  23. M. Sambath and R. Sahadevan, Hopf bifurcation analysis of a diffusive predator-prey model with Monod-Haldane response, Journal of Mathematical Modeling, 5 (2017) 119-136.
  24. X. Li, W. Jiang and J. Shi, Hopf bifurcation and Turing instability in the reaction-diffusion Holling-Tanner predator-prey model, IMA J. Appl. Math (2011) 1-20.
  25. M. Fan and Y. Kuang, Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response, J. Math. Anal. Appl. 295 (2004) 15-39. https://doi.org/10.1016/j.jmaa.2004.02.038
  26. H.B. Shi, W. Tong Li and G. Lin, Positive steady states of a diffusive predator-prey system with modified Holling-Tanner functional response, Nonlinear Anal. RWA. 11 (2010) 3711-3721. https://doi.org/10.1016/j.nonrwa.2010.02.001
  27. B.D. Hassard, N.D. Kazarinoff and Y.H. Wan, Theory and Applications of Hopf Bifurcation. Camb. Univ. Press, Cambridge, (1981).