Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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STABILITY AND BIFURCATION ANALYSIS FOR A TWO-COMPETITOR/ONE-PREY SYSTEM WITH TWO DELAYS

  • Cui, Guo-Hu (Department of Mathematics Lanzhou Jiaotong University) ;
  • Yany, Xiang-Ping (Department of Mathematics Lanzhou Jiaotong University)
  • Received : 2010.07.10
  • Published : 2011.11.01
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Abstract

The present paper is concerned with a two-competitor/oneprey population system with Holling type-II functional response and two discrete delays. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the positive equilibrium and existence of local Hopf bifurcations are investigated. Particularly, by applying the normal form theory and the center manifold reduction for functional differential equations (FDEs) explicit formulae determining the direction of bifurcations and the stability of bifurcating periodic solutions are derived. Finally, to verify our theoretical predictions, some numerical simulations are also included at the end of this paper.

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References

  1. H. Boudjellaba and T. Sari, Stability loss delay in harvesting competing populations, J. Differential Equations 152 (1999), no. 2, 394-408. https://doi.org/10.1006/jdeq.1998.3533
  2. G. J. Butler and P. Waltman, Bifurcation from a limit cycle in a two predator-one prey ecosystem modeled on a chemostat, J. Math. Biol. 12 (1981), no. 3, 295-310. https://doi.org/10.1007/BF00276918
  3. J. M. Cushing, Periodic two-predator, one-prey interactions and the time sharing of a resource niche, SIAM J. Appl. Math. 44 (1984), no. 2, 392-410. https://doi.org/10.1137/0144026
  4. O. Diekmann, R. Nisbet, W. Gurney, and F. Bosch, Simple mathematical models for cannibalism: a critique and a new approach, Math. Biosci. 78 (1986), no. 1, 21-46. https://doi.org/10.1016/0025-5564(86)90029-5
  5. H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel Dekker, New York, 1980.
  6. J. K. Hale, Theory of Functional Differential Equations, Spring-Verlag, New York, 1977.
  7. B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf bifurcation, Cambridge University Press, Cambridge, 1981.
  8. S.-B. Hsu, S. P. Hubbell, and P. Waltman, Competing predators, SIAM J. Appl. Math. 35 (1978), no. 4, 617-625. https://doi.org/10.1137/0135051
  9. T. K. Kar and A. Batabyalb, Persistence and stability of a two prey one predator system, International Journal of Engineering, Science and Technology 2 (2010), 174-190.
  10. A. J. Lotka, Elements of Physical Biology, New York, Williams and Wilkins, 1925.
  11. Z. Liu and R. Yuan, Stability and bifurcation in a harmonic oscillator with delays, Chaos Solitons Fractals 23 (2005), no. 2, 551-562. https://doi.org/10.1016/j.chaos.2004.05.038
  12. Z. Lu and X. Liu, Analysis of a predator-prey model with modified Holling-Tanner functional response and time delay, Nonlinear Anal. Real World Appl. 9 (2008), no. 2, 641-650. https://doi.org/10.1016/j.nonrwa.2006.12.016
  13. S. Muratori and S. Rinaldi, Remarks on competitive coexistence, SIAM J. Appl. Math. 49 (1989), no. 5, 1462-1472. https://doi.org/10.1137/0149088
  14. J. D. Murray, Mathematical Biology, Biomathematics 19, Sringer-Verlag, Berlin Heidel- berg, 1989.
  15. B. Patra, A. Maiti, and G. P. Samanta, Effect of time-delay on a ratio-dependent food chain model, Nonlinear Anal. Model. Control 14 (2009), no. 2, 199-216.
  16. S. Ruan, A. Ardito, P. Ricciardi, and D. DeAngelis, Coexistence in competition models with density-dependent mortality, Comptes Rend. Biol 330 (2007), 845-854. https://doi.org/10.1016/j.crvi.2007.10.004
  17. H. L. Smith, The interaction of steady state and Hopf bifurcations in a two-predator-one-prey competition model, SIAM J. Appl. Math. 42 (1982), no. 1, 27-43.
  18. Y. Song and S. Yuan, Bifurcation analysis in a predator-prey system with time delay, Nonlinear Anal. Real World Appl. 7 (2006), no. 2, 265-284. https://doi.org/10.1016/j.nonrwa.2005.03.002
  19. Y. Takeuchi, Global Dynamical Propertices of Lotka-Volterra Systems, World Scientific, Singapore, 1996.
  20. V. Volterra, Variazionie fluttuazioni del numero dindividui in specie animali conviventi, Mem. Acad. Licei 2 (1926), 31-113.
  21. J. Wei and S. Ruan, Stability and bifurcation in a neural network model with two delays, Phys. D 130 (1999), no. 3-4, 255-272. https://doi.org/10.1016/S0167-2789(99)00009-3
  22. J. Wei and M. Li, Global existence of periodic solutions in a tri-neuron network model with delays, Phys. D 198 (2004), no. 1-2, 106-119. https://doi.org/10.1016/j.physd.2004.08.023
  23. D. Xiao and W. Li, Stability and bifurcation in a delayed ratio-dependent predator-prey system, Proc. Edinb. Math. Soc. (2) 46 (2003), no. 1, 205-220.
  24. Y. Xu, Q. Gan, and Z. Ma, Stability and bifurcation analysis on a ratio-dependent predator-prey model with time delay, J. Comput. Appl. Math. 230 (2009), no. 1, 187-203. https://doi.org/10.1016/j.cam.2008.11.009
  25. X. Yan, Stability and Hopf bifurcation for a delayed prey-predator system with diffusion effects, Appl. Math. Comput. 192 (2007), no. 2, 552-566. https://doi.org/10.1016/j.amc.2007.03.033

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