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BIFURCATIONS OF A PREDATOR-PREY SYSTEM WITH WEAK ALLEE EFFECTS

  • Lin, Rongzhen (The Academy of Fundamental and Interdisciplinary Science, Harbin Institute of Technology) ;
  • Liu, Shengqiang (The Academy of Fundamental and Interdisciplinary Science, Harbin Institute of Technology) ;
  • Lai, Xiaohong (School of Mathematical Sciences, Xiamen University)
  • Received : 2009.12.17
  • Published : 2013.07.01

Abstract

We formulate and study a predator-prey model with non-monotonic functional response type and weak Allee effects on the prey, which extends the system studied by Ruan and Xiao in [Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math. 61 (2001), no. 4, 1445-1472] but containing an extra term describing weak Allee effects on the prey. We obtain the global dynamics of the model by combining the global qualitative and bifurcation analysis. Our bifurcation analysis of the model indicates that it exhibits numerous kinds of bifurcation phenomena, including the saddle-node bifurcation, the supercritical and the subcritical Hopf bifurcations, and the homoclinic bifurcation, as the values of parameters vary. In the generic case, the model has the bifurcation of cusp type of codimension 2 (i.e., Bogdanov-Takens bifurcation).

Keywords

predator-prey;weak Allee effects;bifurcation;limit cycle

Acknowledgement

Supported by : National Natural Science Foundation of China

References

  1. W. C. Allee, Animal Aggregations, a Study in General Sociology, The University of Chicago Press, Chicago, IL., 1931.
  2. W. C. Allee, The Social Life of Animals, Norton, New York, 1938.
  3. W. C. Allee, The Social Life of Animals, Revised Edition, Beacon Press, Boston, MA, 1958.
  4. R. Bogdanov, Bifurcations of a limit cycle for a family of vector fields on the plane, Selecta Math. Soviet. 1 (1981), 373-388.
  5. R. Bogdanov, Versal deformations of a singular point on the plane in the case of zero eigen-values, Selecta Math. Soviet. 1 (1981), 389-421.
  6. C. W. Clark, Mathematical Bioeconomics, The Optimal Management of Renewable Resources, 2nd edn. John Wiley & Sons Inc., New York, 1990.
  7. A. Deredec and F. Courchamp, Extinction thresholds in host-parasite dynamics, Ann. Zool. Fenn. 40 (2003), 115-130.
  8. C. W. Fowler and J. D. Baker, A review of animal population dynamics at extremely reduced population levels, Rep. Int. Whaling Comm. 41 (1991), 545-554.
  9. A. Kent, C. P. Doncaster, and T. Sluckin, Consequences for predators of rescue and Allee effects on prey, Ecol. Model. 162 (2003), 233-245. https://doi.org/10.1016/S0304-3800(02)00343-5
  10. Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol. 36 (1998), no. 4, 389-406. https://doi.org/10.1007/s002850050105
  11. A. J. Lotka, Elements of Physical Biology, Williams & Wilkins, Baltimore, 1926.
  12. L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 1996.
  13. G. D. Ruxton, W. S. C. Gurney, and A. M. de Roos, Interference and generation cycles, Theoret. Population Biol. 42 (1992), 235-253. https://doi.org/10.1016/0040-5809(92)90014-K
  14. S. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, Theo. Pop. Biol. 64 (2003), no. 2, 201-209. https://doi.org/10.1016/S0040-5809(03)00072-8
  15. P. A. Stephens and W. J. Sutherland, Consequences of the Allee effects for behaviour, ecology and conservation, Trends Ecol. Evol. 14 (1999), no. 10, 401-405. https://doi.org/10.1016/S0169-5347(99)01684-5
  16. P. A. Stephens, W. J. Sutherland, and R. P. Freckleton, What is the Allee effects?, Oikos 87 (1999), 185-190. https://doi.org/10.2307/3547011
  17. F. Takens, Forced oscillations and bifurcations, Applications of global analysis, I (Sympos., Utrecht State Univ., Utrecht, 1973), pp. 1-59. Comm. Math. Inst. Rijksuniv. Utrecht, No. 3 - 1974, Math. Inst. Rijksuniv. Utrecht, Utrecht, 1974.
  18. V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature 118 (1926), 558-560. https://doi.org/10.1038/118558a0
  19. M. H. Wang and M. Kot, Speeds of invasion in a model with strong or weak Allee effects, Math. Biosci. 171 (2001), no. 1, 83-97. https://doi.org/10.1016/S0025-5564(01)00048-7
  20. M. H. Wang, M. Kot, and M. G. Neubert, Integrodifference equations, Allee effects, and invasions, J. Math. Biol. 44 (2002), no. 2, 150-168. https://doi.org/10.1007/s002850100116
  21. G. Wang, X. G. Liang, and F. Z. Wang, The competitive dynamics of populations subject to an Allee effect, Ecol. Model 124 (1999), no. 2-3, 183-192. https://doi.org/10.1016/S0304-3800(99)00160-X
  22. D. Xiao, Bifurcations of saddle singularity of codimension three of a planar vector field with nilpotent linear part, Sci. Sinica A 23 (1993), 252-263.
  23. D. Xiao, L. Jennings, Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM J. Appl. Math. 65 (2005), no. 3, 737-753. https://doi.org/10.1137/S0036139903428719
  24. S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math. 61 (2001), no. 4, 1445-1472. https://doi.org/10.1137/S0036139999361896
  25. A. Yakubu, Multiple attractors in juvenile-adult single species models, J. Difference Equ. Appl. 9 (2003), no. 12, 1083-1098. https://doi.org/10.1080/1023619031000146887
  26. Z. Zhang, T. Ding, W. Huang, and Z. Dong, Qualitative Theory of Differential Equations, Science Press, Beijing, 1992 (in Chinese). English edition: Transl. Math. Monogr., vol. 101, Amer. Math. Soc., Providence, RI, 1992.
  27. S. Zhou, Y. Liu, and G. Wang, The stability of predator-prey systems subject to the Allee effects, Theo. Pop. Biol. 67 (2005), 23-31. https://doi.org/10.1016/j.tpb.2004.06.007
  28. H. Zhu and S. A. Campbell, and G. S. K. Wolkowicz, Bifurcation analysis of a predator- prey system with nonmonotonic fuctional response, SIAM J. Appl. Math. 63 (2005), 636-682.

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