Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Global Periodic Solutions in a Delayed Predator-Prey System with Holling II Functional Response

  • Received : 2009.06.16
  • Accepted : 2010.01.27
  • Published : 2010.06.30
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Abstract

We consider a delayed predator-prey system with Holling II functional response. Firstly, the paper considers the stability and local Hopf bifurcation for a delayed prey-predator model using the basic theorem on zeros of general transcendental function, which was established by Cook etc.. Secondly, special attention is paid to the global existence of periodic solutions bifurcating from Hopf bifurcations. By using a global Hopf bifurcation result due to Wu, we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay. Finally, several numerical simulations supporting the theoretical analysis are given.

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References

  1. R. Nussbaum, Periodic solutions of some nonlinear autonomous functional equations, Ann. Mat. Pura Appl., 10(1974), 263-306. https://doi.org/10.1007/BF02417109
  2. M. Baptistiini, P. Tboas, On the existence and global bifurcation of periodic solutions to planar differential delay equations, J. Differential Equations, 127(1996), 391-425. https://doi.org/10.1006/jdeq.1996.0075
  3. J. Hale, S. Lunel, Introduction to Functional Differential Equations, in: Appl.Math. Sci., vol. 99, Spring-Verlag, New York, 1993.
  4. J. Wei, Q. Huang, Global existence of periodic solutions of linard equations with time delay, Dynam. Contin. Discrete Impuls. Systems Ser. A, 6(1999), 603-614.
  5. T. Zhao, Y. Kuang and H. Smith, Global existence of periodic solutions in a class of delayed Gause-type predatorCprey systems, Nonlinear Anal., 28(1997), 1373-1394. https://doi.org/10.1016/0362-546X(95)00230-S
  6. L. Erbe, K. Geba, W. Krawcewicz and J. Wu, $S^1$-degree and global Hopf bifurcations, J. Differential Equations, 98(1992), 277-298. https://doi.org/10.1016/0022-0396(92)90094-4
  7. W. Krawcewicz, J. Wu and H. Xia, Global Hopf bifurcation theory for considering fields and neural equations with applications to lossless transmission problems, Canad. Appl. Math. Quart., 1(1993), 167-219.
  8. W. Krawcewicz, J. Wu, Theory and application of Hopf bifurcations in symmetric functional differential equations, Nonlinear Anal., 35(1999), 845-870. https://doi.org/10.1016/S0362-546X(97)00711-6
  9. S. Ruan, J.Wei, Periodic solutions of planar systems with two delays, Proc. Roy. Soc. Edinburgh Sect. A, 129(1999), 1017C1032.
  10. Y. Song, J.Wei and H. Xi, Stability and bifurcation in a neural network model with delay, Differential Equations Dynamic Systems, 9(2001), 321-339.
  11. Y. Song, J.Wei, Local and global Hopf bifurcation in a delayed hematopoiesis, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 14(2004), 3909-3919. https://doi.org/10.1142/S0218127404011697
  12. J. Wei, Y. Li, Hopf bifurcation analysis in a delayed Nicholson Blowflies equation, Nonlinear Anal., 60(2005), 1351-1367. https://doi.org/10.1016/j.na.2003.04.002
  13. J.Wei, Y. Li, Global existence of periodic solutions in a Tri-Neuron Network model with delays, 198(2004), 106-119. https://doi.org/10.1016/j.physd.2004.08.023
  14. X. Wen, Z. Wang, The existence of periodic solutions for some models with delay, Nonlinear Anal. RWA, 3(2002), 567-581. https://doi.org/10.1016/S1468-1218(01)00049-9
  15. J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc., 35(1998), 4799-4838.
  16. S. Jung, On an asymptotic behavior of exponential functional equation, Acta Mathematica Sinica, English Series, 22(2006), 583-586. https://doi.org/10.1007/s10114-005-0557-x
  17. K. Jun, H. Kim, Stability problem for Jensen-type functional equations of cubic mappings, Acta Mathematica Sinica, English Series, 22(2006), 1781-1788. https://doi.org/10.1007/s10114-005-0736-9
  18. L. Hei, J. Wu, Existence and Stability of Positive Solutions for an Elliptic Cooperative System, Acta Mathematica Sinica, English Series, 21(2005), 1113-1120 . https://doi.org/10.1007/s10114-004-0467-3
  19. S. Ruan, Absolute stability, conditional stability and bifurcation in Kolmogorov-type predatorCprey systems with discrete delays, Quart. Appl. Math., 59(2001), 159-173. https://doi.org/10.1090/qam/1811101
  20. J. Wei, S. Ruan, Stability and bifurcation in a neural network model with two delays, Phys. D, 130(1999), 225-272 . https://doi.org/10.1016/S0167-2789(99)00009-3
  21. S. Nakaoka, Y. Saito and Y. Takeuchi, Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey system, Mathematical Biosciences and Engineering, 3(2006), 173-187.
  22. X. He, Stability and delays in a predatorCprey system, J. Math. Anal. Appl., 198(1996), 355-370. https://doi.org/10.1006/jmaa.1996.0087
  23. W. Wang, Z. Ma, Harmless delays for uniform persistence, J. Math. Anal. Appl., 158(1991), 256-268. https://doi.org/10.1016/0022-247X(91)90281-4

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  1. Global Hopf Bifurcation for a Predator–Prey System with Three Delays vol.27, pp.07, 2017, https://doi.org/10.1142/S0218127417501085