[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Journal of Applied Nonlinear Dynamics
ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Stability and Hopf Bifurcation of a Predator-Prey Model with Discrete and Distributed Delays

Journal of Applied Nonlinear Dynamics 5(1) (2016) 73--91 | DOI:10.5890/JAND.2016.03.006

Canan Çelik$^{1}$; Emine Degirmenci$^{2}$

Department of Mathematical Engineering, Bahçeşehir University, Istanbul, Turkey

Download Full Text PDF

 

Abstract

In this study, a delayed ratio dependent predator-prey model with both discrete and distributed delays is investigated. First, the local stability of a positive equilibrium is studied and then the existence of Hopf bifurcations is established. By using the normal form theory and center manifold theorem, the explicit algorithm determining the stability, direction of the bifurcating periodic solutions are derived. Finally, we perform the numerical simulations for justifying the theoretical results.

References

  1. [1]  Çelik, C. (2011), Dynamical Behavior of a ratio dependent predator-prey system with distributed delay. Discrete and Cont. Dynam. Systems-Series B, 16, No.3, 719-738.
  2. [2]  Çelik, C. (2008), The stability and Hopf bifurcation for a predator-prey system with time delay. Chaos, Solitons & Fractals, 37, 87-99.
  3. [3]  Çelik, C. (2009), Hopf bifurcation of a ratio-dependent predator-prey system with time delay. Chaos, Solitons & Fractals, 42, 1474-1484.
  4. [4]  Çelik, C. Merdan H., (2013), Hopf bifurcation analysis of a system of coupled delayed-differential equations. Applied Mathematics and Computation, 219, 6605-6617.
  5. [5]  Chen, X. (2007), Periodicity in a nonlinear disrete predator-prey system with state dependent delays. Nonlinear Anal. RWA 8, 435-446.
  6. [6]  Hale, J. (1977), Theory of functional differential equations. Springer Verlag, New York Heidelberg Berlin.
  7. [7]  He, X. (1996), Stability and delays in a predator-prey system. J. Math. Anal. Appl. 198 355-370.
  8. [8]  Gopalsamy, K. (1980), Time lags and global stability in two species competition. Bull Math Biol, 42, 728-737.
  9. [9]  Huo, H.-F., Li, W.-T. (2004), Existence and global stability of periodic solutions of a discrete predator-prey system with delays. Appl. Math. Comput. 153 337-351.
  10. [10]  Jiang, G., Lu, Q. (2007), Impulsive state feedback of a predator-prey model. J. Comput. Appl. Math. 200 193-207.
  11. [11]  Karaoglu, E., Merdan, H. (2014), Hopf bifurcations of a ratio-dependent predator-prey model involving two discrete maturation time delays. Chaos, Solitons & Fractals, 68 159-168.
  12. [12]  Karaoglu, E., Merdan, H. (2014), Hopf bifurcations Analysis for a ratio-dependent predator prey system involving two delays. Anziam Journal, 56 Issue 3, 214-231.
  13. [13]  Krise, S., Choudhury, SR. (2003), Bifurcations and chaos in a predator-prey model with delay and a laserdiode system with self-sustained pulsations. Chaos, Solitons & Fractals 16 59-77.
  14. [14]  Leung, A. (1977), Periodic solutions for a prey-predator differential delay equation. J. Differential Equations 26 391-403.
  15. [15]  Liu, Z., Yuan, R. (2006), Stability and bifurcation in a harvested one-predator-two-prey model with delays. Chaos, Solitons & Fractals, 27, Issue 5 1395-1407.
  16. [16]  Liu, B., Teng, Z., Chen, L. (2006), Analysis of a predator-prey model with Holling II functional response concerning impulsive control strategy. J. Comput. Appl. Math. 193 347-362.
  17. [17]  Liu, X., Xiao, D. (2007), Complex dynamic behaviors of a discrete-time predator-prey system. Chaos, Solitons & Fractals 32 80-94.
  18. [18]  Ma, W., Takeuchi, Y. (1998), Stability analysis on a predator-prey system with disributed delays. J. Comput. Appl. Math. 88 79-94.
  19. [19]  Ruan, S. (2001), Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays. Quart Appl Math 59, 159-173.
  20. [20]  Ruan, S., Wei, J. (1999), Periodic solutions of planar systems with two delays. Proc. Roy. Soc. Edinburgh Sect. A 129 1017-1032.
  21. [21]  Sun, C., Han, M., Lin, Y., Chen, Y. (2007), Global qualitative analysis for a predator-prey system with delay. Chaos, Solitons & Fractals 32 1582-1596.
  22. [22]  Teng, Z., Rehim, M. (2006), Persistence in nonautonomous predator-prey systems with infinite delays. J. Comput. Appl. Math. 197 302-321.
  23. [23]  L.-L. Wang, L.-L., Li, W.-T., Zhao, P.-H. (2004), Existence and global stability of positive periodic solutions of a discrete predator-prey system with delays. Adv. Difference Equ. 4 321-336.
  24. [24]  Wang, F., Zeng, G. (2007), Chaos in Lotka-Volterra predator-prey system with periodically impulsive ratioharvesting the prey and time delays. Chaos, Solitons & Fractals. 32 1499-1512.
  25. [25]  Wen, X., Wang, Z. (2002), The existence of periodic solutions for some models with delay. Nonlinear Anal. RWA 3 567-581.
  26. [26]  Xu, R., Wang, Z. (2006), Periodic solutions of a nonautonomous predator-prey system with stage structure and time delays. J. Comput. Appl. Math. 196 70-86.
  27. [27]  Yafia, R. (2007), Hopf bifurcation in differential equations with delay for tumor immune system competition model. SIAM J. Appl.Math. 67, no 6,1693-1703.
  28. [28]  Yafia, R.(2007), Hopf bifurcation analysis and numerical simulations in an ODE model of the immune system with positive immune response. Nonlinear Anal. Real World Appl, no 5, 1359-1369.
  29. [29]  Yan, X.P., Chu, Y.D. (2006), Stability and bifurcation analysis for a delayed Lotka Volterra predator-prey system. J. Comput. Appl. Math. 196 198-210.
  30. [30]  Yu, J., Zhang, K., Fei, S., Li, T. (2008), Simplified exponential stability analysis for recurrent neural networks with discrete and distributed time-varying delays, Applied Mathematics and Computation, 205 465-474.
  31. [31]  Zhou, S.R., Liu, Y.F., Wang, G. (2005), The stability of predator-prey systems subject to the Allee effects. Theor. Population Biol. 67 23-31.
  32. [32]  Zhou, L., Tang, Y. (2002), Stability and Hopf bifurcation for a delay competition diffusion system. Chaos, Solitons & Fractals 14 1201-1225.
  33. [33]  Zhou, X., Wu, Y., Li, Y., Yau, X. (2009), Stability and Hopf Bifurcation analysis on a two neuron network with discrete and distributed delays. Chaos, Solitons & Fractals 40 1493-1505.
  34. [34]  Çelik, C., Duman, O. (2009), Allee effect in a discrete-time predator-prey system. Chaos, Solitons & Fractals, 40 Issue 4 1956-1962.
  35. [35]  Hassard, N.D., Kazarinoff, Y.H. (1981), Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge.

Copyright © 2011-2023 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates