- Volume 49 Issue 4
A stage-structured predator-prey system with time delay and Beddington-DeAngelis functional response is considered. By analyzing the corresponding characteristic equation, the local stability of a positive equilibrium is investigated. The existence of Hopf bifurcations is established. Formulae are derived to determine the direction of bifurcations and the stability of bifurcating periodic solutions by using the normal form theory and center manifold theorem. Numerical simulations are carried out to illustrate the theoretical results.
stability;Hopf bifurcation;predator-prey system;stage structure;time delay;Beddington-DeAngelis functional response
- J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., 44(1975), 331-340. https://doi.org/10.2307/3866
- F. Berezovskaya, G. Karev, R. Arditi, Parametric analysis of the ratio-dependent predator-prey model, J. Math. Biol., 43(2001), 221-246. https://doi.org/10.1007/s002850000078
- K. Cooke, Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl., 86(1982), 592-627. https://doi.org/10.1016/0022-247X(82)90243-8
- D. L. DeAngelis, R. A. Goldstein, R. V. O'Neill, A model for trophic interaction, Ecology, 56(1975), 881-892. https://doi.org/10.2307/1936298
- B. Hassard, N. Kazarinoff, Y. H. Wan, Theory and Applications of Hopf Bifurcation, London Math Soc. Lect. Notes, Series, 41. Cambridge: Cambridge Univ. Press, 1981.
- Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993.
- Y. Kuang, E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36(1998), 389-406. https://doi.org/10.1007/s002850050105
- S. Li, X. Liao and C. Li, Hopf bifurcation in a Volterra prey-predator model with strong kernel, Chaos, Solitons & Fractals, 22(2004), 713-722. https://doi.org/10.1016/j.chaos.2004.02.048
- R. M. May, Stability and Complexity in Model Ecosystem, Princeton Univ. Press, Princeton, 1974.
- C. Sun, M. Han and Y. Lin, Analysis of stability and Hopf bifurcation for a delayed logistic equation, Chaos, Solitons & Fractals, 31(2007), 672-682. https://doi.org/10.1016/j.chaos.2005.10.019
- W. Sun, A stage-structure predator-prey system with Beddington-DeAngelis functional response, J. Southwest China Normal University, 30(5)(2005), 800-804.
- W.Wang, L. Chen, A predator-prey system with stage-structure for predator, Comput. Math. Appl., 33(1997), 83-91.