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Bifurcation analysis in a predator - prey system with time delay. (English) Zbl 1085.92052
Summary: A predator-prey system with a discrete delay and a distributed delay is investigated. We first consider the stability of the positive equilibrium and the existence of local Hopf bifurcations. In succession, using the normal form theory and a center manifold argument, we derive explicit formulas determining stability, direction and other properties of bifurcating periodic solutions. Finally, several numerical simulations for supporting the theoretical analysis are also given.

34K13Periodic solutions of functional differential equations
34K18Bifurcation theory of functional differential equations
37N25Dynamical systems in biology
34K60Qualitative investigation and simulation of models
Full Text: DOI
[1] Beretta, E.; Kuang, Y.: Convergence results in a well-known delayed predator -- prey system. J. math. Anal. appl. 204, 840-853 (1996) · Zbl 0876.92021
[2] Berryman, A. A.: The origins and evolution of predator -- prey theory. Ecology 73, 1530-1535 (1992)
[3] Cushing, J. M.: Integrodifferential equations and delay models in population dynamics. (1977) · Zbl 0363.92014
[4] Dodd, R. K.: Periodic orbits arising from Hopf bifurcations in a Volterra prey -- predator model. J. math. Biol. 35, 432-452 (1997) · Zbl 0866.92017
[5] Faria, T.: Stability and bifurcation for a delayed predator -- prey model and the effect of diffusion. J. math. Anal. appl. 254, 433-463 (2001) · Zbl 0973.35034
[6] Freedman, H. I.; Ruan, S. G.: Uniform persistence in functional differential equations. J. differential equations 115, 173-192 (1995) · Zbl 0814.34064
[7] Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H.: Theory and applications of Hopf bifurcation. (1981) · Zbl 0474.34002
[8] He, X. -Z.: Stability and delays in a predator -- prey system. J. math. Anal. appl. 198, 335-370 (1996) · Zbl 0873.34062
[9] Kuang, Y.: Delay differential equations with applications in population dynamics. (1993) · Zbl 0777.34002
[10] Ruan, S. G.; Wei, J. J.: On the zeros of transcendental functions with applications to stability of delay differential equations with two delays. Dynam. cont. Dis. 10, 863-874 (2003) · Zbl 1068.34072
[11] Y.L. Song, M.A. Han, J.J. Wei, Stability and global Hopf bifurcation for a predator -- prey model with two delays, Chinese Ann. Math. Ser. A 25 (2004) 783 -- 790. · Zbl 1069.34104
[12] Wang, W. D.; Ma, Z. E.: Harmless delays for uniform persistence. J. math. Anal. appl. 158, 256-268 (1991) · Zbl 0731.34085
[13] Zhao, T.; Kuang, Y.; Smith, H. L.: Global existence of periodic solutions in a class of delayed gause-type predator -- prey systems. Nonlinear anal. 28, 1373-1394 (1997) · Zbl 0872.34047