Hopf bifurcations in a predator-prey system with multiple delays. (English) Zbl 1198.34143

Summary: This paper is concerned with a two species Lotka-Volterra predator-prey system with three discrete delays. By regarding the gestation period of two species as the bifurcation parameter, the stability of positive equilibrium and Hopf bifurcations of nonconstant periodic solutions are investigated. Furthermore, the direction of Hopf bifurcations and the stability of bifurcated periodic solutions are determined by applying the normal form theory and the center manifold reduction for functional differential equations (FDEs). In addition, the global existence of bifurcated periodic solutions are also established by employing the topological global Hopf bifurcation theorem, which shows that the local Hopf bifurcations imply the global ones after the second critical value of parameter. Finally, to verify our theoretical predictions, some numerical simulations are also included.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.


34K18 Bifurcation theory of functional-differential equations
37N25 Dynamical systems in biology
Full Text: DOI


[1] Chen, Y.; Song, C., Stability and Hopf bifurcation analysis in a prey – predator system with stage-structure for prey and time delay, Chaos, solitons & fractals, 38, 1104-1114, (2008) · Zbl 1152.34370
[2] 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
[3] Gan, Q.; Xu, R.; Yang, P., Bifurcation and chaos in a ratio-dependent predator – prey system with time delay, Chaos, solitons & fractals, 39, 1883-1895, (2009) · Zbl 1197.37028
[4] Hassard, B.D.; Kazarinoff, N.D.; Wan, Y.H., Theory and applications of Hopf bifurcation, (1981), Cambridge University Press Cambridge · Zbl 0474.34002
[5] He, X., Stability and delays in a predator – prey system, J math anal appl, 198, 355-370, (1996) · Zbl 0873.34062
[6] Huo, H.F.; Li, W.T.; Nieto, J.J., Periodic solutions of delayed predator – prey model with the beddington – deangelis functional response, Chaos, solitons & fractals, 33, 505-512, (2007) · Zbl 1155.34361
[7] Li, W.T.; Yan, X.P.; Zhang, C.H., Stability and Hopf bifurcation for a delayed cooperation diffusion system with Dirichlet boundary conditions, Chaos, solitons & fractals, 38, 227-237, (2008) · Zbl 1142.35471
[8] Li, W.T.; Wu, S.L., Traveling waves in a diffusive predator – prey model with Holling type-III functional response, Chaos, solitons & fractals, 37, 476-486, (2008) · Zbl 1155.37046
[9] Li, Y.; Xiao, D., Bifurcations of a predator – prey system of Holling and Leslie types, Chaos, solitons & fractals, 34, 606-620, (2007) · Zbl 1156.34029
[10] Liu, X.; Xiao, D., Complex dynamic behaviors of a discrete-time predator – prey system, Chaos, solitons & fractals, 32, 80-94, (2007) · Zbl 1130.92056
[11] Lotka, A.J., Elements of physical biology, (1925), Williams and Wilkins New York · JFM 51.0416.06
[12] May, R.M., Time delay versus stability in population models with two and three trophic levels, Ecology, 4, 315-325, (1973)
[13] Song, Y.; Wei, J., Local Hopf bifurcation and global periodic solutions in a delayed predator – prey system, J math anal appl, 301, 1-21, (2005) · Zbl 1067.34076
[14] Volterra, V., Variazionie fluttuazioni del numero d’individui in specie animali conviventi, Mem acad licei, 2, 31-113, (1926) · JFM 52.0450.06
[15] Wang, W.; Ma, Z., Harmless delays for uniform persistence, J math anal appl, 158, 256-268, (1991) · Zbl 0731.34085
[16] Wei, J.; Ruan, S., Stability and bifurcation in a neural network model with two delays, Physica D, 130, 255-272, (1999) · Zbl 1066.34511
[17] Wu, J., Symmetric functional differential equations and neural networks with memory, Trans amer math soc, 350, 4799-4838, (1998) · Zbl 0905.34034
[18] Yan, X.P.; Chu, Y., Stability and bifurcation analysis for a delayed lokta – volterra predator – prey system, J comp appl math, 196, 579-595, (2006)
[19] Yan, X.P.; Li, W.T., Hopf bifurcation and global periodic solutions in a delayed predator – prey system, Appl math comput, 177, 427-445, (2006) · Zbl 1090.92052
[20] Yan, X.P.; Zhang, C.H., Hopf bifurcation in a delayed lokta – volterra predator – prey system, Nonlinear anal RWA, 9, 114-127, (2008) · Zbl 1149.34048
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