Chaos and Hopf bifurcation analysis for a two species predator-prey system with prey refuge and diffusion. (English) Zbl 1222.34099

The authors consider a delayed predator-prey model with Holling type II functional response incorporating a constant prey refuge and diffusion. By analyzing the characteristic equation of the linearized system corresponding to the model, the authors study the local asymptotic stability of the positive equilibrium of the system. By choosing the time delay due to gestation as a bifurcation parameter, the existence of Hopf bifurcations at the positive equilibrium is established. By applying the normal form and the center manifold theory, an explicit algorithm to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived. Finally, using a numerical method, the influence of impulsive perturbations on the dynamics of the system is also investigated.


34K60 Qualitative investigation and simulation of models involving functional-differential equations
34K18 Bifurcation theory of functional-differential equations
34K23 Complex (chaotic) behavior of solutions to functional-differential equations
92D25 Population dynamics (general)
34K20 Stability theory of functional-differential equations
34K17 Transformation and reduction of functional-differential equations and systems, normal forms
34K19 Invariant manifolds of functional-differential equations
34K45 Functional-differential equations with impulses
Full Text: DOI


[1] Gonlez-Olivares, E.; Ramos-Jiliberto, R., Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability, Ecological Modelling, 166, 135-146 (2003)
[2] Chen, L.; Chen, F.; Chen, L., Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a constant prey refuge, Nonlinear Analysis: Real World Applications, 11, 246-252 (2010) · Zbl 1186.34062
[3] Kar, T. K., Modelling and analysis of a harvested prey-predator system incorporating a prey refuge, Journal of Computational and Applied Mathematics, 185, 19-33 (2006) · Zbl 1071.92041
[4] Ji, L.; Wu, C., Qualitative analysis of a predator-prey model with constant-rate prey harvesting incorparating a constant prey refuge, Nonlinear Analysis: Real World Applications, 11, 2285-2295 (2010) · Zbl 1203.34070
[5] Ma, Z., Effects of prey refuges on a predator-prey model with a class of functional responses: the role of refuges, Mathematical Biosciences, 218, 73-79 (2009) · Zbl 1160.92043
[6] Kuang, Y.; Takeuchi, Y., Predator-prey dynamics in models of prey dispersal in two-patch environments, Mathematical Biosciences, 120, 77-98 (1994) · Zbl 0793.92014
[7] Cui, J.; Chen, L., The effect of diffusion on the time varying logistic population growth, Computers & Mathematics with Applications, 36, 1-9 (1998) · Zbl 0934.92025
[8] Cui, J.; Chen, L., Permanence and extinction in logistic and Lotka-Volterra systems with diffusion, Journal of Mathematical Analysis and Applications, 258, 512-535 (2001) · Zbl 0985.34061
[9] Xu, R.; Chen, L., Persistence and stability for a two-species ratio-dependent predator-prey system with time delay in a two-patch environment, Computers and Mathematics with Applications, 40, 577-588 (2000) · Zbl 0949.92028
[10] Hale, J.; Lunel, S. V., Introduction to Functional Differential Equations (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0787.34002
[11] Lian, F.; Xu, Y., Hopf bifurcation analysis of a predator prey system with Holling type IV functional response and time delay, Applied Mathematics and Computation, 215, 1484-1495 (2009) · Zbl 1187.34116
[12] Song, Y.; Yuan, S., Bifurcations analysis in a predator-prey system with time delay, Nonlinear Analysis: Real World Applications, 7, 265-284 (2006) · Zbl 1085.92052
[13] Xu, R.; Gan, Q.; Ma, Z., Stability and bifurcation analysis on a ratio-dependent predator-prey model with time delay, Journal of Computational and Applied Mathematics, 230, 187-203 (2009) · Zbl 1186.34122
[14] Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H., Theory and Applications of Hopf Bifurcation (1981), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0474.34002
[15] Zhang, S.; Chen, L., The study of predator-prey system with defensive ability of prey and impulsive perturbations on the predator, Chaos, Solitons & Fractals, 23, 311-320 (2005)
[16] Xia, Y., Positive periodic solutions for a neutral impulsive delayed Lotka-Volterra competition system with the effect of toxic substance, Nonlinear Analysis: Real World Applications, 8, 204-221 (2007) · Zbl 1121.34075
[17] Meng, X.; Chen, L.; Cheng, H., Two profitless delays for the SEIRS epidemic disease model with nonlinear incidence and pulse vaccination, Applied Mathematics and Computation, 186, 516-529 (2007) · Zbl 1111.92049
[18] Tang, S.; Chen, L., Density-dependent birth rate, birth pulse and their population dynamic consequences, Journal of Mathematical Biology, 44, 185-199 (2002) · Zbl 0990.92033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.