zbMATH — the first resource for mathematics

Stability and bifurcation analysis of a discrete predator-prey model with nonmonotonic functional response. (English) Zbl 1215.92063
Summary: The paper studies the dynamical behavior of a discrete predator-prey system with nonmonotonic functional response. The local stability of equilibria of the model is obtained. The model undergoes flip bifurcations and Hopf bifurcations by using the center manifold theorem and bifurcation theory. Numerical simulations not only illustrate our results, but also exhibit the complex dynamical behavior of the model, such as the period-doubling bifurcation in periods 2, 4 and 8, and quasi-periodic orbits and chaotic sets. The most interesting aspect is choosing the same parameters and the initial value of the model; then we vary the parameter \(K\), and obtain series bifurcations, such as flip bifurcations and Hopf bifurcations.

92D40 Ecology
37N25 Dynamical systems in biology
39A28 Bifurcation theory for difference equations
65C20 Probabilistic models, generic numerical methods in probability and statistics
39A60 Applications of difference equations
Full Text: DOI
[1] Gao, S.; Chen, L., The effect of seasonal harvesting on a single-species discrete population model with stage structure and birth pulse, Chaos solitons fractals, 24, 1013-1023, (2005) · Zbl 1061.92059
[2] Beretta, E.; Kon, R.; Takeuchi, Y., Nonexistence of periodic solutions in delayed lotka – volterra systems, Nonlinear anal., 3, 107-129, (2002) · Zbl 1091.34037
[3] Teng, Z., Nonautonomous lotka – volterra systems with delays, J. differential equations, 179, 538-561, (2002) · Zbl 1013.34072
[4] Redheffer, R., Nonautonomous lotka – volterra systems, I, J. differential equations, 127, 519-541, (1996) · Zbl 0856.34056
[5] Teng, Z., On the non-autonomous lotka – volterra \(N\)-species competing systems, Appl. math. comput., 114, 175-185, (2000) · Zbl 1016.92045
[6] Wang, W.; Mulone, G.; Salemi, F.; Salone, V., Global stability of discrete population models with time delays and fluctuating environment, J. math. anal. appl., 264, 147-167, (2001) · Zbl 1006.92025
[7] Huo, H.; Li, W., Permanence and global stability for nonautonomous discrete model of plankton allelpathy, Appl. math. lett., 17, 1007-1013, (2004) · Zbl 1067.39009
[8] Dai, B.; Zhang, N.; Zou, J., Permanence for the michaelis – menten type discrete three-species ratio-dependent food chain model with delay, J. math. anal. appl., 324, 728-738, (2006) · Zbl 1101.92048
[9] Wang, W.; Lu, Z., Global stability of discrete models of lotka – volterra type, Nonlinear anal., 35, 7, 1019-1030, (1999) · Zbl 0919.92030
[10] Saito, Y.; Ma, W.; Hara, T., A necessary and sufficeint condition for permanence of a lotka – volterra discrete system with delays, J. math. anal. appl., 256, 1, 162-174, (2001) · Zbl 0976.92031
[11] Liao, X.; Zhou, S.; Chen, Y., On permanence and global stability in a general gilpin – ayala competition predator – prey discrete system, Appl. math. comput., 190, 500-509, (2007) · Zbl 1125.39008
[12] Teng, Z.; Chen, L., Permanence and extinction of periodic predator – prey systems in a patchy environment with delay, Nonlinear anal., 4, 335-364, (2003) · Zbl 1018.92033
[13] Song, Y.; Yuan, S.; Zhang, J., Bifurction analysis in the delayed leslie – gower predator – prey system, Appl. math. model., 33, 4049-4061, (2009) · Zbl 1205.34089
[14] Boukal, D.S.; Sabelis, M.W.; Berec, L., How predator functional responses and allee effects in prey affect the paradox of enrichment and population collapses, Theor. popul. biol., 72, 136-147, (2007) · Zbl 1123.92034
[15] Kar, T.K.; Batabyal, A., Stability and bifurcation of a prey – predator model with time delay, C. R. biol., 332, 642-651, (2009)
[16] Lian, F.; Xu, Y., Hopf bifurcation analysis of a predator – prey system with Holling IV functional response and time delay, Appl. math. comput., 215, 1484-1495, (2009) · Zbl 1187.34116
[17] Xu, C.; Tang, X.; Liao, M., Stability and bifurcation analysis of a delayed predator – prey model of prey disperal in two patch environments, Appl. math. comput., 216, 2920-2936, (2010) · Zbl 1191.92077
[18] Xiao, D.; Li, W.; Han, M., Dynamics in a ratio-dependent predator – prey model with predator harvesting, J. math. anal. appl., 324, 14-29, (2006) · Zbl 1122.34035
[19] Liu, X.; Xiao, D., Complex dynamic behaviors of a discrete-time predator – prey system, Chaos solitons fractals, 32, 80-94, (2007) · Zbl 1130.92056
[20] Agiza, H.N.; Elabbasy, E.M.; El-Metwally, H.; Elsadany, A.A., Chaotic dynamics of a discrete prey – predator model with Holling type II, Nonliear anal., 10, 116-129, (2009) · Zbl 1154.37335
[21] Celik, C.; Duman, O., Allee effect in a discrete-time predator – prey system, Chaos solitons fractals, 40, 1956-1962, (2009) · Zbl 1198.34084
[22] Ruan, S.; Xiao, D., Global analysis in a predator – prey system with nonmonotonic functional response, SIAM J. appl. math., 61, 4, 1445-1472, (2001) · Zbl 0986.34045
[23] C. Robinson, Dynamical Models, Stability, Symbolic Dynamics and Chaos, 2nd ed., London, New York, Washington, DC, Boca Raton, 1999. · Zbl 0914.58021
[24] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical model and bifurcation of verctor field, (1983), Springer-Verlag New York, pp. 160-165
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.