Analysis of an impulsive predator-prey system with Monod-Haldane functional response and seasonal effects. (English) Zbl 1180.92085

Summary: For a class of impulsive predator-prey systems with Monod-Haldane functional response and seasonal effects, conditions for local and global stability of prey-free solutions and for the permanence of the system by using Floquet theory of impulsive differential equations and comparison techniques are investigated. In addition, we analyze numerically the phenomena caused by seasonal effects and impulsive perturbations. This will be applicable to the controllability of the prey population and the predator population.


92D40 Ecology
34A37 Ordinary differential equations with impulses
34D20 Stability of solutions to ordinary differential equations
Full Text: DOI EuDML


[1] C. S. Holling, “The functional response of predator to prey density and its role in mimicry and population regulations,” Mem. Ent. Sec. Can, vol. 45, pp. 1-60, 1965.
[2] J. F. Andrews, “A mathematical model for the continuous culture of macroorganisms untilizing inhibitory substrates,” Biotechnology and Bioengineering, vol. 10, pp. 707-723, 1968.
[3] W. Sokol and J. A. Howell, “Kineties of phenol oxidation by ashed cell,” Biotechnology and Bioengineering, vol. 23, pp. 2039-2049, 1980.
[4] S.-B. Hsu and T.-W. Huang, “Global stability for a class of predator-prey systems,” SIAM Journal on Applied Mathematics, vol. 55, no. 3, pp. 763-783, 1995. · Zbl 0832.34035
[5] S. Ruan and D. Xiao, “Global analysis in a predator-prey system with nonmonotonic functional response,” SIAM Journal on Applied Mathematics, vol. 61, no. 4, pp. 1445-1472, 2001. · Zbl 0986.34045
[6] E. Sáez and E. González-Olivares, “Dynamics of a predator-prey model,” SIAM Journal on Applied Mathematics, vol. 59, no. 5, pp. 1867-1878, 1999. · Zbl 0934.92027
[7] J. Sugie, R. Kohno, and R. Miyazaki, “On a predator-prey system of Holling type,” Proceedings of the American Mathematical Society, vol. 125, no. 7, pp. 2041-2050, 1997. · Zbl 0868.34023
[8] J. M. Cushing, “Periodic time-dependent predator-prey systems,” SIAM Journal on Applied Mathematics, vol. 32, no. 1, pp. 82-95, 1977. · Zbl 0348.34031
[9] S. Gakkhar and R. K. Naji, “Chaos in seasonally perturbed ratio-dependent prey-predator system,” Chaos, Solitons & Fractals, vol. 15, no. 1, pp. 107-118, 2003. · Zbl 1033.92026
[10] G. C. W. Sabin and D. Summers, “Chaos in a periodically forced predator-prey ecosystem model,” Mathematical Biosciences, vol. 113, no. 1, pp. 91-113, 1993. · Zbl 0767.92028
[11] G. J. Ackland and I. D. Gallagher, “Stabilization of large generalized Lotka-Volterra foodwebs by evolutionary feedback,” Physical Review Letters, vol. 93, no. 15, Article ID 158701, 4 pages, 2004.
[12] G. Jiang and Q. Lu, “The dynamics of a prey-predator model with impulsive state feedback control,” Discrete and Continuous Dynamical Systems. Series B, vol. 6, no. 6, pp. 1301-1320, 2006. · Zbl 1120.34047
[13] X. Liu and L. Chen, “Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator,” Chaos, Solitons & Fractals, vol. 16, no. 2, pp. 311-320, 2003. · Zbl 1085.34529
[14] B. Liu, Y. Zhang, and L. Chen, “Dynamic complexities in a Lotka-Volterra predator-prey model concerning impulsive control strategy,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 15, no. 2, pp. 517-531, 2005. · Zbl 1080.34026
[15] K. Negi and S. Gakkhar, “Dynamics in a Beddington-DeAngelis prey-predator system with impulsive harvesting,” Ecological Modelling, vol. 206, no. 3-4, pp. 421-430, 2007.
[16] X. Song and Y. Li, “Dynamic behaviors of the periodic predator-prey model with modified Leslie-Gower Holling-type II schemes and impulsive effect,” Nonlinear Analysis: Real World Applications, vol. 9, no. 1, pp. 64-79, 2008. · Zbl 1142.34031
[17] D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, vol. 66 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Science & Technical, Harlo, UK, 1993. · Zbl 0815.34001
[18] V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific, Teaneck, NJ, USA, 1989. · Zbl 0719.34002
[19] H. Baek, “Dynamic complexites of a three-species Beddington-DeAngelis system with impulsive control strategy,” Acta Applicandae Mathematicae, pp. 1-16, 2008. · Zbl 1194.34087
[20] W. Wang, H. Wang, and Z. Li, “The dynamic complexity of a three-species Beddington-type food chain with impulsive control strategy,” Chaos, Solitons & Fractals, vol. 32, no. 5, pp. 1772-1785, 2007. · Zbl 1195.92066
[21] Z. Xiang and X. Song, “The dynamical behaviors of a food chain model with impulsive effect and Ivlev functional response,” Chaos, Solitons & Fractals, vol. 39, no. 5, pp. 2282-2293, 2009. · Zbl 1197.34012
[22] S. Zhang, D. Tan, and L. Chen, “Chaos in periodically forced Holling type IV predator-prey system with impulsive perturbations,” Chaos, Solitons & Fractals, vol. 27, no. 4, pp. 980-990, 2006. · Zbl 1097.34038
[23] S. Zhang and L. Chen, “A study of predator-prey models with the Beddington-DeAnglis functional response and impulsive effect,” Chaos, Solitons & Fractals, vol. 27, no. 1, pp. 237-248, 2006. · Zbl 1102.34032
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.