Dynamical Behaviors for a Predator-Prey System with State-Dependent Impulsive Effects

Abstract:

With some theory about continuous and impulsive dynamical system, an impulsive model based on a special predator-prey system is considered. We assume that the impulsive effects occur when the density of the prey reaches a given value. For such a state-dependent impulsive system, the existence, uniqueness and orbital asymptotic stability of an order-1 periodic solution are discussed. Further, the existence of an order-2 periodic solution is also obtained, and persistence of the system is investigated.

• Periodical: Applied Mechanics and Materials (Volumes 130-134)
• Pages: 385-390
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.130-134.385
• Online since: October 2011
• Authors: Ling Zhen Dong, Lan Sun Chen
• Keywords: Extinction, Order-1 Periodic Solution, Order-2 Periodic Solution, Periodic Solution, Persistence, Poincaré Map, State-Dependent Impulsive Differential System

References

[1] D. Bainov, P. Simeonov, Impulsive differential equations: Periodic solution and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics, (1993)66.

[2] K. Chen, Uniqueness of a limit cycle for a predator-prey system, SIAM J. Math. Anal, 1981(12): 541-548.

DOI: 10.1137/0512047

[3] Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J., Knuth, D. E., On the Lambert W function, Advances in Computational Mathematics, 1996(5): 329-359.

DOI: 10.1007/bf02124750

[4] J. M. Cushing, Periodic time-dependent predator-prey system, SIAM J. Appl. Math, 1977(32): 82-95.

DOI: 10.1137/0132006

[5] Eric Funasaki, Mark Kor, Invasion and chaos in a periodically pulsed mass-action chemostat, Theoretical population Biology, 1993(44): 203-224.

DOI: 10.1006/tpbi.1993.1026

[6] H. I. Freedman, P. Waltman, Uniformly persistent systems, Amer. Math. Soci., 1986(3).

[7] B. E. Kendall, Cycles, chaos, and noise in predator-prey dynamics, Chaos, Solitons & Fractals, ~2001(12): 321-332.

DOI: 10.1016/s0960-0779(00)00180-6

[8] V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of Impulsive Differential Equations, Singapore: World Scientific, (1989).

DOI: 10.1142/0906

[9] J. C. Panetta, A mathematical model of periodically pulsed chemotherapy: tumor recurrence and metastasis in a competition environment, Bull. Math. Biol., 1996(58): 425-447.

DOI: 10.1007/bf02460591

[10] S. Tang, L. Chen, Multiple attractors in stage-structured population models with birth pulses, Bull. Math. Biol., 2003: 1-17.

[11] S. Tang, L. Chen, The periodic predator-prey Lotka-Volterra model with impulsive effect, Journal of Mechanics in Medicine and Biology, (2002)2, Nos, 3 & 4: 267-296.

DOI: 10.1142/s021951940200040x

[12] V. S. Varma, Exact solution for a special prey-predator or competing species system, Bull. Math. Biol., 1977(39): 619-622.

DOI: 10.1016/s0092-8240(77)80064-5