Complex Dynamic Behaviors in an Ivlev-type Stage-structured Predator-prey System Concerning Impulsive Control Strategy

Authors: Shunyi Li, Zhifang He, Xiangui Xue

Abstract:

An Ivlev-type predator-prey system and stage-structured for predator concerning impulsive control strategy is considered. The conditions for the locally asymptotically stable prey-eradication periodic solution is obtained, by using Floquet theorem and small amplitude perturbation skills——when the impulsive period is less than the critical value. Otherwise, the system is permanence. Numerical examples show that the system considered has more complicated dynamics, including high-order quasi-periodic and periodic oscillating, period-doubling and period-halving bifurcation, chaos and attractor crisis, etc. Finally, the biological implications of the results and the impulsive control strategy are discussed.

Keywords: Stage-structured predator-prey system, Impulsive, Permanence, Bifurcation, Chaos.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1088296

References:

[1] W.D. Wang, L.S. Chen, A predator-prey system with stage-structure for predator, Comp. Math. Appl., 33 (1997) 83-91.
[2] S.Q. Liu, L.S. Chen, Models and Methods in Stage-structured Biological Population, Beijing, Science Press.
[3] X.Y. Song, Z.Y. Xiang, The prey-dependent consumption two-prey onepredator models with stage structure for the predator and impulsive effects. J. Theor. Biol., 242 (2006) 683-698.
[4] L.S. Wang, R. Xu, G.H. Feng, A Stage-Structured Predator-Prey System with Impulsive Effect and Holling Type-II Functional Response. J. Math. Rese. Expo., 31 (2011) 147-156.
[5] L.S. Wang, R. Xu, G.H. Feng, Analysis of a Stage-Structured Predator- Prey System Concerning Impulsive Control Strategy. Differ. Equ. Dyn. Syst., 19 (2011) 303-313.
[6] S. Y. Li, Z. L. Xiong, X. Wang, The study of a predator-prey system with group defense and impulsive control strategy, Appl. Math. Model., 34 (2010) 2546-2561.
[7] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of impulsive differential equations. Singapore, World Scientific, 1989.
[8] D.D. Bainov, P.S. Simeonov, Impulsive differential equations: asymptotic properties of the solutions. Singapore, World Scientific, 1993.
[9] Z.F. He, Z.L. Xiong, W.H. Wang, Stochastic persistence and stationary distribution in a Beddington-deAngelis type prey-predator model, J.Comp. Info. Syst., 8 (2012) 4743-4750.