A delayed Holling type III functional response predator-prey system with impulsive perturbation on the prey. (English) Zbl 1418.92108
Summary: A Holling type III functional response predator-prey system with constant gestation time delay and impulsive perturbation on the prey is investigated. The sufficient conditions for the global attractivity of a predator-extinction periodic solution are obtained by the theory of impulsive differential equations, i.e. the impulsive period is less than the critical value \(T_{1}^{*}\). The conditions for the permanence of the system are investigated, i.e. the impulsive period is larger than the critical value \(T_{2}^{*}\). Numerical examples show that the system has very complex dynamic behaviors, including (1) high-order periodic and quasi-periodic oscillations, (2) period-doubling and -halving bifurcations, and (3) chaos and attractor crises. Further, the importance of the impulsive period, the gestation time delay, and the impulsive perturbation proportionality constant are discussed. Finally, the impulsive control strategy and the biological implications of the results are discussed.
MSC:
| 92D25 | Population dynamics (general) |
| 34K60 | Qualitative investigation and simulation of models involving functional-differential equations |
| 34A37 | Ordinary differential equations with impulses |
| 34K20 | Stability theory of functional-differential equations |
| 34K13 | Periodic solutions to functional-differential equations |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
References:
| [1] | Li, S; Xue, Y; Liu, W, Hopf bifurcation and global periodic solutions for a three-stage-structured prey-predator system with delays, Int. J. Inf. Syst. Sci., 8, 142-156, (2012) · Zbl 1343.34183 |
| [2] | Li, S; Liu, W; Xue, X, Bifurcation analysis of a stage-structured prey-predator system with discrete and continuous delays, Appl. Math., 4, 1059-1064, (2013) · doi:10.4236/am.2013.47144 |
| [3] | Li, S; Xiong, Z, Bifurcation analysis of a predator-prey system with sex-structure and sexual favoritism, Adv. Differ. Equ., 2013, (2013) · Zbl 1379.92050 · doi:10.1186/1687-1847-2013-219 |
| [4] | Wang, L; Xu, R; Feng, G, A stage-structured predator-prey system with time delay and Holling type-III functional response, Int. J. Pure Appl. Math., 48, 53-66, (2008) |
| [5] | Zou, W; Xie, J; Xiong, Z, Stability and Hopf bifurcation for an eco-epidemiology model with Holling-III functional response and delays, Int. J. Biomath., 1, 377-389, (2008) · Zbl 1166.34050 · doi:10.1142/S179352450800031X |
| [6] | Zhang, X; Xu, R; Gan, Q, Periodic solution in a delayed predator prey model with Holling type III functional response and harvesting term, World J. Model. Simul., 7, 70-80, (2011) |
| [7] | Das, U; Kar, TK, Bifurcation analysis of a delayed predator-prey model with Holling type III functional response and predator harvesting, J. Nonlinear Dyn., 2014, (2014) · Zbl 1407.92140 |
| [8] | Zhang, Z; Yang, H; Fu, M, Hopf bifurcation in a predator-prey system with Holling type III functional response and time delays, J. Appl. Math. Comput., 44, 337-356, (2014) · Zbl 1301.92069 · doi:10.1007/s12190-013-0696-7 |
| [9] | Fan, Y; Wang, L, Multiplicity of periodic solutions for a delayed ratio-dependent predator-prey model with Holling type III functional response and harvesting terms, J. Math. Anal. Appl., 365, 525-540, (2010) · Zbl 1188.34112 · doi:10.1016/j.jmaa.2009.11.009 |
| [10] | Cai, Z; Huang, L; Chen, H, Positive periodic solution for a multi species competition-predator system with Holling III functional response and time delays, Appl. Math. Comput., 217, 4866-4878, (2011) · Zbl 1206.92041 · doi:10.1016/j.amc.2010.10.014 |
| [11] | Li, G; Yan, J, Positive periodic solutions for neutral delay ratio-dependent predator-prey model with Holling type III functional response, Appl. Math. Comput., 218, 4341-4348, (2011) · Zbl 1260.92023 · doi:10.1016/j.amc.2011.10.009 |
| [12] | Xu, C; Wu, Y; Lu, L, Permanence and global attractivity in a discrete Lotka-Volterra predator-prey model with delays, Adv. Differ. Equ., 2014, (2014) · Zbl 1417.92161 · doi:10.1186/1687-1847-2014-208 |
| [13] | Li, S, Complex dynamical behaviors in a predator-prey system with generalized group defense and impulsive control strategy, Discrete Dyn. Nat. Soc., 2013, (2013) · Zbl 1417.92137 |
| [14] | Li, S; Xiong, Z; Wang, X, The study of a predator-prey system with group defense and impulsive control strategy, Appl. Math. Model., 34, 2546-2561, (2010) · Zbl 1195.34019 · doi:10.1016/j.apm.2009.11.019 |
| [15] | Su, H; Dai, B; Chen, Y; etal., Dynamic complexities of a predator-prey model with generalized Holling type III functional response and impulsive effects, Comput. Math. Appl., 56, 1715-1725, (2008) · Zbl 1152.34309 · doi:10.1016/j.camwa.2008.04.001 |
| [16] | Fan, X; Jiang, F; Zhang, H, Dynamics of multi-species competition-predator system with impulsive perturbations and Holling type III functional responses, Nonlinear Anal., Theory Methods Appl., 74, 3363-3378, (2011) · Zbl 1213.92059 · doi:10.1016/j.na.2011.02.012 |
| [17] | Liu, Z; Zhong, S, An impulsive periodic predator-prey system with Holling type III functional response and diffusion, Appl. Math. Model., 36, 5976-5990, (2012) · Zbl 1349.34047 · doi:10.1016/j.apm.2012.01.032 |
| [18] | Yan, C; Dong, L; Liu, M, The dynamical behaviors of a nonautonomous Holling III predator-prey system with impulses, J. Appl. Math. Comput., 47, 193-209, (2015) · Zbl 1334.92381 · doi:10.1007/s12190-014-0769-2 |
| [19] | Tan, R; Liu, W; Wang, Q, Uniformly asymptotic stability of almost periodic solutions for a competitive system with impulsive perturbations, Adv. Differ. Equ., 2014, (2014) · Zbl 1417.34120 · doi:10.1186/1687-1847-2014-2 |
| [20] | Xu, L; Wu, W, Dynamics of a nonautonomous Lotka-Volterra predator-prey dispersal system with impulsive effects, Adv. Differ. Equ., 2014, (2014) · Zbl 1348.34137 · doi:10.1186/1687-1847-2014-264 |
| [21] | Ju, Z; Shao, Y; Kong, W, An impulsive prey-predator system with stagestructure and Holling II functional response, Adv. Differ. Equ., 2014, (2014) · Zbl 1444.92089 · doi:10.1186/1687-1847-2014-280 |
| [22] | Jia, J; Cao, H, Dynamic complexities of Holling type II functional response predator-prey system with digest delay and impulsive, Int. J. Biomath., 2, 229-242, (2009) · Zbl 1342.92177 · doi:10.1142/S179352450900056X |
| [23] | Bainov, DD, Simeonov, DD: Impulsive Differential Equations: Periodic Solutions and Applications. Longman, Harlow (1993) · Zbl 0815.34001 |
| [24] | Lakshmikantham, V, Bainov, DD, Simeonov, PS: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) · Zbl 0719.34002 · doi:10.1142/0906 |
| [25] | Kuang, Y: Delay Differential Equations with Applications in Population Dynamics. Academic Press, New York (1993) · Zbl 0777.34002 |
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.
