An impulsive prey-predator system with stage-structure and Holling II functional response. (English) Zbl 1444.92089
Summary: Taking into account that individual organisms usually go through immature and mature stages, in this paper, we investigate the dynamics of an impulsive prey-predator system with a Holling II functional response and stage-structure. Applying the comparison theorem and some analysis techniques, the sufficient conditions of the global attractivity of a mature predator periodic solution and the permanence are investigated. Examples and numerical simulations are shown to verify the validity of our results.
MSC:
| 92D25 | Population dynamics (general) |
| 37N25 | Dynamical systems in biology |
| 34A37 | Ordinary differential equations with impulses |
References:
| [1] | Liu B, Zhang YJ, Chen LS: The dynamical behaviors of a Lotka-Volterra predator-prey model concerning integrated pest management.Nonlinear Anal., Real World Appl. 2005, 6:227-243. · Zbl 1082.34039 · doi:10.1016/j.nonrwa.2004.08.001 |
| [2] | Baek HK: Qualitative analysis of Beddington-DeAngelis type impulsive predator-prey models.Nonlinear Anal., Real World Appl. 2010, 11:1312-1322. · Zbl 1204.34062 · doi:10.1016/j.nonrwa.2009.02.021 |
| [3] | Zhang SW, Chen LS: A study of predator-prey models with the Beddington-DeAngelis functional response and impulsive effect.Chaos Solitons Fractals 2006, 27:237-248. · Zbl 1102.34032 · doi:10.1016/j.chaos.2005.03.039 |
| [4] | Song XY, Li YF: Dynamic complexities of a Holling II two-prey one-predator system with impulsive effect.Chaos Solitons Fractals 2007, 33:463-478. · Zbl 1136.34046 · doi:10.1016/j.chaos.2006.01.019 |
| [5] | Georgescu P, Morosanu G: Impulsive perturbations of a three-trophic prey-dependent food chain system.Math. Comput. Model. 2008, 48:975-997. · Zbl 1187.34071 · doi:10.1016/j.mcm.2007.12.006 |
| [6] | Wang WM, Wang HL, Li ZQ: The dynamic complexity of a three-species Beddington-type food chain with impulsive control strategy.Chaos Solitons Fractals 2007, 32:1772-1785. · Zbl 1195.92066 · doi:10.1016/j.chaos.2005.12.025 |
| [7] | Pei YZ, Li CG, Fan SH: A mathematical model of a three species prey-predator system with impulsive control and Holling functional response.Appl. Math. Comput. 2013, 219:10945-10955. · Zbl 1302.92111 · doi:10.1016/j.amc.2013.05.012 |
| [8] | Jiang GR, Lu QS, Qian LN: Complex dynamics of a Holling type II prey-predator system with state feedback control.Chaos Solitons Fractals 2007, 31:448-461. · Zbl 1203.34071 · doi:10.1016/j.chaos.2005.09.077 |
| [9] | Shao YF, Li Y: Dynamical analysis of a stage structured predator-prey system with impulsive diffusion and generic functional response.Appl. Math. Comput. 2013, 220:472-481. · Zbl 1329.92110 · doi:10.1016/j.amc.2013.05.063 |
| [10] | Jiang XW, Song Q, Hao MY: Dynamics behaviors of a delayed stage-structured predator-prey model with impulsive effect.Appl. Math. Comput. 2010, 215:4221-4229. · Zbl 1184.92054 · doi:10.1016/j.amc.2009.12.044 |
| [11] | Song XY, Chen LS: Optimal harvesting and stability for a two-species competitive system with stage structure.Math. Biosci. 2001, 170:173-186. · Zbl 1028.34049 · doi:10.1016/S0025-5564(00)00068-7 |
| [12] | Shao YF, Dai BX: The dynamics of an impulsive delay predator-prey model with stage structure and Beddington-type functional response.Nonlinear Anal., Real World Appl. 2010, 11:3567-3576. · Zbl 1218.34099 · doi:10.1016/j.nonrwa.2010.01.004 |
| [13] | Kuang Y: Delay Differential Equations: With Applications in Population Dynamics. Academic Press, New York; 1993. · Zbl 0777.34002 |
| [14] | Li YK, Kuang Y: Periodic solutions of periodic delay Lotka-Volterra equations and systems.J. Math. Appl. 2001, 255:260-280. · Zbl 1024.34062 |
| [15] | Zhu HT, Zhu WD, Zhang ZD: Persistence of competitive ecological mathematic model.J. Central South Univ. For. Technol. 2011,3(4):214-218. |
| [16] | Lakshmikantham V, Bainov DD, Simeonov P: Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989. · Zbl 0719.34002 · doi:10.1142/0906 |
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