Permanence and global attractivity of stage-structured predator-prey model with continuous harvesting on predator and impulsive stocking on prey. (English) Zbl 1231.34021
Summary: We investigate a stage-structured delayed predator-prey model with impulsive stocking on prey and continuous harvesting on predator. According to the fact of biological resource management, we improve the assumption of a predator-prey model with stage structure for predator population that each individual predator has the same ability to capture prey. It is assumed that the immature and mature individuals of the predator population are divided by a fixed age, and immature predator population does not have the ability to attach prey. Sufficient conditions are obtained, which guarantee the global attractivity of predator-extinction periodic solution and the permanence of the system. Our results show that the behavior of impulsive stocking on prey plays an important role for the permanence of the system, and provide tactical basis for the biological resource management. Numerical analysis is presented to illuminate the dynamics of the system.
Keywords:
stage-structured; impulsive stocking; continuous harvesting; global attractivity; permanenceReferences:
| [1] | Nieto J J, Rodriguez-Lopez R. Periodic boundary value problems for non-Lipschitzian impulsive functional differential equations[J]. J Math Anal Appl, 2006, 31(8):593–610. · Zbl 1101.34051 · doi:10.1016/j.jmaa.2005.06.014 |
| [2] | Saker S H. Oscillation and global attractivity of impulsive periodic delay respiratory dynamiscs model[J]. Chinese Ann Math Ser B, 2005, 26(4):511–522. · Zbl 1096.34053 · doi:10.1142/S0252959905000403 |
| [3] | d’Onofrio A. A general framework for modeling tumor-immune system competition and immunotherapy: mathematical analysis and biomedical inferences[J]. Physica D: Nonlinear Phenomena, 2005, 20(8):220–235. · Zbl 1087.34028 · doi:10.1016/j.physd.2005.06.032 |
| [4] | Gao Shujing, Chen Lansun. Pulse vaccination strategy in a delayed SIR epidemic model with vertical transmission[J]. Discrete and Continuous Dynamical Systems SERIES B, 2007, 7(1):77–86. · Zbl 1191.34062 |
| [5] | Clark C W. Mathematical bioeconomics[M]. New York: Wiley, 1990. · Zbl 0712.90018 |
| [6] | Jiao Jianjun, Meng Xinzhu, Chen Lansun. A stage-structured Holling mass defence predator-prey 3 model with impulsive perturbations on predators[J]. Applied Mathematics and Computation, 2007, 189(2):1448–1458. · Zbl 1117.92053 · doi:10.1016/j.amc.2006.12.043 |
| [7] | Song Xinyu, Li Yongfeng. Dynamic complexities of a Holling II two-prey one-predator system with impulsive effect[J]. Chaos, Solitons and Fractals, 2007, 33(2):463–478. · Zbl 1136.34046 · doi:10.1016/j.chaos.2006.01.019 |
| [8] | Aiello W G, Freedman H I. A time-delay model of single-species growth with stage-structured[J]. Math Biosci, 1990, 101(2):139. · Zbl 0719.92017 · doi:10.1016/0025-5564(90)90019-U |
| [9] | Freedman H I, Gopalsamy K. Global stability in time-delayed single species dynamics[J]. Bull Math Biol, 1986, 48(5/6):485–492. · Zbl 0606.92020 |
| [10] | Beretta E, Kuang Y. Global analysis in some delayed ratio-dependent predator-prey system[J]. Nonlinear Anal, 1998, 32(3):381–408. · Zbl 0946.34061 · doi:10.1016/S0362-546X(97)00491-4 |
| [11] | Yang Kuang. Delay differential equation with application in population dynamics[J]. New York: Academic Press, 1993, 67–70. |
| [12] | Wang Wendi, Chen Lansun. A predator-prey system with stage structure for predator[J]. Comput Math Appl, 1997, 33(8):83–91. · doi:10.1016/S0898-1221(97)00056-4 |
| [13] | Jiao Jianjun, Pang Guoping, Chen Lansun, Luo Guilie. A delayed stage-structured predator – prey model with impulsive stocking on prey and continuous harvesting on predator[J]. Applied Mathematics and Computation, 2008, 195(1):316–325. · Zbl 1126.92052 · doi:10.1016/j.amc.2007.04.098 |
| [14] | Song Xinyu, Chen Lansun. Optimal harvesting policy and stability for a single-species growth model with stage structure[J]. Journal of System Sciences and Complex, 2002, 15(2):194–201. · Zbl 0996.92038 |
| [15] | Dong Lingzhen, Chen Lansun, Sun Lihua. Extinction and permanence of the predator-prey system with stocking of prey and harvesting of predator impulsively[J]. Math Meth Appl Sci, 2006, 29 (4):415–425. · Zbl 1086.92051 · doi:10.1002/mma.688 |
| [16] | Wang W, Mulone G, Salemi F, Salone V. Permanence and stability of a stage-structured predatorprey model[J]. J Math Anal Appl, 2001, 262(2):499–528. · Zbl 0997.34069 · doi:10.1006/jmaa.2001.7543 |
| [17] | Lakshmikantham V, Bainov D D, Simeonov P. Theory of impulsive differential equations[M]. Singapor: World Scientific, 1989. · Zbl 0719.34002 |
| [18] | Bainov D, Simeonov P. Impulsive differential equations: periodic solutions and applications[M]. New York: Longman Scientific and Technical, 1993, 66. · Zbl 0815.34001 |
| [19] | Caltagirone L E, Doutt R L. Global behavior of an SEIRS epidemic model with delays, the history of the vedalia beetle importation to California and its impact on the development of biological control[J]. Ann Rev Entomol, 1989, 34:1–16. · doi:10.1146/annurev.en.34.010189.000245 |
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