具Markov转换和脉冲扰动的捕食-食饵系统的动力学

Abstract

Abstract:

In this paper, a stochastic delay predator-prey system with Markovian switching and impulsive perturbations is studied. We establish conditions for the existence of a global positive solution for the considered system. The superior limit of expectations for the solution of this system is estimated. Afterwards we obtain certain asymptotic results regarding long-time behavior of trajectories of the solution and prove stochastically ultimately boundedness of the system. Furthermore, by constructing a suitable Lyapunov function and using comparison theorem of stochastic differential equation, a set of sufficient conditions for extinction, non-persistence in the mean for every positive solution of the system are obtained. Finally, we give the conclusion.

Key words: Predator-prey system, Stochastic perturbations, Impulsive effects, Extinction, Non-persistence in the mean

CLC Number:

O231

Zhang Shuwen. Dynamics of a Predator-Prey System with Impulsive Perturbations and Markovian Switching[J].Acta mathematica scientia,Series A, 2016, 36(3): 569-583.

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[1] Guo H J, Chen L S. The effects of impulsive harvest on a predator-prey system with distributed time delay. Communications in Nonlinear Science and Numerical Simulation, 2009, 14(5):2301-2309
[2] He M G, Li Z, Chen F D. Permanence extinction and global attractivity of the periodic Gilpin-Ayala competition system with impulses. Nonlinear Analysis:Real World Applications, 2010, 11:1537-1551
[3] Hou J, Teng Z D, Gao S J. Permanence and global stability for nonautonomous N-species Lotka-Valterra competitive system with impulses. Nonlinear Analysis:Real World Applications, 2010, 11:1882-1896
[4] Liu B, Zhang Q, Gao Y H. The dynamics of pest control pollution model with age structure and time delay. Applied Mathematics and Computation, 2010, 216:2814-2823
[5] Jiao J J, Ye K L, Chen L S. Dynamical analysis of a five-dimensioned chemostat model with impulsive diffusion and pulse input environmental toxicant. Chaos, Solitons and Fractals, 2011, 44:17-27
[6] Chen L J, Chen F D. Dynamic behaviors of the periodic predator-prey system with distributed time delays and impulsive effect. Nonlinear Analysis:Real World Applications, 2011, 12:2467-2473
[7] Jiao J J, Yang X S, Chen L S. Harvesting on a stage-structured single population model with mature individuals in a polluted environment and pulse input of environment toxin. BioSystems, 2009, 97:186-190
[8] Liu B, Zhang L. Dynamics of a two-species Lotka-Volterra competition system in a polluted environment with pulse toxicant input. Applied Mathematics and Computation, 2009, 214:155-162
[9] Wen X, Wang Z. Persistence and extinction in two species models with impulse. Acta Math Sin (Engl Ser), 2006, 2:447-454
[10] Jin Z, Han M, Li G. The persistence in a Lokta-Volterra competitive systems with impulsive. Chaos, Solitons and Fractals, 2011, 24:1105-1117
[11] Wang W, Shen J, Luo Z. Partial survival and extinction in two competing species with impulses. Nonlinear Analysis:Real World Applications, 2009, 10:1243-1254
[12] Liu M, Qiu H, Wang K. A remark on a stochastic predator-prey system with time delays. Applied Mathematics Letters, 2013, 26:318-323
[13] Maja Vasilova. Asymptotic behavior of a stochastic Gilpin-Ayala predator-prey system with time-dependent delay. Mathematical and Computer Modelling, 2013, 57:764-781
[14] May R M. Stability and Complexity in Model Ecosystems. Princeton:Princeton University Press, 1973
[15] Liu M, Wang K. Global stability of a nonlinear stochastic predator-prey system with Beddington-DeAngelis functional response. Communications in Nonlinear Science and Numerical Simulation, 2011, 16:1114-1121
[16] Lian B S, Hu S G. Asymptotic behavior of the stochastic Gilpin-Ayala competition models. J Math Anal Appl, 2008, 339:419-428
[17] Ji C Y, Jiang D Q, Li X Y. Qualitative analysis of a stochastic ratio-dependent predator-prey system. Journal of Computational and Applied Mathematics, 2011, 235:1326-1341
[18] Li C X, Sun J T, Sun R Y. Stability analysis of a class of stochastic differential delay equations with nonlinear impulsive effects. Journal of Frankin Institute, 2010, 347:1186-1198
[19] Cheng P, Deng F Q. Global exponential stability of impulsive stochastic functional differential systems. Statistics and Probability Letters, 2010, 80:1854-1862
[20] Lu Chun, Ding X H. Persistence and extinction of stochastic logistic model with delays and impulsive perturbation. Acta Mathematica Scientia, 2014, 34:1551-1570
[21] Li C X, Sun J T. Stability analysis of nonlinear stochastic differential delay systems under impulsive control. Phys Lett A, 2010, 374:1154-1158
[22] Li C X, Shi J P, Sun J T. Stability of impulsive stochastic differential delay systems and its application to impulsive stochastic neural networks. Nonlinear Analysis, 2011, 74:3099-3111
[23] Liu M, Wang K. On a stochastic logistic equation with impulsive perturbations. Computers and Mathematics with Applications, 2012, 63:871-886
[24] Lao Q, Mao X. Stochastic population dynamics under regime switching. Journal of Mathematical Analysis and Applications, 2007, 334:69-84
[25] Slatkin M. The dynamics of a population in a Markovian environment. Ecology, 1978, 59:249-256
[26] Jeffries C. Stability of predation ecosystem models. Ecology, 1976, 57:1321-1325
[27] Arnold L. Stochastic Differential Equation:Theory and Application. New York:Wiley, 1972
[28] Mao X R. Stochastic Differential Equations and Application. Chichester:Horwood, 1997
[29] Mao X R, Marion G, Renshaw E. Environmental noise suppresses explosion in population dynamics. Stochastic Process Appl, 2002, 97:95-110
[30] Mao X, Yuan C. Stochastic Differential Equations with Markovian Switching. London:Imperial College Press, 2006

Dynamics of a Predator-Prey System with Impulsive Perturbations and Markovian Switching

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