In this paper, astochastic predator-prey systems with nonlinear harvesting
and impulsive effect are investigated. Firstly, we show the existence and uniqueness
of the global positive solution of the system. Secondly, by constructing
appropriate Lyapunov function and using comparison theorem with an
impulsive differential equation, we study that a positive periodic solution exists.
Thirdly, we prove that system is globally attractive. Finally, numerical
simulations are presented to show the feasibility of the obtained results.
References
[1]
Heggerud, C. and Lan, K. (2015) Local Stability Analysis of Ratio-Dependent Predator-Prey Models with Predator Harvesting Rates. Applied Mathematics and Computation, 270, 349-357. https://doi.org/10.1016/j.amc.2015.08.062
[2]
Liu, M. and Wang, K. (2012) On a Stochastic Logistic Equation with Impulsive Perturbations. Computers & Mathematics with Applications, 63, 871-886. https://doi.org/10.1016/j.camwa.2011.11.003
[3]
Li, Y. and Gao, H. (2008) Existence, Uniqueness and Global Asymptotic Stability of Positive Solutions of a Predator-Prey System with Holling II Functional Response with Random Perturbation. Nonlinear Analysis, 68, 1694-1705. https://doi.org/10.1016/j.na.2007.01.008
[4]
Ji, C., Jiang, D. and Li, X. (2010) Qualitative Analysis of a Stochastic Ratio-Dependent Predator-Prey System. Journal of Computational and Applied Mathematics, 235, 1326-1341. https://doi.org/10.1016/j.cam.2010.08.021
[5]
Zuo, W. and Jiang, D. (2016) Stationary Distribution and Periodic Solution for Stochastic Predator-Prey Systems with Nonlinear Predator Harvesting. Communications in Nonlinear Science and Numerical Simulation, 36, 65-80. https://doi.org/10.1016/j.cnsns.2015.11.014
[6]
Ji, C., Jiang, D. and Shi, N. (2009) Analysis of a Predator-Prey Model with Modified Leslie-Gower and Holling-Type II Schemes with Stochastic Perturbation. Journal of Mathematical Analysis and Applications, 359, 482-498. https://doi.org/10.1016/j.jmaa.2009.05.039
[7]
Zhang, Y., Chen, S.H., Gao, S.J. and Wei, X. (2017) Stochastic Periodic Solution for a Perturbed Non-Autonomous Predator-Prey Model with Generalized Nonlinear Harvesting and Impulses. Physica A, 486, 347-366. https://doi.org/10.1016/j.physa.2017.05.058
[8]
Lu, C. and Ding, X. (2014) Persistence and Extinction of a Stochastic Logistic Model with Delays and Impulsive Perturbation. Acta Mathematica Scientia, 34B, 1551-1570. https://doi.org/10.1016/S0252-9602(14)60103-X
[9]
Cheng, S. (2009) Stochastic Population Systems. Stochastic Analysis and Applications, 27, 854-874. https://doi.org/10.1080/07362990902844348
[10]
Ji, C. and Jiang, D. (2009) Analysis of a Predator-Prey Model with Modified Leslie-Gower and Holling-Type II Schemes with Stochastic Perturbance. Journal of Mathematical Analysis and Applications, 359, 482-498. https://doi.org/10.1016/j.jmaa.2009.05.039
[11]
Ji, C. and Jiang, D. (2011) Dynamics of a Stochastic Density Dependent Predator-Prey System with Beddington-DeAngelis Functional Response. Journal of Mathematical Analysis and Applications, 381, 441-453. https://doi.org/10.1016/j.jmaa.2011.02.037
[12]
Zhang, S.W. and Tan, D.J. (2015) Dynamics of a Stochastic Predator-Prey System in a Polluted Environment with Pulse Toxicant Input and Impulsive Perturbations. Applied Mathematical Modelling, 39, 6319-6331. https://doi.org/10.1016/j.apm.2014.12.020
[13]
Zuo, W.J. and Jiang, D.Q. (2016) Periodic Solutions for a Stochastic Non-Autonomous Holling-Tanner Predator-Prey System with Impulses. Nonlinear Analysis, 22, 191-201. https://doi.org/10.1016/j.nahs.2016.03.004
[14]
Li, D. and Xu, D. (2013) Periodic Solutions of Stochastic Delay Differential Equations and Applications to Logistic Equation and Neural Networks. Journal of the Korean Mathematical Society, 50, 1165-1181. https://doi.org/10.4134/JKMS.2013.50.6.1165
[15]
Liu, Z.J., Guo, S.L., Tan, R.H. and Liu, M. (2016) Modeling and Analysis of a Non-Autonomous Single-Species Model with Impulsive and Random Perturbations. Applied Mathematical Modelling, 40, 5510-5531. https://doi.org/10.1016/j.apm.2016.01.008
[16]
Higham, D.J. (2001) An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations. SIAM Review, 43, 525-546. https://doi.org/10.1137/S0036144500378302
