Dynamics of a stochastic predator-prey coupled system with modified Leslie-Gower and Holling type II schemes

Year 2018, Volume: 31 Issue: 1, 236 - 249, 01.03.2018

Safia Slımanı , Azzedine Benchettah

Abstract

We study a stochastic predator-prey system with modified Leslie-Gower and Holling type II functional response among n patches. The existence and uniqueness as well as boundedness of solution are obtained. Furthermore, we obtain sufficient conditions for stochastic permanence, and by the Lyapunov functional technique we obtain sufficient conditions for the existence of the stationary distribution. Finally, we illustrate our conclusions through numerical simulation.

Keywords

Stochastic permanence, Stationary distribution, Lyapunov functional

References

• M. A. Aziz-Alaoui and M. Daher Okiye. Boundedness and global stabil- ity for a predator-prey model with modified Leslie-Gower and Holling- type II schemes. Appl. Math. Lett., 16(7):1069–1075, 2003.
• Nirav Dalal, David Greenhalgh, and Xuerong Mao. A stochastic model for internal HIV dynamics. J. Math. Anal. Appl., 341(2):1084–1101, 2008.
• Chunyan Ji, Daqing Jiang, and Ningzhong Shi. Analysis of a predator- prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation. J. Math. Anal. Appl., 359(2):482–498, 2009.
• Rafail Khasminskii. Stochastic stability of differential equations, vol- ume 66 of Stochastic Modelling and Applied Probability. Springer, Hei- delberg, second edition, 2012. With contributions by G. N. Milstein and M. B. Nevelson.
• Peter E. Kloeden and Eckhard Platen. Numerical solution of stochastic differential equations, volume 23 of Applications of Mathematics (New York). Springer-Verlag, Berlin, 1992.
• Michael Y. Li and Zhisheng Shuai. Global-stability problem for coupled systems of differential equations on networks. J. Differential Equations, 248(1):1–20, 2010.
• Xiaoyue Li and Xuerong Mao. Population dynamical behavior of non- autonomous Lotka-Volterra competitive system with random perturba- tion. Discrete Contin. Dyn. Syst., 24(2):523–545, 2009.
• Jingliang Lv and Ke Wang. Analysis on a stochastic predator-prey model with modified Leslie-Gower response. Abstr. Appl. Anal., pages Art. ID 518719, 16, 2011.
• Xuerong Mao, Chenggui Yuan, and Jiezhong Zou. Stochastic differ- ential delay equations of population dynamics. J. Math. Anal. Appl., 304(1):296–320, 2005.
• Long Zhang and Zhidong Teng. The dynamical behavior of a predator- prey system with Gompertz growth function and impulsive dispersal of prey between two patches. Math. Methods Appl. Sci., 39(13):3623–3639, 2016.
• Li Zu, Daqing Jiang, and Donal O’Regan. Stochastic permanence, sta- tionary distribution and extinction of a single-species nonlinear diffu- sion system with random perturbation. Abstr. Appl. Anal., pages Art. ID 320460, 14, 2014.