Properties of a Stochastic Predator-Prey System with Holling II Functional Response
In this paper, a stochastic predator-prey system with Holling II functional response is studied. First, we show that there is a unique positive solution to the system for any given positive initial value. Then, stochastically bounded of the positive solution to the stochastic system is derived. Moreover, sufficient conditions for global asymptotic stability are also established. In the end, some simulation figures are carried out to support the analytical findings.
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[5] C. Ji, D. Q. Jiang and X. Li, Qualitative analysis of a stochastic ratiodependent
predator-prey system, Journal of Computational and Applied
Mathematics, 2011, 235, 1326-1341.
[6] J. Lv and K. Wang, Asymptotic properties of a stochastic predator-prey
system with Holling II functional response, Commun Nonlinear Sci
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[7] D. Q. Jiang, N. Shi and X. Li, Global stability and stochastic permanence
of a non-autonomous logistic equation with random perturbation, J.
Math. Anal. Appl, 2008, 340, 588-597.
[8] C. Ji, D. Q. Jiang and N. Jiang, Analysis of a predator-prey model
with modified Leslie-Gower and Holling-type II schemes with stochastic
perturbation, J Math Anal Appl, 2009, 359, 482-498.
[9] X. Mao, Stochastic differential equations and applications, Chichester,
England: Horwood Publishing, 1997.
[10] Q. Luo and X. Mao, Stochastic population dynamics under regime
switching, J Math Anal Appl, 2007, 334, 69-84.
[11] X. Li and X. Mao, Population dynamical behavior of non-autonomous
Lotka-Volterra competitive system with random perturbation, Discrete
Contin Dyn Syst, 2009, 24, 523-545.
[12] D. Higham, An algorithmic introduction to numerical simulation of
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<p>[1] A. Lotka, Elements of Physical Biology, Baltimore, USA: Williams
and Wilkins, 1925.
[2] V. Volterra, Variazioni e fluttuazioni del numero dindividui in specie
danimali conviventi, Mem. Acad. Lincei, 1926, 2, 31-113.
[3] Y. Li and H. Gao, Existence, uniqueness and global asymptotic stability
of positive solutions of a predator-prey system with Holling II functional
response with random perturbation, Nonlinear Anal, 2008, 68, 1694-
1705.
[4] M. Liu and K. Wang, Global stability of a nonlinear stochastic predatorprey
system with Beddington-DeAngelis functional response, Commun.
Nonlinear Sci. Numer. Simulat, 2011, 16, 1114-1121.
[5] C. Ji, D. Q. Jiang and X. Li, Qualitative analysis of a stochastic ratiodependent
predator-prey system, Journal of Computational and Applied
Mathematics, 2011, 235, 1326-1341.
[6] J. Lv and K. Wang, Asymptotic properties of a stochastic predator-prey
system with Holling II functional response, Commun Nonlinear Sci
Numer Simulat, 2011, 16, 4037-4048.
[7] D. Q. Jiang, N. Shi and X. Li, Global stability and stochastic permanence
of a non-autonomous logistic equation with random perturbation, J.
Math. Anal. Appl, 2008, 340, 588-597.
[8] C. Ji, D. Q. Jiang and N. Jiang, Analysis of a predator-prey model
with modified Leslie-Gower and Holling-type II schemes with stochastic
perturbation, J Math Anal Appl, 2009, 359, 482-498.
[9] X. Mao, Stochastic differential equations and applications, Chichester,
England: Horwood Publishing, 1997.
[10] Q. Luo and X. Mao, Stochastic population dynamics under regime
switching, J Math Anal Appl, 2007, 334, 69-84.
[11] X. Li and X. Mao, Population dynamical behavior of non-autonomous
Lotka-Volterra competitive system with random perturbation, Discrete
Contin Dyn Syst, 2009, 24, 523-545.
[12] D. Higham, An algorithmic introduction to numerical simulation of
stochastic differential equations, SIAM Rev, 2001, 43, 525-546.</p>
@article{"International Journal of Engineering, Mathematical and Physical Sciences:65375", author = "Xianqing Liu and Shouming Zhong and Fuli Zhong and Zijian Liu", title = "Properties of a Stochastic Predator-Prey System with Holling II Functional Response", abstract = "In this paper, a stochastic predator-prey system with Holling II functional response is studied. First, we show that there is a unique positive solution to the system for any given positive initial value. Then, stochastically bounded of the positive solution to the stochastic system is derived. Moreover, sufficient conditions for global asymptotic stability are also established. In the end, some simulation figures are carried out to support the analytical findings.
", keywords = "stochastically bounded, global stability, Holling II
functional response, white noise, Markovian switching.", volume = "7", number = "4", pages = "735-7", }
