Asymptotic properties of a stochastic predator-prey system with Holling II functional response. (English) Zbl 1218.92072

Lv, Jingliang; Wang, Ke

Commun. Nonlinear Sci. Numer. Simul. 16, No. 10, 4037-4048 (2011).

Summary: A stochastic predator-prey system with Holling II functional response is proposed and investigated. We show that there is a unique positive solution to the model for any positive initial value, and that the positive solution to the stochastic system is stochastically bounded. Moreover, under some conditions, we conclude that the stochastic model is stochastically permanent and persistent in mean.

MSC:

92D40	Ecology
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60H30	Applications of stochastic analysis (to PDEs, etc.)

Keywords:

stochastically ultimately bounded; stochastic permanence; persistent in mean

References:

[1]	Li, Y.; Gao, H., Existence,uniqueness and global asymptotic stability of positive solutions of a predator-prey system with Holling II functional response with random perturbation, Nonlinear Anal, 68, 1694-1705 (2008) · Zbl 1143.34033
[2]	Chen, L.; Chen, J., Nonlinear Biological Dynamical System (1993), Science Press: Science Press Beijing
[3]	Liu, M.; Wang, K., Global stability of a nonlinear stochastic predator-prey system with Beddington-DeAngelis functional response, Commun Nonlinear Sci Numer Simulat, 16, 1114-1121 (2011) · Zbl 1221.34152
[4]	Ji, C.; Jiang, D.; Shi, N., Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation, J Math Anal Appl, 359, 482-498 (2009) · Zbl 1190.34064
[5]	Mao, X.; Marion, G.; Renshaw, E., Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Process Appl, 97, 95-110 (2002) · Zbl 1058.60046
[6]	Jiang, D.; Shi, N.; Li, X., Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J Math Anal Appl, 340, 588-597 (2008) · Zbl 1140.60032
[7]	Mao, X.; Sabanis, S.; Renshaw, E., Asymptotic behaviour of stochastic Lotka-Volterra model, J Math Anal Appl, 287, 141-156 (2003) · Zbl 1048.92027
[8]	Soboleva, T. K.; Pleasants, A. B., Population growth as a nonlinear stochastic process, Math Comput Model, 38, 1437-1442 (2003) · Zbl 1044.60047
[9]	Du, N. H.; Sam, V. H., Dynamics of a stochastic Lotka-Volterra model perturbed by white noise, J Math Anal Appl, 32, 482-497 (2006)
[10]	Li, X.; Mao, X., Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation, Discrete Contin Dyn Syst, 24, 523-545 (2009) · Zbl 1161.92048
[11]	Ji, C.; Jiang, D.; Li, X., Qualitative analysis of a stochastic ratio-dependent predator-prey system, J Comput Appl Math, 235, 1326-1341 (2010) · Zbl 1229.92076
[12]	Higham, D. J., An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev, 43, 525-546 (2001) · Zbl 0979.65007