Stochastic permanence, stationary distribution and extinction of a single-species nonlinear diffusion system with random perturbation. (English) Zbl 1406.92546

Summary: We analyze the influence of stochastic perturbations on a single-species logistic model with the population’s nonlinear diffusion among \(n\) patches. First, we show that this system has a unique positive solution. Then we obtain sufficient conditions for stochastic permanence and persistence in mean, stationary distribution, and extinction. Finally, we illustrate our conclusions through numerical simulation.


92D25 Population dynamics (general)
92D40 Ecology
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
Full Text: DOI


[1] Allen, L. J. S., Persistence and extinction in single-species reaction-diffusion models, Bulletin of Mathematical Biology, 45, 2, 209-227 (1983) · Zbl 0543.92020 · doi:10.1016/S0092-8240(83)80052-4
[2] Lu, Z. Y.; Takeuchi, Y., Global asymptotic behavior in single-species discrete diffusion systems, Journal of Mathematical Biology, 32, 1, 67-77 (1993) · Zbl 0799.92014 · doi:10.1007/BF00160375
[3] Allen, L., Persistence, extinction, and critical patch number for island populations, Bulletin of Mathematical Biology, 65, 1-12 (1987) · Zbl 0603.92019
[4] Freedman, H. I.; Takeuchi, Y., Global stability and predator dynamics in a model of prey dispersal in a patchy environment, Nonlinear Analysis: Theory, Methods & Applications, 13, 8, 993-1002 (1989) · Zbl 0685.92018 · doi:10.1016/0362-546X(89)90026-6
[5] Mao, X. R.; Yuan, C. G.; Zou, J., Stochastic differential delay equations of population dynamics, Journal of Mathematical Analysis and Applications, 304, 1, 296-320 (2005) · Zbl 1062.92055 · doi:10.1016/j.jmaa.2004.09.027
[6] Liu, M.; Wang, K., Persistence and extinction in stochastic non-autonomous logistic systems, Journal of Mathematical Analysis and Applications, 375, 2, 443-457 (2011) · Zbl 1214.34045 · doi:10.1016/j.jmaa.2010.09.058
[7] Li, X. Y.; Gray, A.; Jiang, D. Q.; Mao, X. R., Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, Journal of Mathematical Analysis and Applications, 376, 1, 11-28 (2011) · Zbl 1205.92058 · doi:10.1016/j.jmaa.2010.10.053
[8] Jiang, D. Q.; Shi, N. Z., A note on nonautonomous logistic equation with random perturbation, Journal of Mathematical Analysis and Applications, 303, 1, 164-172 (2005) · Zbl 1076.34062 · doi:10.1016/j.jmaa.2004.08.027
[9] Jiang, D. Q.; Shi, N. Z.; Li, X. Y., Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, Journal of Mathematical Analysis and Applications, 340, 1, 588-597 (2008) · Zbl 1140.60032 · doi:10.1016/j.jmaa.2007.08.014
[10] Ji, C. Y.; Jiang, D. Q.; Liu, H.; Yang, Q. S., Existence, uniqueness and ergodicity of positive solution of mutualism system with stochastic perturbation, Mathematical Problems in Engineering, 2010 (2010) · Zbl 1204.34065 · doi:10.1155/2010/684926
[11] Mao, X. R.; Yuan, C. G., Stochastic Differential Equations with Markovian Switching (2006), London, UK: Imperial College Press, London, UK · Zbl 1126.60002
[12] Ji, C.; Jiang, D.; Shi, N., 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, 2, 482-498 (2009) · Zbl 1190.34064 · doi:10.1016/j.jmaa.2009.05.039
[13] Mao, X. R., Stochastic Differential Equations and Applications (1997), New York, NY, USA: Horwood, New York, NY, USA · Zbl 0892.60057
[14] Ikeda, N.; Watanabe, S., Stochastic Differential Equations and Diffusion Processes, 24 (1989), Tokyo, Japan: North-Holland, Amsterdam, The Netherlands; Kodansha Ltd., Tokyo, Japan · Zbl 0684.60040
[15] Strang, G., Linear Algebra and Its Applications (1988), Thomson Learning Inc.
[16] Higham, D. J., An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43, 3, 525-546 (2001) · Zbl 0979.65007 · doi:10.1137/S0036144500378302
[17] Hasminskii, R. Z., Stochastic Stability of Differential Equations, 7 (1980), Alphen aan den Rijn, The Netherlands: Sijthoff & Noordhoff, Alphen aan den Rijn, The Netherlands · Zbl 0441.60060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.