Czechoslovak Mathematical Journal, Vol. 67, No. 4, pp. 867-890, 2017

Population dynamical behavior of a single-species nonlinear diffusion system with random perturbation

Li Zu, Daqing Jiang, Donal O'Regan

Received June 28, 2015.   First published October 24, 2017.

Abstract:  We consider a single-species stochastic logistic model with the population's nonlinear diffusion between two patches. We prove the system is stochastically permanent and persistent in mean, and then we obtain sufficient conditions for stationary distribution and extinction. Finally, we illustrate our conclusions through numerical simulation.
Keywords:  stochastic permanence; persistent in mean; extinction; stationary distribution
Classification MSC:  34F05, 92D25
DOI:  10.21136/CMJ.2017.0350-15

[1] L. J. S. Allen: Persistence and extinction in single-species reaction-diffusion models. Bull. Math. Biol. 45 (1983), 209-227. DOI 10.1007/BF02462357 | MR 0707172 | Zbl 0543.92020
[2] S. Cerrai: Second Order PDE's in Finite and Infinite Dimension. A Probabilistic Approach. Lecture Notes in Mathematics 1762, Springer, Berlin (2001). DOI 10.1007/b80743 | MR 1840644 | Zbl 0983.60004
[3] L. S. Chen, J. Chen: Nonlinear Biological Dynamical System. Science Press, Beijing (1993).
[4] G. Da Prato: Kolmogorov Equations for Stochastic PDEs. Advanced Courses in Mathematics-CRM Barcelona, Birkhäuser, Basel (2004). DOI 10.1007/978-3-0348-7909-5 | MR 2111320 | Zbl 1066.60061
[5] T. C. Gard: Introduction to Stochastic Differential Equations. Pure and Applied Mathematics 114, Marcel Dekker, New York (1988). DOI 10.1002/asm.3150040209 | MR 0917064 | Zbl 0628.60064
[6] D. J. Higham: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43 (2001), 525-546. DOI 10.1137/S0036144500378302 | MR 1872387 | Zbl 0979.65007
[7] N. Iked, S. Watanabe: Stochastic Differential Equations and Diffusion Processes. North-Holland Mathematical Library 24, North-Holland, Amsterdam; Kodansha, Tokyo (1989). MR 1011252 | Zbl 0684.60040
[8] C. Ji, D. Jiang, H. Liu, Q. Yang: Existence, uniqueness and ergodicity of positive solution of mutualism system with stochastic perturbation. Math. Probl. Eng. 2010 (2010), Article ID 684926, 18 pages. DOI 10.1155/2010/684926 | MR 2670476 | Zbl 1204.34065
[9] C. Ji, D. Jiang, N. Shi: Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation. J. Math. Anal. Appl. 359 (2009), 482-498. DOI 10.1016/j.jmaa.2009.05.039 | MR 2546763 | Zbl 1190.34064
[10] D. Jiang, N. Shi: A note on nonautonomous logistic equation with random perturbation. J. Math. Anal. Appl. 303 (2005), 164-172. DOI 10.1016/j.jmaa.2004.08.027 | MR 2113874 | Zbl 1076.34062
[11] D. Jiang, N. Shi, X. Li: Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation. J. Math. Anal. Appl. 340 (2008), 588-597. DOI 10.1016/j.jmaa.2007.08.014 | MR 2376180 | Zbl 1140.60032
[12] R. Khas'minskiĭ: Stochastic Stability of Differential Equations. Monographs and Textbooks on Mechanics of Solids and Fluids. Mechanics: Analysis, 7. Sijthoff Noordhoff, USA; Alphen aan den Rijn, The Netherlands (1980). MR 0600653 | Zbl 0441.60060
[13] X. Li, A. Gray, D. Jiang, X. Mao: Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching. J. Math. Anal. Appl. 376 (2011), 11-28. DOI 10.1016/j.jmaa.2010.10.053 | MR 2745384 | Zbl 1205.92058
[14] H. Liu, Q. Yang, D. Jiang: The asymptotic behavior of stochastically perturbed DI SIR epidemic models with saturated incidences. Automatica 48 (2012), 820-825. DOI 10.1016/j.automatica.2012.02.010 | MR 2912805 | Zbl 1246.93117
[15] M. Liu, K. Wang: Persistence and extinction in stochastic non-autonomous logistic systems. J. Math. Anal. Appl. 375 (2011), 443-457. DOI 10.1016/j.jmaa.2010.09.058 | MR 2735535 | Zbl 1214.34045
[16] Z. Lu, Y. Takeuchi: Global asymptotic behavior in single-species discrete diffusion systems. J. Math. Biol. 32 (1993), 67-77. DOI 10.1007/BF00160375 | MR 1256831 | Zbl 0799.92014
[17] X. Mao: Stochastic Differential Equations and Their Applications. Ellis Horwood Series in Mathematics and Its Applications, Horwood Publishing, Chichester (1997). DOI 10.1533/9780857099402 | MR 1475218 | Zbl 0892.60057
[18] X. Mao, C. Yuan: Stochastic Differential Equations with Markovian Switching. Imperial College Press, London (2006). DOI 10.1142/p473 | MR 2256095 | Zbl 1126.60002
[19] X. Mao, C. Yuan, J. Zou: Stochastic differential delay equations of population dynamics. J. Math. Anal. Appl. 304 (2005), 296-320. DOI 10.1016/j.jmaa.2004.09.027 | MR 2124664 | Zbl 1062.92055
[20] A. Okubo: Diffusion and Ecological Problems: Mathematical Models. Biomathematics, vol. 10, Springer, Berlin (1980). DOI 10.1002/bimj.4710240311 | MR 0572962 | Zbl 0422.92025
[21] G. Strang: Linear Algebra and Its Applications. Academic Press (A Subsidiary of Harcourt Brace Jovanovich, Publishers), New York (1980). MR 0575349 | Zbl 0463.15001
[22] C. Zhu, G. Yin: Asymptotic properties of hybrid diffusion systems. SIAM J. Control Optim. 46 (2007), 1155-1179. DOI 10.1137/060649343 | MR 2346378 | Zbl 1140.93045

Affiliations:   Li Zu, College of Mathematics and Statistics, Hainan Normal University, Longkun South Road, Hainan 571158, Haikou, Hainan, China, e-mail:; Daqing Jiang (corresponding author), School of Science, China University of Petroleum, 66 Changjiang West Road, Qingdao 266580, Huangdao, Shandong, China, and Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, King Abdulaziz University, Umm Al Muminin, Al-Sharafeyah, Jeddah 23218, Saudi Arabia, e-mail:; Donal O'Regan, School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, University Road, Galway H91 TK33, Ireland, e-mail:

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