## Abstract and Applied Analysis

### Stochastic Delay Logistic Model under Regime Switching

#### Abstract

This paper is concerned with a delay logistical model under regime switching diffusion in random environment. By using generalized Itô formula, Gronwall's inequality, and Young's inequality, some sufficient conditions for existence of global positive solutions and stochastically ultimate boundedness are obtained, respectively. Also, the relationships between the stochastic permanence and extinction as well as asymptotic estimations of solutions are investigated by virtue of $V$-function technique, $M$-matrix method, and Chebyshev's inequality. Finally, an example is given to illustrate the main results.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 241702, 26 pages.

Dates
First available in Project Euclid: 1 April 2013

https://projecteuclid.org/euclid.aaa/1364846439

Digital Object Identifier
doi:10.1155/2012/241702

Mathematical Reviews number (MathSciNet)
MR2947738

Zentralblatt MATH identifier
1251.34099

#### Citation

Wu, Zheng; Huang, Hao; Wang, Lianglong. Stochastic Delay Logistic Model under Regime Switching. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 241702, 26 pages. doi:10.1155/2012/241702. https://projecteuclid.org/euclid.aaa/1364846439

#### References

• K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol. 74, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992.
• V. Kolmanovskiĭ and A. Myshkis, Applied Theory of Functional-Differential Equations, vol. 85, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992.
• Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191, Academic Press, Boston, Mass, USA, 1993.
• A. Bahar and X. Mao, “Stochastic delay population dynamics,” International Journal of Pure and Applied Mathematics, vol. 11, no. 4, pp. 377–400, 2004.
• X. Mao, S. Sabanis, and E. Renshaw, “Asymptotic behaviour of the stochastic Lotka-Volterra model,” Journal of Mathematical Analysis and Applications, vol. 287, no. 1, pp. 141–156, 2003.
• X. Mao, G. Marion, and E. Renshaw, “Environmental Brownian noise suppresses explosions in population dynamics,” Stochastic Processes and their Applications, vol. 97, no. 1, pp. 95–110, 2002.
• D. Jiang and N. Shi, “A note on nonautonomous logistic equation with random perturbation,” Journal of Mathematical Analysis and Applications, vol. 303, no. 1, pp. 164–172, 2005.
• D. Jiang, N. Shi, and X. Li, “Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation,” Journal of Mathematical Analysis and Applications, vol. 340, no. 1, pp. 588–597, 2008.
• T. C. Gard, Introduction to Stochastic Differential Equations, vol. 114, Marcel Dekker, New York, NY, USA, 1988.
• A. Bahar and X. Mao, “Stochastic delay Lotka-Volterra model,” Journal of Mathematical Analysis and Applications, vol. 292, no. 2, pp. 364–380, 2004.
• X. Mao, “Delay population dynamics and environmental noise,” Stochastics and Dynamics, vol. 5, no. 2, pp. 149–162, 2005.
• S. Pang, F. Deng, and X. Mao, “Asymptotic properties of stochastic population dynamics,” Dynamics of Continuous, Discrete & Impulsive Systems A, vol. 15, no. 5, pp. 603–620, 2008.
• I. A. Dzhalladova, J. Baštinec, J. Diblík, and D. Y. Khusainov, “Estimates of exponential stability for solutions of stochastic control systems with delay,” Abstract and Applied Analysis, vol. 2011, Article ID 920412, 14 pages, 2011.
• Z. Yu, “Almost surely asymptotic stability of exact and numerical solutions for neutral stochastic pantograph equations,” Abstract and Applied Analysis, vol. 2011, Article ID 143079, 14 pages, 2011.
• Y. Takeuchi, N. H. Du, N. T. Hieu, and K. Sato, “Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment,” Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 938–957, 2006.
• Q. Luo and X. Mao, “Stochastic population dynamics under regime switching,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 69–84, 2007.
• X. Li, D. Jiang, and X. Mao, “Population dynamical behavior of Lotka-Volterra system under regime switching,” Journal of Computational and Applied Mathematics, vol. 232, no. 2, pp. 427–448, 2009.
• X. Li, A. Gray, D. Jiang, and X. Mao, “Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching,” Journal of Mathematical Analysis and Applications, vol. 376, no. 1, pp. 11–28, 2011.
• C. Zhu and G. Yin, “On hybrid competitive Lotka-Volterra ecosystems,” Nonlinear Analysis A, vol. 71, no. 12, pp. e1370–e1379, 2009.
• N. H. Du, R. Kon, K. Sato, and Y. Takeuchi, “Dynamical behavior of Lotka-Volterra competition systems: non-autonomous bistable case and the effect of telegraph noise,” Journal of Computational and Applied Mathematics, vol. 170, no. 2, pp. 399–422, 2004.
• M. Slatkin, “The dynamics of a population in a Markovian environment,” Ecology, vol. 59, no. 2, pp. 249–256, 1978.
• M. Liu and K. Wang, “Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment,” Journal of Theoretical Biology, vol. 264, no. 3, pp. 934–944, 2010.
• M. Liu and K. Wang, “Asymptotic properties and simulations of a stochastic logistic model under regime switching,” Mathematical and Computer Modelling, vol. 54, no. 9-10, pp. 2139–2154, 2011.
• X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, UK, 2006.