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Article

Keywords:
persistence; extinction; Markov switching; delay; stochastic perturbations
Summary:
The paper is concerned with a stochastic delay predator-prey model under regime switching. Sufficient conditions for extinction and non-persistence in the mean of the system are established. The threshold between persistence and extinction is also obtained for each population. Some numerical simulations are introduced to support our main results.
References:
[1] Cheng, S. R.: Stochastic population systems. Stochastic Anal. Appl. 27 (2009), 854-874. DOI 10.1080/07362990902844348 | MR 2541380 | Zbl 1180.92071
[2] Gard, T. C.: Introduction to Stochastic Differential Equations. Pure and Applied Mathematics 114 Marcel Dekker, New York (1988). MR 0917064 | Zbl 0628.60064
[3] Higham, D. J.: 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
[4] Li, X., Gray, A., Jiang, D., Mao, X.: 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
[5] Liu, M., Li, W., Wang, K.: Persistence and extinction of a stochastic delay logistic equation under regime switching. Appl. Math. Lett. 26 (2013), 140-144. DOI 10.1016/j.aml.2012.04.010 | MR 2971415 | Zbl 1270.34188
[6] Liu, M., Wang, K.: Persistence, extinction and global asymptotical stability of a non-autonomous predator-prey model with random perturbation. Appl. Math. Modelling 36 (2012), 5344-5353. DOI 10.1016/j.apm.2011.12.057 | MR 2956748 | Zbl 1254.34074
[7] Luo, Q., Mao, X.: Stochastic population dynamics under regime switching II. J. Math. Anal. Appl. 355 (2009), 577-593. DOI 10.1016/j.jmaa.2009.02.010 | MR 2521735 | Zbl 1162.92032
[8] Mao, X., Marion, G., Renshaw, E.: Environmental Brownian noise suppresses explosions in population dynamics. Stochastic Processes Appl. 97 (2002), 95-110. DOI 10.1016/S0304-4149(01)00126-0 | MR 1870962 | Zbl 1058.60046
[9] Mao, X., Yuan, C.: Stochastic Differential Equations with Markovian Switching. World Scientific Hackensack (2006); Imperial College Press, London, 2006. MR 2256095 | Zbl 1126.60002
[10] May, R. M.: Stability and Complexity in Model Ecosystems. With a new introduction by the author. 2nd ed. Princeton Landmarks in Biology Princeton University Press, Princeton (2001). Zbl 1044.92047
[11] Samanta, G. P.: Influence of environmental noise in Gompertzian growth model. J. Math. Phys. Sci. 26 (1992), 503-511. Zbl 0778.92017
[12] Samanta, G. P.: Logistic growth under colored noise. Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 37 (1993), 115-122. MR 1375084 | Zbl 0840.92019
[13] Samanta, G. P., Chakrabarti, C. G.: On stability and fluctuation in Gompertzian and logistic growth models. Appl. Math. Lett. 3 (1990), 119-121. DOI 10.1016/0893-9659(90)90153-3 | Zbl 0707.92021
[14] Zhu, C., Yin, G.: On hybrid competitive Lotka-Volterra ecosystems. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71 (2009), e1370--e1379. DOI 10.1016/j.na.2009.01.166 | MR 2671923 | Zbl 1238.34059
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