Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching. (English) Zbl 1205.92058

Summary: We prove that a stochastic logistic population under regime switching controlled by a Markov chain is either stochastically permanent or extinctive, and we obtain sufficient and necessary conditions for stochastic permanence and extinction under some assumptions. In the case of stochastic permanence we estimate the limit of the average in time of the sample path of the solution by two constants related to the stationary probability distribution of the Markov chain and the parameters of the subsystems of the population model. Finally, we illustrate our conclusions through two examples.


92D25 Population dynamics (general)
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
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[1] Ahmad, S.; Lazer, A.C., Average conditions for global asymptotic stability in a nonautonomous Lotka-Volterra system, Nonlinear anal., 40, 37-49, (2000) · Zbl 0955.34041
[2] Mao, X.; Marion, G.; Renshaw, E., Environmental Brownian noise suppresses explosions in population dynamics, Stochastic process. appl., 97, 95-110, (2002) · Zbl 1058.60046
[3] Mao, X.; Marion, G.; Renshaw, E., Asymptotic behavior of the stochastic Lotka-Volterra model, J. math. anal. appl., 287, 141-156, (2003) · Zbl 1048.92027
[4] Mao, X., Delay population dynamics and environmental noise, Stoch. dyn., 5, 2, 149-162, (2005) · Zbl 1093.60033
[5] Jiang, D.; Shi, N., A note on nonautonomous logistic equation with random perturbation, J. math. anal. appl., 303, 164-172, (2005) · Zbl 1076.34062
[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] Gard, T.C., Persistence in stochastic food web models, Bull. math. biol., 46, 357-370, (1984) · Zbl 0533.92028
[8] Gard, T.C., Stability for multispecies population models in random environments, Nonlinear anal., 10, 1411-1419, (1986) · Zbl 0598.92017
[9] Gard, T.C., Introduction to stochastic differential equations, (1998), Marcel Dekker · Zbl 0682.92018
[10] Bahar, A.; Mao, X., Stochastic delay Lotka-Volterra model, J. math. anal. appl., 292, 364-380, (2004) · Zbl 1043.92034
[11] Bahar, A.; Mao, X., Stochastic delay population dynamics, Int. J. pure appl. math., 11, 377-400, (2004) · Zbl 1043.92028
[12] Pang, S.; Deng, F.; Mao, X., Asymptotic property of stochastic population dynamics, Dyn. contin. discrete impuls. syst. ser. A math. anal., 15, 602-620, (2008)
[13] Takeuchi, Y.; Du, N.H.; Hieu, N.T.; Sato, K., Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment, J. math. anal. appl., 323, 938-957, (2006) · Zbl 1113.34042
[14] Luo, Q.; Mao, X., Stochastic population dynamics under regime switching, J. math. anal. appl., 334, 69-84, (2007) · Zbl 1113.92052
[15] Du, N.H.; Kon, R.; Sato, K.; Takeuchi, Y., Dynamical behavior of Lotka-Volterra competition systems: non-autonomous bistable case and the effect of telegraph noise, J. comput. appl. math., 170, 399-422, (2004) · Zbl 1089.34047
[16] Slatkin, M., The dynamics of a population in a Markovian environment, Ecology, 59, 249-256, (1978)
[17] Freedman, H.I.; Ruan, S., Uniform persistence in functional differential equations, J. differential equations, 115, 173-192, (1995) · Zbl 0814.34064
[18] Mao, X., Stochastic differential equations and applications, (1997), Horwood Publishing Chichester · Zbl 0874.60050
[19] Mao, X.; Yuan, C., Stochastic differential equations with Markovian switching, (2006), Imperial College Press · Zbl 1126.60002
[20] Mao, X.; Yin, G.; Yuan, C., Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica, 43, 264-273, (2007) · Zbl 1111.93082
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