Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching

Xiaoyue Li, Alison Gray, Daqing Jiang, Xuerong Mao

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In this paper, we prove that a stochastic logistic population under regime switching controlled by a Markov chain is either stochastically permanent or extinctive, and we obtain the 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.
Original languageEnglish
Pages (from-to)11-28
Number of pages18
JournalJournal of Mathematical Analysis and Applications
Issue number1
Publication statusPublished - 1 Apr 2011


  • brownian motion
  • stochastic differential equation
  • generalized Itô's formula
  • markov chain
  • stochastic permanence

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