Exponential Stability Of Stochastic Population System

The theory of stochastic differential equations(SDE) has been widely applied to control theory, Biology, communication, financial, etc. For many years, some of authors has done a great deal of research on deterministic population system. However, the study of stochastic population model is seldom seen.This paper mainly consider stochastic population system model with Markovian switching and fractional Brown motion which are external environment disturbance. The Euler-Maruyama numerical solutions method of stochastic population system and conditions of exponential stability of numerical solutions are given. The main contents are following:1. The exponential stability of numerical solutions to the stochastic competitive population system with Brown motion and Markovian switching is discussed. Euler-Maruyama method is used to define the numerical solutions. Using Ito’s formula, Barkholder-Davis-Gundy’s inequality and Gronwall’s Lemma are obtained for the exponential stability of numerical solutions to the stochastic competitive population system.2. The exponential stability of stochastic age-dependent population with diffusion in Hilbert space is discussed. The definitions of strong solution and almost surely exponential stability are given. Using Ito’s formula, exponential martingale formula, Borel-Cantelli lemma, Holder inequality,some criteria are obtained the sufficient conditions of exponential stability of stochastic age-dependent population with diffusion.3. The exponential stability of numerical solutions to the stochastic competitive population system with diffusion and fraction Brown motion is discussed. Numerical solutions are given. Using Ito’s formula, some properties of fractional Brown motion, Lipschitz condition, Doob inequality, Gronwall’s Lemma, Cauchy-Schwarz inequality, some criteria are obtained for the exponential stability of numerical solutions to the stochastic competitive population system.