Dynamical Analysis Of Stochastic Logistic Equations With Multiplicative Noise

Microbial fermentation is a key link in the production of many products.In the actual production process,the growth of microorganisms is inevitably affected by stochasticity such as environmental noise(temperature and so on).However,the stochastic Logistic equation(SLE)can better simulate the random fluctuations of cell growth at fermentation process.Therefore,this thesis used Lyapunov functions and It(?) formula to analyze the stability,persistence and extinction(PES)and stationary distribution in SLE with multiplicative noise and stochastic delay Logistic equation(SDLE)with multiplicative noise.The main contents of this thesis are:Chapter 1 contains theories and overview of stochastic differential equations.Firstly,this chapter presents an overview to Logistic equations,an overview to fermentation process and an overview to the stability,persistence and extinction for Logistic equations.In the same way,an overview of numerical methods and theorems for stochastic equations are presented to analyze the PES and stationary distribution in Logistic equations.Last section of this chapter is the main research contents and innovations of this thesis.Chapter 2 analyzes the SLE with multiplicative noise.Firstly,this chapter presents the preliminaries which are the core of this chapter.Secondly,it presents the SLE with multiplicative noise for cell growth at fermentation process(CGFP)and proves it has a positive solution.In addition,Lyapunov functions and sufficient conditions are introduced to analyze the stability in SLE with multiplicative noise for CGFP.Lyapunov functions and the adequate criteria are obtained to analyze the persistence in the mean,stochastic persistence,extinction in the mean and stationary distribution in SLE with multiplicative noise.Subsequently,the 4th stage stochastic Runge-Kutta(SRK4)method is used for numerical solution of SLE with multiplicative noise and the last section is chapter summary.Chapter 3 analyzes the SDLE with multiplicative noise.Firstly,presents the SDLE with multiplicative noise for CGFP and proves it has a positive solution.In the same way,the adequate criteria and Lyapunov functions are introduced to dismember the stability.Moreover,the sufficient conditions and Lyapunov functions are obtained to commute the persistence in the mean,stochastic persistence,extinction in the mean and stationary distribution in SDLE with multiplicative noise.Finally,the 4-stage Runge-Kutta method(RK4)and 4-stage stochastic Runge-Kutta(SRK4)are used for the numerical solution of delay Logistic model and SDLE with multiplicative noise respectively.The final section of this chapter is chapter summary.Chapter 4 presents the summary contents of the study briefly,and provides the direction for future research.