Asymptotic Behavior And Dissipative Control For Two Kinds Of Stochastic Biological Models

This thesis investigates the dynamic behavior of two kinds of stochastic biological mod-els.Considering the effect of white noise,Markov switching and diffusion on the asymptotic behavior of systems,constructing proper Lyapunov-Krasvoskii functions,using Ito calculus theory,stochastic process theory and linear matrix inequality(LMI)technology,we investi-gate the dynamic behaviors of these two kinds of stochastic biological models,including the long-time behavioral characteristics,stationary distribution and dissipative control and so on.Numerical simulations showing the complex dynamics of stochastic biological systems are given.The full text consists of seven chapters and the main results are described as follows:In the first chapter,the research background is introduced on three-species food chain model and gene regulation network model.The development history and current situation of these two kinds of biological mathematical models are mainly reviewed.Finally,the main research work and structure of this paper are given.In chapter 2,preliminary knowledge and basic mathematical tools is presented briefly.Especially the conclusions of the lemma,theorem and matrix inequality needed in this paper.Chapter 3 is concerned with the long-time behavioral characteristics of the three-species food chain stochastic model.By Markov semigroup theory,it is proved that the distribution density of the solution converges to a constant density or weak convergence to a singular measure with L1.And the behavioral characteristics of the solution are analyzed by numeri-cal simulation.In chapter 4,we study the asymptotic behavior and stationary distribution of the three-species food chain stochastic model under Markov switching,and prove the existence of global positive solutions and obtain the sufficient conditions of persistence in time average and extinction in probability.In the relatively weak white noise,we prove that there exists a unique stationary distribution and its ergodicity.Finally,numerical simulations are given to verify the correctness of the theory.Chapter 5 deals with the dissipative control problem of the three-species food chain stochastic model under hidden Markov switching.For the hidden Markov chain,we use Wonham filter to estimate the unobservable stochastic process effectively.The H? control and passive control were introduced into the three-species food chain stochastic model,and the stability of this model was analyzed by passive control and H? control respectively.Finally,numerical simulation is given.Chapter 6 focuses on the dissipative control of the genetic regulatory network model with fractional Brownian motion,time-varying delay,Markov chain and reaction diffusion processes.By constructing the appropriate Lyapunov-Krasovskii function,using the linear matrix inequality technique(LMI),we obtain several sufficient conditions for the global dissipation and ? dissipation.Finally we give two numerical examples to simulate.At the end of the paper,the research results are summarized and further research plan is put forward.