Numerical Investigation Of Several Classes Of Biological Dynamical Systems

The goal of dynamical systems is to investigate the evolution of complex systems.The research of biological dynamical systems is to construct reasonable biological mathematical model,and investigate it by mathematical methods,then deal with problems in biology.On the one hand,stochastic fluctuations are ubiquitous in the complex system,and they may affect the properties of dynamical systems in a profound way.On the other hand,there are often time delays in dynamical systems.It needs time to transport the relevant material,energy and information of the dynamical systems.Therefore,to make the research more realistic,it is crucial to take randomness and delay into account in the study of dynamical systems.This thesis focuses on:(i)we discuss numerical method of the deterministic quanti-ties(e.g.the mean first exit time and the first escape probability)in stochastic dynamical systems,and make use of these deterministic quantities to study the transition of gene tran-scriptional regulatory network;(ii)we consider the compact difference schemes for the delay diffusion problems and nonlinear diffusion problems of the biological dynamical systems.The thesis is organised as follows.In Chapter 1,we discuss research background,current status analysis,research con-tents,importance and highlights of the thesis.Chapter 2 is devoted to the review of relevant definitions and concepts of the determin-istic dynamical systems and the stochastic dynamical systems.In Chapter 3,we study numerical method of the deterministic quantities(the mean first exit time and the first escape probability)in stochastic dynamical systems,and apply these deterministic quantities to the gene transcriptional regulatory network.First,we present the definitions and concepts of these deterministic quantities.Then,we discuss the numerical method of the mean first exit time and the first escape probability,which are solutions of some differential-integral equations.Finally,these deterministic quantities are utilised to quantify the transition in a two-dimensional gene transcriptional regulatory network.In Chapter 4,we introduce compact difference method to solve one class of biological diffusion problem.After discussing the research background and current research analysis of this problem,we construct the numerical scheme of the compact difference method and prove the stability and convergence of the proposed method.We finally perform numerical simulations to verify the effectiveness of the proposed method.In Chapter 5,we study compact alternating direction implicit method which is used to solve the two-dimensional nonlinear biological diffusion problem.First,we introduce the research background and existing works of this problem.Next,we present the construction of the compact alternating direction implicit method.Then,we analyse the convergence and stability of the proposed method in the sense of H1 and L? norms.Finally,some numerical examples are conducted to validate the effectiveness of the proposed method.In Chapter 6,we summarize the contributions of the thesis,and present some future research topics.