Dynamic Analysis Of Two Types Of Gene Regulatory Network Model

In this paper,gene regulatory network models are studied by using the theory and methods of nonlinear dynamics.We mainly consider the following contents:first of all,the impacts of multiple time delays on a gene regulatory network mediated by small noncoding RNA is studied.Second,we study the bifurcation behaviors of a continuous gene regulation model,mainly considering Hopf bifurcation and Bogdanov-Takens bifurcation.The last of all,we discretize the above continuous gene regulation model by using the forward Euler method,and study some codimension-one and codimensiontwo bifurcations of the discrete system.The dissertation is summarized as follows:In chapters 1 and 2,we mainly introduce the research background,the current situation of research status,the background knowledge of gene regulation network,and the relevant concepts,theorems and conclusions of some nonlinear dynamic systems that will be used in this paper.In chapter 3,the impacts of multiple time delays on a gene regulatory network mediated by small noncoding RNA is studied.By analyzing the associated characteristic equation of the corresponding linearized system,the asymptotic stability of the positive equilibrium is investigated and Hopf bifurcation is demonstrated.Furthermore,the explicit formulae for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are given by the center manifold theorem and the normal form theory for functional differential equations.Finally,some numerical simulations are demonstrated for supporting the theoretical results.In chapter 4,the stability and two kinds of bifurcations of a genetic regulatory network are considered.We give a complete stability analysis involved in mentioned model.The Hopf bifurcation of codimension 1 and Bogdanov-Takens bifurcation of codimension 2 for the nonhyperbolic equilibria of the model is characterized analytically.In order to determine the stability of limit cycle of Hopf bifurcation,the first Lyapunov number is calculated and a numerical example is given to illustrate graphically.Three bifurcation curves related to Bogdanov-Takens bifurcation,namely the saddle-node,Hopf and homoclinic bifurcation curves,are given explicitly by calculating a universal unfolding near the cusp.Moreover,the numerical simulation results show that the model has other bifurcation types,including saddle-node and cusp bifurcations.The bifurcation diagram and phase portraits are also given to verify the validity of the theoretical results.These results show that there exists rich bifurcation behavior in the genetic regulatory network.In chapter 5,we study the bifurcation and stability of a discrete gene regulatory network.Firstly,we discuss the existence and stability conditions of the fixed points.Secondly,the conditions for existence of three cases of codimension-one bifurcation(fold bifurcation,flip bifurcation and Naimark-Sacker bifurcation)are derived by using the center manifold theorem and bifurcation theory.Then,the conditions for the occurrence of codimension-two bifurcation(fold-flip bifurcation,1:2,1:3 and 1:4 strong resonance)are investigated by using several variable substitutions and introduction of new parameters.Meanwhile,these bifurcation curves are returned to the original variables and parameters to express for easy verification.Finally,numerical simulations not only show the validity of the proposed results,but also exhibit the interesting and complex dynamical behaviors.In chapter 6,the summarizes of the dissertation are given and some meaningful issues worth studying in the future are also raised.