Research Of Certain Dynamical Behaviors Of A Class Of Biochemical Reactions

In this thesis,according to the problem of between two important members of the Rho GTPase family on inhibiting crosstalk,we choose a kind of reaction-diffusion enzyme reaction model as the research object.By using the technique of invariant rectangle,the comparison principle which is applicable for the parabolic partial differential equations,the Poincare inequality,as well as the global steady state bifurcation theory,we study the global existence and boundedness of the in-time solutions,the existence and non-existence of the non-constant positive steady state solutions for this class of reaction diffusion equations.The specific research content is as follows:Firstly,by using the technique of the invariant rectangle,we prove the global existence and boundedness of the in-time solutions.Then,by combining the comparison principle which is applicable to the parabolic partial differential equations,we show the existence of the attraction region of the solutions.Secondly,by using Poincaré inequality and Cauchy-Schwartz inequality,we show the existence of the non-existence of the non-constant steady state solutions under certain conditions.Finally,by using abstract global steady state bifurcation theorem,we prove that,under certain conditions,the reaction diffusion equation will have the global steady state bifurcation curve.Then,we show the existence of the non-constant positive steady state solutions from the steady state bifurcation point of view.The theoretical results obtained in the thesis will allow for clearer understanding of the dynamics of this particular reaction diffusion system.