Dynamics And Pattern Formations Of The Sporns-Seelig Model: Linear Diffusion Versus Nonlinear Diffusion

This paper studies a reaction-diffusion substrate-enzyme Sporns-Seelig system which was used to model the genetic regulatory mechanism of enzyme induction. To study the influences of diffusions on the emergence of spatiotemporal patterns, we use the qualitative theory of infinite dimensional dynamical system and the branch theory and other methods, we consider the problem in two cases: classical linear diffusion and nonlinear density-dependent diffusion.Dynamics and pattern formations of both classical linear diffusion problem and the nonlinear diffusion problem are considered in details.The main contents of the paper are as follows:Chapter one: This article mainly from the two aspects of domestic and foreign to this the-sis research background and the latest research situation. Then it gives a brief introduction the research significance and research papers need to master the basic knowledge and summarized the research work of this thesis.Chapter two: This paper studies the dynamics and pattern formations of this model in linear diffusion. We use the knowledge of invariant rectangle to prove that an attraction region which attracts all the solutions of the system. Then by constructing suitable Lyapunov func-tions, we are able to derive precise conditions so that the unique constant equilibrium solution is globally asymptotically stable. In particular, the stability of the solution is proved by using the stability theory of ordinary differential equations. Then, a series of useful inequalities are used to prove the nonexistence of non-constant steady states. Finally, we use the Hopf bi-furcation theorem and the theorem of steady-state to perform detailed spatiotemporal patterns bifurcation analysis under the one-dimensional case.Chapter three: This paper studies the dynamics and pattern formations of this model in nonlinear diffusion. Since the system with nonlinear diffusion is the degenerate system, we construct the corresponding perturbed non-degenerate system to the original one and prove the existence and boundedness of weak solutions. Then, This paper can get a priori estimates for the positive non-constant steady states by the user-friendly maximum principle and use the Leray-Schuder topological degree theory and homotopy invariance to prove the existence of non-constant steady state solutions of the system.