The Research On Spatiotemporal Patterns Of A Reaction-Diffusion Substrate-inhibition Seelig Model

This paper mainly studies the spatiotemporal pattern of a reaction-diffusion substrate-inhibition Seelig model.Firstly, the dynamical behavior of ordinary differential equations are analyzed, and the global existence and boundedness of the solutions are proved by using invariant rectangle method. The existence conditions of locally asymptotic steady state solution, locally asymptotic stable periodic solution and global asymptotic stable steady state solution are obtained respectively.Secondly, the existence of attraction region of the system is proved by using the maximal principle of the parabolic system and the Hopf boundary lemma for elliptic systems. That is,regardless of the initial value of the system, all the solutions will be drawn to a fixed rectangle,provided that the time is sufficiently large.Then, we derive suitable conditions, so that the solutions of the parasolic solutions converge to either the unique positive constant equilibrium solution or to the stable pariodic solutions.Finally, we study the existence and nonexistence of Turing patterns by using global steady state bifurcation theorems. The theoretical results provide a theoretical evidence for the clearer understanding of the rich dynamical behavior of the system and the formation mechanism of the pattern formations.