Dynamics Analysis In A Gierer-meinhardt Reaction-diffusion Model With Homogeneous Neumann Boundary Condition

Biological development is a complicated process.However,the formation of organism organs is the core stage in the process of development of embryo.The effect of concentration of “morphogen” on the formation of organism organs in the body was intensively investigated by some researchers.In order to further study the process of organ tissue formation in organs,a diffusive Gierer-Meinhardt model subject to homogeneous Neumann boundary condition is considered in the present thesis.This thesis is arranged as follows:The first chapter summarizes the research background and current situation of GiererMeinhardt reaction-diffusion model.Furthermore,the main contents of this paper are pointed out and some theorems used in this paper are introduced.Chapter 2 mainly studies the local asymptotic stability and Hopf bifurcation of the positive equilibrium of local ODE system.Some theoretical predictions are verified by numerical simulations.In the third chapter,the local asymptotical stability and Turing instability of the constant positive equilibrium of the diffusion system are discussed.Finally,numerical simulations are also provided in order to check the obtained theoretical conclusions.In Chapter 4,the bifurcation direction of homogeneous Hopf bifurcation and the stability of bifurcation periodic solution are carried out by employing the normal form method and the center manifold technique for reaction-diffusion equations.In addition,the existence of spatial inhomogeneous Hopf bifurcation is discussed.