Research On Bifurcation And Turing Pattern Of Gierer-Meinhardt Model

The patterns in nature are colorful and varied.The reaction-diffusion system is a vital paradigm for explaining the objective universe.The analysis of how its pattern dynamics change with system parameters will allow us to gain a thorough understanding of the essence.The Gierer-Meinhardt model is a well-known and widely used reaction-diffusion system model developed by biologists.This model has long piqued the interest of researchers.Related theories of differential equations and dynamical systems will help to better realize the transition from finite-dimensional to infinite-dimensional dynamical systems,deepening our understanding of finite-dimensional to infinite-dimensional.This paper investigates the Hopf bifurcation and Turing instability of the Gierer-Meinhardt model under Neumann boundary conditions,incorporating eigenvalue analysis of characteristic polynomials,stability theorems,normal form,and central manifold theorems.The following is a synopsis of the basic content:1.The activator-inhibitor model of the Gierer-Meinhardt model is investigated.Following derivation,the existence and stability conditions of the model’s positive equilibrium are obtained,demonstrating that the Hopf bifurcation that the model experiences within a certain parameter range is supercritical.Secondly,the effect of diffusion on the stability of the equilibrium and the stability of the periodic solution bifurcating from the Hopf bifurcation of the system is discussed,and Turing instability conditions are determined.Furthermore,through the numerical simulation of the theoretical results,the specific pattern form of the system is verified.2.The Gierer-Meinhardt model with activator saturation is investigated.The stability of the model’s unique positive equilibrium is demonstrated,as well as the precise parameter conditions for the Hopf bifurcation.The results show that the model’s Hopf bifurcation is supercritical or subcritical within a certain parameter range.It is also confirmed that,in addition to the periodic solution bifurcating from the Hopf bifurcation,there is at least one stable limit cycle.Finally,the conditions for the model’s Turing instability are addressed,and the results are confirmed by numerical simulation.According to the observations,the system’s Turing pattern is striped or spotted.