Bifurcation And Pattern Analysis Of Two Kinds Of Reaction-diffusion Models

In this paper,the reaction diffusion equation in two-dimensional space region is considered,and its dynamic behavior is analyzed and discussed.Specifically,two types of models are studied,namely,forest restoration-population pressure model and FitzHugh-Nagumo(FHN)neural network model.We discuss the Turing instability of the two systems and obtain the corresponding patterns.First of all,to restore the forest logistic model,on the basis of considering the influence of the population pressure on forest restoration,under Dirichlet boundary conditions are established with Holling ? function feature of forest restoration-population pressure model.In the square region of space,the existence of the Hopf bifurcation and the Turing instability near the positive equilibrium point are analyzed.Mixed patterns are obtained when the steady-state bifurcation occurs,and hyperhexagonal patterns are obtained when the Hopf bifurcation occurs.In addition,the critical condition for the Turing instability of the FHN model with reaction diffusion under the Neumann boundary condition is derived.Unlike previous studies FHN model,interaction in different patterns,different bifurcations occur respectively to get the different patterns,namely,simple bifurcation of the pattern is stripe and rectangular,double bifurcation of the pattern is spot,square and the four-fold bifurcation occurs quadrilateral pattern.At the same time,the theoretical results are applied to the diffusion FHN model with two coupled strengths,and the influence of coupling strength on the stability of the model is given by numerical simulation.