Divergence Between Two Types Of Reaction Diffusion Model And Stability Study

Many reaction diffusion models involving biology, chemistry and physics have been built in modern science research. By learning these equations, we can do some reasonable explanations and predictions, also, the work efficiency can be greatly improved. The pioneer-climax model and the Gray-Scott model are investigated in this paper. The former is used to study biological growth and the latter is about chemical reaction.pioneer-climax modelGray-Scott modelThe main contents in this paper are as follows:In Chapter1, we study the global bifurcation of the pioneer-climax model. The priori estimates of steady-state solution are given in section1. In section2, the structure of local bifurcation solution is obtained by using bifurcation theory, the degree theory and the maximum principle. At last, we describe the structure of global bifurcation solution.In chapter2, the bifurcation and the stability of the steady-state solutions of the pioneer-climax model is researched. Firstly, we gain the sufficient condition for the existence of positive solution. Secondly, the stability results of the bifurcation solution are obtained through using perturbation theory of linear operators and stability theory of bifurcation solutions. In chapter3, the Gray-Scott model is investigated. In section1, the priori esti-mates of positive solution are gotten by using the maximum principle and Harnack inequality. In section2, turing instability and global stability of positive constant steady-state solution are given by specturm theory and Turing theory. In section3, the structure of local bifurcation solution is given by bifurcation theory. In section4, we research the structure of global bifurcation solution by using degree theory and prove that the local branch can be extended to the global branch. In section5, the simulation is mainly done.