Morphogenesis Of A Generalized Plant And Water Diffusion Model

Plant and water model mainly studies the interaction between plant and water.It can give early warning signals of land desertification in semi-arid areas.Many scholars have done a lot of research on this model.This paper studies the morphogenesis of a generalized plant and water diffusion model,and discusses the cases where the nonlinear term is sublinear,linear and superlinear.Firstly,the dynamic behavior of its ordinary differential equation is studied,the number of solutions and stability interval are obtained.Then we obtain the prior estimate of the reaction-diffusion equation and the existence of the global solution.Finally,the local stability of constant equilibrium solutions of reaction-diffusion equations,and the global stability in the case of sublinearity are discussed.The nonexistence of positive solutions is proved,and the existence of non constant positive solutions under the superlinear condition is proved by using topological degree theory.Our results in reality can be explained as pattern formation of the model.