Dynamical Properties Of A Class Of Reaction-diffusion-ordinary Differential Systems

Reaction diffusion equation models mainly investigate the interaction among species,which has important significance on the protection of species diversity and ecological balance.Because of the natural ecological law among species,we notice that the competition and cooperation relationship among species with quiescent and active species state plays an important role in the dynamical behaviors of population.Therefore,the dynamical properties of a class of reaction-diffusion-ordinary differential equations is studied in this paper.The trivial stability of the positive equilibria in a convex domain is described.In this paper,we analyze the stability of the reaction-diffusion-ordinary systems by using linearization and Lyapunov functionals.Firstly,we obtain the existence of the non-negative solution and the equilibria of the system,and prove the globally asymptoticial stability of the positive constant equilibrium of the reaction-diffusion-ordinary differential system by constructing Lyapunov functionals;Finally,we use the basic elliptic equation tools to determine that the non-constant equilibrium is unstable.