On A Class Of Neural Networks And Food Chains Model With Reaction-diffusion Terms

Reaction diffusion equation with delays is a class of abstract functional differential equations, it is widely applied in many areas. The following is the general reaction diffusion equation:whereΩis a bounded domain in Rⁿ with smooth boundary, d(x)≥0,α(x)∈C^1+α((?)Ω)α∈(0,1). (?) denotes outward normal derivative on (?)Ω,φis a given vector function. The equation applied in population ecology, control theory, climate models and epidemic models and so on. In this paper, we dedicate ourselves to dynamic behavior of a class of neural networks and prey-predator models with reaction diffusion terms.This paper includes in two parts. In the first part, we discuss dynamic behavior of a reaction-diffusion delayed neural networks with Dirichlet boundary conditions. First of all, we consider exponential stability of a time-varying delayed reaction-diffusion cellular neural networks with Dirichlet boundary conditions. Without assuming monotonicity of activation functions, we obtain sufficient conditions guaranteeing globally exponential stability of the system by using variation parameters and inequality techniques. Those conditions play important roles in designing exponential stability of the system and applications. Nextly, we investigate dynamic behavior of a reaction-diffusion delayed neural networks with mixed boundary conditions. Using upper and lower solutions method, we obtain that if activation functions in the neural networks have mixed quasi-monotonic property and the corresponding elliptic system exists upper and lower solutions, the neural networks exist unique non-constant solution. In additional conditions, we further obtain that the unique non-constant solution of the neural networks is convergent to unique solution of corresponding elliptic system. Employing drive-response concept and Hardy inequality and Liapunov functional method, we give the sufficient conditions guaranteeing ex- ponential synchronization on drive-response systems. Meanwhile, we find that the conditions are easy verified and dependent on diffusive coefficients and controller gain matrix in response system.In the second part, we discuss dynamic behavior of a class of prey-predator model with reaction-diffusion terms. In view of invariant principle in reaction diffusion equation and Liapunov functional method, we analyze the globally asymptotical stability of unique equilibrium in this system. Nextly, we give the criterion of persistence using the related results by Dunbar(1986), Hutson and Moran(1987).