Existence Of Travelling Wave Solutions To Delayed Lattice Differential Equations

In the developing of modern science and technology, disciplinarian veracity is an important assurance for their development. The disciplinarian veracity is usually realised by establishing mathematical models. And reaction diffusion equations are common models in the description of linear or nonlinear systems such as physical phenomena, chemical processes, ecological systems. At the present time a large number of papers are devoted to reaction diffusion equations and this number continues to increase. Especially in the field of physical phenomena, chemical processes, ecological systems and so on, lots of models of these fileds can be reduced to problems of the existence, uniqueness, stability and estimate of wave speed of the travelling wave solutions for the reaction diffusion equations.This article discusses mostly the existence of travelling wavefronts of a delayed lattice model for population and the existence of travelling waves for a diffusive population model with discrete time delay. The existence of such solutions is proved by the technique developed by J.Wu and X.Zou. Thus the existence of travelling wave fronts is reduced to the existence of an admissible pair of upper and lower solutions. For delayed lattice reaction-diffusion systems, the existence of travelling wavefronts is provided. Firstly, the monotone iteration scheme, together with the upper-lower solutions technique, is applied to establish the existence of wavefronts for the delayed-lattice reaction diffusion systems. Secondly, we construct a pair of properly upper and lower solutions for the systems.The upper and lower solutions method (or monotone method) is a very effective method in dealing with the existence of travelling wave solutions in some material reaction-diffusion equations. And the key of monotone method is constructing appropriate upper and lower solutions. The reaction-diffusion systems which can be dealed with monotone method have following feactures: their reaction terms must have some monotonicity or quasi-monotonicity.