Traveling Waves And Entire Solutions For Lattice Differential Equation

Lattice dynamical systems usually refer to infinite systems of ordinary differ-ential equations on discrete space or infinite systems of difference equations (such as the D-dimensional integer lattice ZD). Such systems arise, on the one hand, from practical backgrounds, such as modeling in biology, electrical circuit theory, material science, image processing and chemical kinetics can be induced to lattice dynamical systems. On the other hand, they also arise as the spatial discretization of partial differential equations. Therefore, it is more meaningful and valuable in theory and practice to study such equations. In this thesis, we consider traveling wave solutions and entire solutions of lattice dynamical systems. Here, the entire solutions are defined in the whole space and for all time t∈R.Our thesis firstly considers the entire solutions of a lattice reaction-diffusion-convection equation with bistable nonlinearity in periodic media. Using a priori estimates of the exponential decaying of the traveling wave solution at infinity, we can construct suitable sub-super solutions. Then the existence of entire solutions is obtained via sub-super solutions method and the comparison principle. With the appearing of the convection, the discretization of space and the periodic media, two traveling waves solutions with opposite directions may admit different speeds (that is, C1≠C1), and thus lose their symmetry, which requires us to construct the different sub-super solutions. On the basis of existence, uniqueness and Liapunov stability of entire solutions are established further.Next, we study the entire solutions of a two-dimensional (2-D) lattice dy-namical system with monostable nonlinearity. Due to the discretization of space, c* depends on the directionθof the traveling wave. Traveling waves coming from different directionsθandθmay admit c*(θ)≠c*(θ). Therefore, for a limited number of traveling wave solutions with different propagation speeds from differ-ent directions, the propagation speed may be smaller than c*(θ) and c*(θ). While the propagation speeds are larger than, smaller than or equal to c*(θ) and c*(θ). by solving sequence of Cauchy problems starting at times -k with suitable initial conditions, and using the maximum principle and the comparison principle, we gain the existence of entire solutions and the continuous dependence of entire solutions on the parameters.In the population dynamics, lattice dynamical system is used to describe pro-cesses of the growth and invasion of species in the space of discrete plaque environ-ment. A reaction-diffusion system with a quiescent stage on a 2D spatial lattice for a single-species population with two separate mobile and stationary states, de-scribes a species population of which the individuals alternate between mobile and stationary states, and only the mobile ones reproduce. In the third part, using the comparison principle with appropriate sub-solutions and upper estimates, some new entire solutions are constructed by combining spatially independent solutions and traveling fronts with different wave speeds and directions of propagation.Finally, we consider the periodic traveling wave solutions of a nonlocal integro-differential equation with bistable nonlinearity. This kind of equation describes the diffusion process by the convolution operator. It can arise from the biological populations, ecological and infectious diseases and many other research areas. Under the bistable hypotheses, using the sub- and upper- solutions method, comparison principle and squeezing technique, we prove the stability and the uniqueness of periodic traveling wave solutions with phase shift. Then, we prove the existence of traveling wave solutions by the theory of monotone dynamical systems.