Spatial Dynamics Of Two Types Of Lattice Differential Systems

Lattice dynamical systems(i.e.spatially discrete reaction diffusion systems)arise from many different fields,such as biological invasion,disease spread,image processing and crystal growth and so on.As a special solution,the traveling wave solution is a class of important solution of such systems.In this thesis,we concerned with spatial dynamics of two types for lattice differential systems in the population dynamics,including the existence,asymptotic behavior,monotonicity,uniqueness and stability of traveling wave solutions.For a delayed lattice differential system with quiescent stage,we study the qualitative prop-erties of the traveling waves.Under monostable and monotone assumptions,the existence of traveling wave fronts with wave speed c≥ c*is first established via the classical monotone iteration technique coupled with lower and upper solutions and the limiting argument.Then,by utilizing Ikehara’s theorem,the asymptotic behaviors of all wave profiles at minus/plus infinity are proved.Meanwhile the non-existence of traveling wave solutions for O<c<c*is obtained,using two-side Laplace transform.With the help of sliding method combined with strong comparison theorem,the monotonicity and uniqueness of traveling fronts are further obtained.The exponential asymptotic stability of all non-critical traveling fronts are finally proved by comparison theorem.For a Lotka-Volterra nonlinear competitive system in a lattice,we study the stability of the traveling wave front.Under the bistable assumption,we first transform the competitive sys-tem to a equivalent cooperation system.Then,with the help of Banach fixed point theorem and interval monotone condition,the existence and uniqueness of solutions and correspond-ing comparison theorem of the Cauchy problem are investigated.Next,by introducing a linear auxiliary system and utilizing the characteristic vectors of coefficient matrix,a pair of supper and lower solutions are constructed.The global asymptotic stability and uniqueness of all traveling fronts are finally proved via monotone semiflow theorem.