Existence Of Monostable Waves In Non-Class Ical Reaction-Diffusion Systems With Delays

Reaction-Diffusion equation is the classical model in describing the spatio-temporal pattern. However, it is not sufficiently accurate in formulating some evolutionary process. Thus, some nonclassical diffusion equations are needed to model these processes. In this thesis, the dynamics of several nonclassical diffusion models are concerned from the viewpoint of monostable case solutions.In chapter 2, the existence of traveling wavefronts in the nonlocal dispersal and cooperative Lotka-Volterra system with delays is investigated. By constructing upper and lower solutions, the existence of traveling wavefronts of system is proved. Such as wavefronts admit asymptotic behavior like exponential functions near the unstable equilibrium state, which is very important in further studying the properties of traveling wavefronts.Then, the traveling wavefronts of a nonlocal diffusion model are investigated, which formulates the existence of monotone traveling wavefronts in a single species model with nonlocal diffusion and age-structure. When the wave speed is larger than a threshold depending only on the basic parameters, then we will prove the existence by upper and lower solution technique. If the wave speed equals to the threshold, then we show the conclusion by passing to a limit function. In addition, if the wave speed is smaller than a threshold, the nonexistence is proved by asymptotic speeds of spread theory.Finally, a diffusion system which discrete time is considered by its wavefronts. Applying the Schauder's fixed point theorem and a cross-iteration scheme, we reduce the existence of traveling wave solutions to the existence of an admissible pair of upper-lower solution which are easy to construct in practice. Moreover, these results are applied to temporary discrete and diffusion-competition Lotka-Volterra system with delays, and obtain its existence of monostable traveling wave solutions.