Traveling Wave Solutions Of Lotka-Volterra Systems With Delays

It seems that the time delay and diffusion are inevitable in many evolutionary processes, thus the delayed reaction diffusion equations and reaction diffusion equations with nonlocal delay attracted much attentions, one important topic is its travelling wave solutions. But the result concerning with the system is scarce, and many important properties can not be displayed by the existed results. This paper is focused on the travelling wave solution of Lotka-Volterra systems with (nonlocal) delays, and the main interesting is the travelling wave solutions relating the coexistence equilibrium due to the importance of coexistence of species in ecology. The main result is divided into four chapters.Firstly, the existence of the travelling wave solutions of delayed reaction diffusion systems are considered by Schauder's fixed point theorem and upper-lower solutions. The result is also applied to consider the two-species diffusion-competition models with delays and establish the existence of travelling wave solution connecting 0 with coexistence equilibrium. Especially, the result implies that the delays appeared in the interspecific competition terms do not affect the existence of traveling waves while the delays appeared in the intraspecific competition terms do. Moreover, its positive travelling wave solution grows like an exponential function near 0.Then, the existence, asymptotic stability and minimal wave speed of travelling wave fronts of cooperative Lotka-Volterra system with delay is studied. By constructing proper upper-lower solutions, the existence of travelling wave fronts connecting 0 with coexistence equilibrium was established and the result indicates that the travelling wave fronts grows exponentially near 0. Then the asymptotic stability of the travelling wave fronts is established by developing the so-called squeezing technique based on upper-lower solutions technique and comparison principle, which implies that even for delayed systems, its travelling wave front also can determine the long time behavior of the corresponding initial value problem. Moreover, the minimal wave speed and nonexistence of travelling wave fronts are established and is equivalent to the faster spreading speed of two species.Subsequently, we consider the bistable waves of a diffusive Lotka-Volterra type model with nonlocal delays for two competitive species. By introducing an undelayed reaction diffusion system with more variables, the existence, asymptotic stability and the uniqueness of wave speed are established for the system with nonlocal delay. It is interesting that the bistable wave is persistent even for large delay and the stability of travelling wave fronts implies that the bistable wave of nonlocal delayed system also can determine the long term behavior of the corresponding initial value problem which indicates the impor- tance of travelling wave fronts of reaction diffusion system with nonlocal delay. Especially, the result concerning with the bistable waves can be generalized in mathematics and implies the spatial isolation or species invasion in ecology. The corresponding monostable case is also be studied.In the last, the monostable travelling wave fronts of Lotka-Volterra cooperative diffusion systems with nonlocal infinite delay is studied by an idea similar to that in Chapter 4, but the technique is different to that. The existence, nonexistence and stability of the travelling wave fronts of the system is established by the upper-lower solutions, spectral theory and squeezing technique. The stability result of travelling wave fronts also extends the importance of travelling wave fronts of monostable reaction diffusion systems with nonlocal infinite delay.