Traveling Wave Solutions Of Two Species Competition Systems In Heterogeneous Environments

Reaction-diffusion equations and systems are widely used in many fields, including biology, chemistry and physics. Traveling wave solutions are one important type of solutions of reaction-diffusion equation and system and they have been used to model many physical problems. By experimental observations and numerical calculations, there exist non-planar traveling wave solutions with a variety of different shaped level sets in combustion theory, chemical reaction and so on. Note that there exists various heterogeneities in nature environments. It becomes very important to study the non-planar traveling wave solutions of reaction-diffusion system in heterogeneous environments. The two species competition models are a class of models that used to interpret the interactions amongst species in which two or more species interact. In this thesis we first study the time periodic non-planar traveling wave solutions of time periodic two species competition system. Furthermore, dispersal evolution systems are also widely used to model the population dynamics of many species which exhibit nonlocal internal interactions. In this thesis we also study space periodic traveling wave solutions of two species competition with nonlocal dispersal and space periodic dependence.We first study time periodic traveling curved fronts for time periodic two species competition system with diffusion in two dimensional spatial space. We establish the asymptotic behaviors of one-dimensional traveling wave solutions of the system at infinity. Using these asymptotic behaviors, we then construct appropriate superand subsolutions and prove the existence of two-dimensional time periodic traveling curved fronts. We also show that the time periodic traveling curved front is asymptotically stable and unique.We then study the time periodic traveling solutions of pyramidal shapes for time periodic two species competition-diffusion system in RNwith N ≥ 3. By the supersolution and subsolution technique and comparison principle, we construct the time periodic pyramidal traveling front solution of time periodic two species competition-diffusion system in RN.Finally, we study space periodic traveling solutions of two species competition system with nonlocal dispersal and space periodic dependence. Under suitable assumptions, the system admits two semitrivial space periodic equilibria(u*1(x), 0) and(0, u*2(x)), where(u*1(x), 0) is linearly and globally stable and(0, u*2(x)) is linearly unstable with respect to space periodic perturbations. By sub- and supersolutions techniques and comparison principle, we show that, for any ξ ∈ SN-1, there exists a continuous space periodic traveling wave solution connecting(u*1(x), 0) and(0, u*2(x)) and propagating in the direction of ξ with speed c > c*(ξ), where c*(ξ) is the spreading speed of the system in the direction of ξ. Moreover, for c < c*(ξ) there is no such solution. When the wave speed c > c*(ξ), we also prove the asymptotic stability and uniqueness of traveling wave solution using squeezing techniques.