Traveling Wave Solutions And Asymptotic Spreading For A Time Periodic Lotka-Volterra System With Weak Competition

Since the interspecies competitive behavior can not be avoided, much attention has been paid to the dynamics behavior of different competition systems to describe this phenomena. Furthermore, there are many factors varying in time in the natural envi-ronment, then a time periodic competition system is arisen. Due to the importance of coexistence state in ecology, in this thesis, we will study the travelling wave solutions as well as asymptotic spread relating the coexistence state of a time periodic Lotka-Volterra competition system with weak competition.Firstly, we investigate the existence and nonexistence of traveling wave solutions for the system. The existence of periodic traveling wave solutions is proved by using cross iteration and sub-and super-solutions methods. Moreover, by combining classical theory of asymptotic spread with the conclusion of a scalar equation, the existence of nontrivial time periodic traveling wave solutions is established. Then we obtain the nonexistence of time periodic traveling wave solutions by using the theory of asymptotic spread. These results illustrate that the co-invasions of two competitors are successful in periodic envi-ronment, and they can eventually attain to a coexistence state.Secondly, we consider the asymptotic speeds of spread. By combining auxiliary sys-tems with sub-and super-solutions methods, we deduce some results about the asymptotic speeds of spread of the periodic system. The results indicate that the interspecific com-petition can decrease the invasion speed of one species, which implies that the nontrivial effect of nonlinear term for the invasion speed from up bound of the invasion speed.