Traveling Wave Solutions And Entire Solutions Of The Lotka-Volterra Competition System With Nonlocal Dispersal

The theory of reaction-diffusion equations is an important part of modern mathematics. Compared to the Laplace diffusion of classical diffusion-reaction e-quation, the nonlocal dispersal represented by convolution operator has attracted many researcher’s attention because it is more precies to describe many practi-cal problems. In recent years, a lot of nonlocal dispersal equations have been derived from many subjects such as biology, epidemiology, material science, neu-ral network and so on. Therefore, research on nonlocal dispersal equation and system has very important theoretical and practical significance. In this article, we study the travelling wave solutions and entire solutions of the Lotka-Volterra competition system with nonlocal dispersal.Firstly, we consider the traveling wave solutions of the Lotka-Volterra com-petition system with nonlocal dispersal in monotone situation. With the aid of a solution sequence of a truncated problem, we prove the existence of travelling wave solutions by constructing an appropriate super solution. It shows that the existence of travelling wave solutions can be ensured only by a super solution which is easy to construct in reality.Secondly, we study the asymptotic behavior of the traveling wave solutions. By using a proposition established in Zhang[63], we prove the exponential decay of the traveling wave solutions at infinity. Then by the bilateral Laplace trans-form and Ikehara’s theorem, we obtain the precise exponential decay rate of the travelling wave solutions at. infinity.Finally, we study the entire solutions of the system. We first get some priori estimates by using the asymptotic behavior, and then construct a suitable pair of sub-super solution. After that, by combing sub-super solution method with comparison principle, the entire solutions which behave like two traveling wave solutions propagating from both sides of the x-axis is established.