Traveling Wave Solutions In A Diffusive System With Two Preys And One Predator

Predator-prey principle is a basic relationship between different populations (see Murray[40]). In recent decades, to illustrate and predict some ecological phenomena, much attention has been paid to the dynamics of different predator-prey systems, and traveling wave solution becomes a concerning subject they study. However, most of them only consider the traveling wave solutions of two species predator-prey systems, there are little research results for the traveling wave solutions of three species predator-prey systems. Therefore, this paper is concerned with the traveling wave solutions in a diffusive system with two preys and one predator.Firstly, we study the existence and the asymptotic behavior of traveling wave solu-tions for three species predator-prey systems. By using Schauder’s fixed point theorem and upper and lower solutions method, we prove the existence of traveling wave solu-tions for the three species predator-prey system. At the same time, by combining the classical theory of asymptotic spreading with the idea of contracting rectangles, we prove the asymptotic behavior of traveling wave solutions at infinity. The results show that Schauder’s fixed point theorem may reduce the existence of traveling wave solutions to find an admissible pair of upper and lower solutions.Secondly, we consider the minimal speed of traveling wave solutions. We still use Schauder’s fixed point theorem and upper and lower solutions method to establish the existence of traveling wave solutions for the three species predator-prey system. Then we prove the nonexistence of traveling wave solutions by applying the theory of asymptotic spreading. Combining the contents in the second chapter, we obtain the the minimal speed of traveling wave solutions. The result shows that the minimal wave speed provides an estimation on the invasion speed of the three species.Finally, we discuss the paper and make a summary. In fact,[25,32] established the existence of traveling wave solutions for the two species predator-prey system by constructing upper and lower solutions. But they did not answer the nonexistence of traveling wave solutions, and then they also did not answer the minimal speed of traveling wave solutions. Thus we complete the results in Li and Li [25], Lin et al.[32] in some wav.