兩物種的羅特卡-弗爾特拉擴散競爭方程組之非單調行波解

Abstract

本篇論文主要研究兩物種的羅特卡-弗爾特拉擴散競爭方程組。我們透過研究行波解來了解此系統。我們成功地證明了非單調解的存在性，其中此解連接了兩個平衡態(0,0)和(0,1)。在過去的文獻中，鮮少有這方面相關的研究。然而，此類型的非單調解在生態學中扮演著重要的角色，可以啟發我們發現一些特別現象。我們主要的研究方法是用不動點定理配合適當的上下解去證明解的存在性。我們也運用了縮小區間的手法證明解的右半部逼近行為。此外，藉由證明速度小於某個特定值時不存在解，我們也刻畫了此系統行波的最小速度。

We study the two-species Lotka–Volterra competitive system with diffusion. To understand the dynamics of the system, it is fundamental to investigate the traveling wave solutions. We successfully show the existence of non-monotone pulse-front travelling waves connecting the two equilibria (0,0) and (0,1). In the literature, fewer results are known for the existence of such waves. These waves play an important role in ecology and may motivate us to explore other interesting phenomena in the Lotka– Volterra system. Our approach to prove the existence of traveling waves is based on a method by applying Schauder’s fixed point theorem with the help of suitable upper-lower solutions. One of our main breakthroughs is the construction of such appropriate upper-lower solutions for the competition system. We also apply the idea of shrinking rectangles to the derivation of the asymptotic behavior of the right-hand tail. Moreover, by proving the non-existence of traveling wave solutions with speed less than a critical value, we characterize the minimal wave speed of traveling waves for this model.