多物種的羅特卡-弗爾特拉擴散競爭方程組之非單調行波解

Abstract

這篇論文主要研究多物種的羅特卡-弗爾特拉擴散競爭系統(Lotka-Volterra competitive system with diffusion)。我們透過研究行波解以了解該系統，並成功證明連接 O := (0,0,··· ,0) 和 e1 := (1,0,··· ,0) 兩個平衡態的非單調解的存在性。關於這方面的研究在過去的文獻中相當稀少。然而，這類非單調解在生態學中具有重要意義，它可以啟發我們發現一些特殊現象。我們主要的研究方法為利用 Schauder 不動點定理，以及合適的上下解來證明解的存在性，並通過縮小區間的方法來描述z→∞時的漸近行為。另外，透過證明不存在速度小於某個特定值 s∗ 的解，我們找出該系統行波解的最小速度。

This focuses on the n-species Lotka-Volterra competitive system with diffusion. Understanding traveling wave solutions is essential for gaining insights into this dynamical system. We successfully show the existence of non-monotonic pulse-front traveling wave solutions that connect two equilibriums O := (0,··· ,0) and e1 := (1,0,··· ,0). These solutions are significant in ecology and can inspire the exploration of other intriguing phenomena within the Lotka-Volterra system. To prove the existence of traveling wave solutions, we rely on the application of the Schauder fixed-point theorem and appropriate upper-lower solutions. A key breakthrough in our work is the construction of these suitable upper-lower solutions for the competition system. Additionally, the concept of shrinking rectangles is employed to deduce the asymptotic behavior when z → ∞. Furthermore, by proving the non-existence of traveling wave solutions at speeds below a critical threshold s∗, we identify the minimum speed of traveling wave solutions for this model.