4物種 Lotka-Volterra 擴散競爭模型解的存在性與2物種行波解波速之研究

Abstract

這篇論文分成兩個部分。第一個部份我們將介紹4物種的生物競爭模型，包含兩原生物種以及兩外來種之生物競爭模型，並設定原生物種為彼此強競爭之關係，外來物種為弱勢族群(弱競爭)的狀態。我們將證明，在適當條件之下此模型存在非0之行波解。第二部份，我們將利用變分形式之最小最大公式，將兩強競爭物種的生物模型中,行波解之波速零值條件表示出來。此外，我們即可間接利用此公式得到行波解波速的正負號判準。

This article is divided into two parts. In the first part, we explores the question of whether coexistence can persist over time when a third and forth species, denoted as w1 and w2, invade an ecosystem which is comprised of two species u and v, within the domain R. In this scenario, u, v, w1,and w2 engage in competition with each other. Assuming that the impact of wi on u and v are very small, along with other appropriate conditions, we demonstrate that these four species can coexist in the form of a non-monotone traveling wave. Our new technique, using the method of iteration argument, Schauder’s Fixed point theory, sub- and super-solutions and the bifurcation theory, provides methods for constructing small perturbation types of non-monotonic waves. In the second part, we use a min-max variational approach to represent the sign of the traveling wave speed in the two species Lotka-Volterra system with strong competition.