競爭擴散系統的行波解

Abstract

這篇博士論文分為兩個部分，以考慮兩種由生態學而來的偏微分方程的行波解。第一部份我們考慮三個物種的競爭擴散系統。第二部分是考慮關於雙物種的自由邊界問題。 關於三個物種的競爭擴散系統，其行波解可以考慮為 R^6 中某向量場的異宿軌道 (heteroclinic orbit)。方程的參數在適當的假設下，我們利用異宿軌道的分歧理論來證明，三個物種的行波解可由兩個連接到某個相同平衡點(equilibrium)的雙物種行波解分歧出來。此三物種行波解的每一個部分都是正解。其波形為：其中一個物種連接某個正態(positive state)到零，另一物種連接零到某個正態， 第三物種為介於以上兩物種中間，在某個長區間內靠近某正態的脈衝(pulse)。我們對方程的參數，在哪些範圍會有三物種的行波解，找到了某些明確的表現形式，來作為此結果的具體應用。 關於雙物種競爭模型的自由邊界問題，最早是由 Mimura, Yamdada 及 Yotsutani 所提出。基於 Du 和 Lin 的散佈--消滅二分性(spreading-vanishing dichotomy)的結果，我們假設，當時間趨於無窮，自由邊界的散佈速度會趨於某個常數，而由此考慮對應於原自由邊界問題的的行波解問題。我們得出此問題的行波解的存在性與唯一性。

We divide the thesis into two parts to investigate the travelling wave of two types partial differential equations coming from ecology. In Part 1, we consider the 3-species Lotka-Volterra competition-diffusion systems. In Part 2, we consider a free boundary problem for a two-species competitive model. For the 3-species Lotka-Volterra competition-diffusion system, a travelling wave solution can be considered as a heteroclinic orbit of a vector field in R^6. Under suitable assumptions on the parameters of the equations, we apply a bifurcation theory of heteroclinic orbits to show that a 3-species travelling wave can bifurcate from two 2-species waves which connect to a common equilibrium. The three components of the 3-species wave obtained are positive and have the profiles that one component connects a positive state to zero, one component connects zero to a positive state, and the third component is a pulse between the previous two with a long middle part close to a positive constant. As concrete examples of application of our result, we find several explicit regions of the parameters of the equations where the bifurcations of 3-species travelling waves occur. The free boundary problem for a two-species competitive model in ecology was proposed by Mimura, Yamada and Yotsutani. Motivated by the spreading-vanishing dichotomy obtained by Du and Lin, we suppose the spreading speed of the free boundary tends to a constant as time tends to infinity and consider the corresponding travelling wave problem. We establish the existence and uniqueness of a travelling wave solution for this free boundary problem.