論洛特卡-佛爾特拉方程組之解

Abstract

本作品討論二物種與三物種擴散型洛特卡-佛爾特拉方程組。對二競爭物種組，雙曲函數法用來構建嚴密行進波解。在藤田型結果的基礎之上，發展共存位移法來找二合作物種組的爆破解(和廖嫻)。對競爭-合作混合型三物種組，我們用上下解法來構築行進波解的存在性。以廣義雙曲函數法，可以證明三競爭物種組有嚴密(和三村昌泰等人)和半嚴密行進波解(和裘愉生)。此外，藉由極大值原理，三競爭物種組的行進波解不存在性也可以建立。最終，我們證明由熱傳方程的解可以構造出三競爭物種擴散型組的解。更進一步工作包含如何研究由擴散項引發的長期共存，這是從熱傳方程的解構築出來之解所發現的有趣新現象。

In the present work, we study diffusive Lotka-Volterra systems of two-species and three-species. For competitive systems of two species, the tanh method is applied to construct exact traveling wave solutions. Based on the Fujita-type results, the method of shifted coexistence is developed to find blow-up solutions of cooperative systems of two species (with Xian Liao). For competitive-cooperative and competitive systems of three species, we employ the method of super- and subsolutions to establish the existence of traveling wave solutions. By using the generalized tanh method, it is shown that exact (with M. Mimura et al.) and semi-exact (with Yu-Sheng Chiou) traveling wave solutions exist for competitive systems of three species. In addition, nonexistence of traveling wave solutions to competitive systems of three species is also established by the maximum principle. Finally, we show solutions to competitive systems of three species can be constructed from the solutions of the heat equation. Further investigations include how to study diffusion-enhanced long-term coexistence, which is an interesting new phenomenon discovered by means of the solutions constructed from the heat equation.