在生態學上的資源預算模型之數值模擬與分析

Abstract

本論文運用數學理論和電腦輔助來決定系統是否會具有混沌現象，使用拓樸熵(topological entropy)以及黎阿普諾夫指數(lyapunov exponents)，還有運用穩定性分析找出產生週期解的參數的分界。系統擁有正拓樸熵則具有李約克定義中的混沌，而正黎阿普諾夫指數則表示系統具有德瓦尼(Devaney)混沌定義中的敏感性。在生態學上，佐竹曉子(Akiko Satake)和巖佐庸(Yoh Iwasa)於2000年修改了井鷺裕司(Yuji Isagi)等人的資源預算模型 (resource budget model)，建立更一般化的資源預算模型(generalized resource budget model)。本論文從數學和數值的角度去分析佐竹曉子(Akiko Satake)和巖佐庸(Yoh Iwasa)的資源預算模型，找出該模型產生同步的充分條件以及找出在某些情況下具有正拓樸熵與正黎阿普諾夫指數，再利用拓樸熵去證明此模型會有李約克定義的混沌現象。

In our work, we use mathematical theorem and computer-assists to determine whether maps or systems are chaotic. We use topological entropy and lyapunov exponents, and use stability analysis to find the boundary of parameters that has periodic solution. If a system have positive topological entropy means that system is chaotic according to the definition of Li and Yorke and if system have positive Lyapunov exponent means sensitivity in Devaney's chaos. In ecology, Satake and Iwasa's generalized resource budget model that modified from Isagi et al.'s resource budget model in 2000. In this work, mathematical views and numerical analysis are presented to discover the sufficient condition that synchronicity will happened and to discover the conditions that the system have positive topological entropy and positive Lyapunov exponent on Satake and Iwasa's generalized resource budget model. Subsequently, topological entropy are utilized to prove that the model is chaotic in Li and Yorke's sense.