混沌理論之速返斥子在生態學上的應用

Abstract

本論文簡短地回顧混沌理論(chaos theory)的歷史，並利用關係圖描述混沌(chaos)和黎阿普諾夫指數(Lyapunov exponent)、拓樸熵(topological entropy)、奇異吸引子(strange attractor)、速返斥子(snapback repeller)以及薩可夫斯基定理(Sarkovskii's theorem)之間的關係。這些數學和電腦輔助的理論及工具能夠藉由計算某個量或證明其存在性來決定某個系統是否會有混沌現象。在生態學上，佐竹曉子(Akiko Satake)和巖佐庸(Yoh Iwasa)修改了井鷺裕司(Yuji Isagi)的能量預算模型(resource budget model)，並建立更一般化的能量預算模型 (generalized resource budget model)，利用正的黎阿普諾夫指數證明如果消耗係數(depletion coefficient)大於一，則系統會產生混沌現象。然而，正的黎阿普諾夫指數只意謂德瓦尼(Devaney)混沌定義中的敏感性(sensitivity)而已。因此，本論文從數學和數值的角度去分析佐竹曉子和巖佐庸的模型，利用速返斥子方法去證明此模型會有德瓦尼定義的混沌現象。此外，本論文也修正了他們論文在探討消耗係數是整數時所遺漏的部份，並更進一步區分奇數與偶數的差異性。

This work briefly reviews the history of chaos theory and elucidates the relationship among chaos, Lyapunov exponent, topological entropy, strange attractor, snapback repeller, and Sarkovskii's theorem, connecting them to each other using a relational graph. Mathematical and computer-assisted tools can be used to determine whether maps or systems are chaotic by finding a quantity or sometimes identifying the existence of a property. In ecology, Satake's generalized resource budget model that modified from Isagi's resource budget model, Satake and Iwasa proved by computing the positive Lyapunov exponent that if the depletion coefficient k is greater than one, then the system is chaotic. However, a positive Lyapunov exponent means only sensitivity in Devaney's chaos. Therefore, this work presents mathematical views and a numerical analysis on Satake's model, using the "snapback repeller method" to prove that the model is chaotic in Devaney's sens (involving transitivity, density, and sensitivity). Moreover, this work also overcomes the omission of Satake's paper (Satake & Iwasa, 2000) when the depletion coefficient k is a positive integer. Furthermore, the end of this work investigates the difference between odd depletion coefficients and even depletion coefficients.