擴散跟有向流在競爭物種演化下的效應：一個關於Lotka-Volterra競爭模型的回顧

Abstract

本論文完整地回顧了一個具有生態學意義的問題：兩個競爭物種在資源異質分布的孤立環境中將如何演化？本研究透過再分布機制由相互競爭、隨機擴散跟有向移動組成的假設建立一個特別的 Lotka-Volterra 競爭模型，用以分析競爭物種的長期演化結果，亦即決定該模型均衡解的穩定性。本研究以標準程序應用諸如最大值原則（maximum principles）、變分法（calculus of variation）和單調動力系統理論（the theory of monotone dynamical systems）等數學方法。主要結論是隨機擴散和有向移動共同決定了演化結果，因此不同擴散速率和有向流傾向的組合可能影響演化結果。據此，研究者建立了一個初步的分歧圖以提供理論上可信賴的預測。

This thesis is a rather complete survey concerning an ecologically meaningful problem: how would two competing species evolve in a given spatially heterogeneous and isolated environment? A special kind of the Lotka-Volterra competition model is derived by assuming that the mechanisms of redistribution consist of mutual competition, random diffusion, and advective motion. The main task is to analyze the evolutionary results of the competing species in the long run, or equivalently, to determine the stability of equilibria of the model. The mathematical methods such as maximum principles, calculus of variation, and the theory of monotone dynamical systems are utilized as the standard procedure. The main conclusion is that both random diffusion and advective motion decide the evolutionary results; thus different combinations of diffusion rates and advective tendencies may influence the evolutionary results. Accordingly, a preliminary bifurcation diagram can be established to provide certain theoretically reliable predictions.