演化動力學為我們提供的數學框架去探索不同系統中隨時間的變化以及其可能的原因，如理論生態學、種群遺傳學，流行病學，免疫學等。演化動力學的基本元素是繁殖，天擇和突變。討論天擇和突變的研究課題中，本論文只探討及解決其中兩個重要問題：生命的起源和生態系統的生物多樣性。從準種方程式出發，描述天擇及突變過程中基因組頻率隨時間的變化，我們發現適者生存的演化優勢存在的條件和突變率、基因組長度和相對競爭力的關係，此關係式被稱為突變臨界門檻。將這個概念及數學技巧應用到更複雜的競爭關係中，我們得到一個通用相圖，可以解釋不同條件下的突變臨界門檻。接下來，我們忽略突變只考慮頻率依存天擇下的演化動力學。利用能模擬繁殖方程式中人口動力學的"環境－個體－取代"演化演算法，個體間交互作用的隨機性被自然的涵蓋在演化演算法中。計算結果顯示不同空間網路結構的廣義剪刀－石頭－布模型仍具有相同的滅絕模式。滅絕相圖中，人口動態可被區分為兩個不同的相以及一個介於兩相相變中間的臨界點。因此，滅絕模式的相圖可以被看作是生態系統中生態穩定性（生物多樣性）的重要指標。

Evolutionary dynamics provides us the mathematical framework to explore the dynamics of evolution in diverse research topics, such as theoretical ecology, population genetics, epidemiology, immunology and etc. The basic building blocks of evolutionary dynamics are reproduction, selection and mutation. To discuss the basic properties and research topics on selection and mutation, we only address two important questions about the origin of life and the biodiversity in ecological systems in the thesis. Start from the quasispecies equations which describes the dynamics of genome frequency under mutation-selection processes, an important relationship among mutation rate, genome length and relative fitness called error threshold is discovered to maintain the evolutionary advantage of the fittest genome in the single-peak fitness landscape. Apply the idea and the techniques to the peak-mesa-background fitness landscape, the universal phase diagram is found to explain the error threshold under different conditions. Next, we switch the focus to the evolutionary dynamics in population level by considering the frequency-dependent selection without mutation in replicator equations. The Reference-Gamble-Birth (RGB) algorithm, which has the same dynamics as replicator equations in infinite population limit, is introduced to handle the stochastic interactions in agent-based simulation. The numerical results of RGB algorithm shows that the population dynamics shares the same universal extinction patterns in generalized rock-paper-scissors communities under different types of spatial networks. The universal extinction patterns show that the population dynamics can be divided into two phases with a critical point at the phase transition, and therefore the extinction patterns can be viewed as the indicator of ecological stability (biodiversity) for ecological systems.

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