反應擴散方程式（reaction diffusion equation）已經被證實能夠真實地描述生態系統中族群的演變過程，有名的例子如：Fisher’s 方程式（Fisher’s equation）、Nagumo模型（Nagumo model）或是非局部Nagumo模型（non-local Nagumo model）預測出的一些重要特性都已經在自然環境中被觀察到，奠基在這些成功的基石上，我們更近一步的探討族群動態學中兩個重要的議題。
我們第一個探討的議題是環境的臨界尺度（critical size）：系統大於（小於）臨界尺度將會決定族群最終會生存（滅亡），在應用的層面，因為老鼠被認為是散播漢他病毒的媒介，所以我們專注在老鼠的動態，臨界尺度的概念給了我們一些靈感，讓我們在消滅漢他病毒的同時得以保持食物鏈的完整性，此外，我們在原本使用的模型中放入了兩種修正項，並且計算放入修正項之後臨界尺度的變化。
第二個討論的議題是源於植被的非局部交互作用而在空間中觀察到的週期性圖形，之前使用非局部的耦合Nagumo模型的計算告訴我們植物會展現出兩種不同的生長模式，一個是共存，另一個是隔離，植物會選擇在空間中形成同相位的生長（共存）或是反相位的生長（隔離），先前的研究發現當非局部的範圍由小變到大會使得系統經歷一個從隔離到共存的相變，在這份論文中我們會更近一步探討相變。

It has been demonstrated that the reaction diffusion equation is capable of describing evolution of population in ecosystems realistically. For example, Fisher's equation, Nagumo model or non-local Nagumo model are well known models for which important properties predicted by these models were observed in nature. Based on these successes, we further explore the following two issues in the field of population dynamics.

The first topic that we investigate is the critical size of environment, above which the population survive and below which the population will become extinct. We focus on the dynamics of rodents, which are generally considered as the media to spread the Hantavirus disease. The concept of critical size might shed light on eliminating the disease without destroying the integrity of food web. Furthermore, two different modifications on the previous models are made respectively and the consequences of the critical size with respect to the modification are calculated.

The second topic is about the periodic spatial pattern observed in vegetation which originates from the non-local interactions between plants of the system. Previous calculation based on a two-species non-local Nagumo model has shown two possible modes of vegetation distributions, namely, co-existence and isolation. The two species would form either localized in-phase structures (coexistence) or localized out-of-phase structure (isolation). It was found that the domination of isolated mode would transition into co-existed state when the interaction range is varied. A detailed investigation of mode transition in the two-species non-local Nagumo model is discussed.

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