運用反應擴散方程式來研究自然界中的圖案形成已經有數十年了，而獵物--掠食者的雙物種系統在生態系中是很常見且基礎的架構。然而，在許多文獻中所使用的擴散項往往不是過於簡單，不然就是缺乏生物/物理上的詮釋。因此，我們透由考慮物種之間的交互作用，提出一個合理且全面的擴散項以建立一個符合實際生態系統的獵物--掠食者模型。此外，我們也會探討物種之間的交互作用對於圖案形成的影響。
在自然界中，獵物所需要的資源並不是永遠都充足的。資源在空間上的分布會隨著被獵物消耗的多寡而有所起伏。因此，資源在空間上的分布應該被考慮，所以我們將資源當作第三個物種加到模型中。我們將會研究資源在空間有密度變化對於圖案形成的影響，也會跟雙物種模型做比較，研究差異的來源。

Reaction-diffusion equations have been used to study the pattern formation in nature for decades. The two-species prey-predator system is the most common and basic example in ecosystems. However, the diffusion terms many studies adopt either are too simple or lack biological/physical interpretations. Hence, we propose a more comprehensive and reasonable self/cross diffusion term derived by considering inter/intra-species interactions in order to build a more realistic prey-predator model. Moreover, the influence of the inter/intra-species interactions on pattern formation is studied as well.

In nature, the resources are not always sufficient for prey. Resources will vary in space due to the consumption of prey. Therefore, the influence of the variation of resources should be taken in to consideration, and resources are added into our model as the third species. The influence of the variation of resources on pattern formation is studied, and we compare this three-species model to the two-species prey-predator model to see the effect due to resources variation.

Contents ii
List of Figures vi
1 Introduction 1
2 Two-Species Model: Prey-Predator System 3
2.1 Reaction-Diusion Equation of Prey-Predator System . . . . . . . . 3
2.1.1 Modied Diusion Terms . . . . . . . . . . . . . . . . . . . . 4
2.1.2 Formation of Turing Pattern . . . . . . . . . . . . . . . . . . 6
2.2 Model I: Holling Type II Functional Response . . . . . . . . . . . . 9
2.2.1 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . 11
2.2.2 Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.3 Eect of Intraspecies Interactions . . . . . . . . . . . . . . . 15
2.2.4 Eect of Interspecies Interactions . . . . . . . . . . . . . . . 19
2.2.5 Turing Space . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.6 Simulation Result . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Model II: Modied Holling Type II Functional Response . . . . . . 32
2.3.1 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . 33
2.3.2 Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.3 Simulation Result . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.4 Dierence between Model I and Model II . . . . . . . . . . . 41
3 Three-Species Model: Resource-Prey-Predator System 45
3.1 Model: Modied Holling Type II Functional Response with Three
Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.1 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . 47
3.1.2 Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1.3 Eect of the Diusion Length of Resources . . . . . . . . . . 51
3.2 Comparison to the Two-Species Model . . . . . . . . . . . . . . . . 55
3.2.1 Simulation Result . . . . . . . . . . . . . . . . . . . . . . . . 57
4 Conclusion 60
Reference 62

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