植披之圖案形成已經藉由非局域Nagumo 模型研究，例如出現在南非與納米
比亞乾旱草原地帶的自然景觀” 仙女圈”, 而以往非局域Nagumo 模型皆是單物
種模型，但在自然界中植披是包含多種的植物，因此吾人將非局域Nagumo 模
型擴展至非局域耦合Nagumo 模型，以此模型研究多物種植披間的競爭行為，
並藉由線性穩定性分析，吾人觀察到若核函數是截止函數形式，隨著非局域性
競爭長度經過相變點將會發生相變現象，偏好的生長型態將會改變，在長度尺
度上也出現不連續的變化。

Vegetation pattern formation has been successfully described by the non-local
Nagumo model in the past decade. For example, it can be used to explain the
vegetation formation named ”fairy circles” that are observed in vast territories in
southern Angola, Namibia and South Africa. The non-local Nagumo model has
been investigated thoroughly in the past and it exhibits various solutions including
kink/anti-anik, traveling wave and patterns. Since the vegetation of multiple
species is commonly observed in the realistic ecosystems, therefore, we employ the
non-local coupled Nagumo model for multiple species that considers the vegetation
competition between two plant species with different non-local competition
range. We investigate the non-local coupled Nagumo model using a linear stability
analysis and we find the growth patterns changes rapidly as the non-local
range exceeds a critical value. This mode transition phenomenon is investigated
extensively to determine how the mode transition point varies with the coupling
strength. In addition, we discuss how the kernal function affect the mode transition
phenomenon.

1 Introduction 1
2 Model 5
2.1 The Fisher-KPP equation . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 The Nagumo model . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 The non-local Nagumo model . . . . . . . . . . . . . . . . . . . . . 6
2.4 The non-local coupled Nagumo model . . . . . . . . . . . . . . . . 7
3 The linear stability analysis 12
3.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Identical nonlocal range . . . . . . . . . . . . . . . . . . . . . . . . 14
4 Pattern formation in non-local Nagumo model 17
4.1 The mode transition phenomenon . . . . . . . . . . . . . . . . . . . 17
4.2 Influence of non-local function on mode transition . . . . . . . . . . 24
5 The Gaussian approximation method 29
6 Conclusion

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