蝶翅斑紋形成之數學模式研究

Mathematical Modeling of Pattern Formation in Butterfly Wings

葉正烽(Cheng-Feng Yeh)

指導教授 ： 林達德

國立臺灣大學/生物資源暨農學院/生物產業機電工程學研究所/碩士(2005年)

https://doi.org/10.6342/NTU.2005.01128

摘要

本研究結合L系統與涂林系統，進行蝴蝶翅膀斑紋形成數學模式之研究。在L系統部分，本研究先以化學方式去除蝴蝶翅膀上的鱗片以取得翅脈影像，經過簡單影像處理後藉由本研究所開發之程式以細線化、找出翅脈分歧點、邊緣、翅脈，再依據生長點建立似翅脈之樹狀結構、然後據以產生L系統字串，翅脈之樹狀結構另以三次曲線以及自動分段逼近等程序達到以L系統字串重建並描述蝴蝶翅脈的目標，在自動分段逼近的部份甚至能比不分段重建與原圖差異點數少了四倍。在這過程中也以降解析度與從翅脈分歧點螺旋向外搜尋的方法解決了兩個問題：翅脈影像線寬不為單一像素及不能完全搜尋或取出影像中所有翅脈。而在涂林系統方面，本研究所開發的程式應用了四種不同的涂林系統，其中包含了Joakim Linde、眼斑、G-M涂林系統與Schnakenberg涂林系統，將此四種系統拓展至二維空間上來展現，在初始設定方面，程式能以一般圖檔填入灰階值的方式經過轉換來設定初始值，邊界條件也能以圖檔或L系統字串的方式輸入，程式也能以批次的方式一次處理大量實驗，並在系統出現發散時給予例外處理；本研究比較並討論了各項係數對花紋發展的影響，找出程式提供的四種系統在何種係數組合下能發展成有意義的花紋。本研究最後結合以L系統字串重建的翅脈與涂林系統，配合不同係數並將重建之翅脈作為邊界條件進行蝴蝶斑紋形成研究，發現邊界條件對我們所使用的四種涂林系統所產生的斑紋型態沒有顯著影響，但對斑紋分布則是會有一定程度的影響。本研究能模擬出蛺蝶科中具有同心圓眼斑以及沿著翅膀邊緣上分布之波浪花紋，另外也能模擬分布於翅膀全域較小較無規則的波浪斑紋；藉由本研究我們發現蝶翅斑紋可以分為屬於背景的全域斑紋與以翅室中線為準屬於區域性的眼斑花紋，而本研究皆能分別以電腦模擬出來。

關鍵字

L系統 ； 涂林系統 ； 生物斑紋形成 ； 影像處理

並列摘要

This study combines L-system and Turing system for the research of the mathematical modeling of pattern formation in butterfly wings. In the part of L-system, we develope a program includes some features to realize the objective of reconstruct and describe butterfly veins by L-system strings. Furthermore, the number of different points between the original image and reconstruction of auto sectioned is lesser than the one of non-sectioned by 4 times. There are two problems solved in this process, 1. The line width of vein image is not 1 pixel. 2. Unable to find or extract all of the veins from image completely. We use the method of lower down the resolution and the method of spiral searching from every cross-point to solve these problems. In the part of Turing system, the program developed by this study provides four kinds of Turing system, including activator-inhibitor system (Joakim Linde), eyespot system, G-M Turing system, and Schnakenberg Turing system. On the hand of boundary condition, it could be inputted by the format of bitmap or L-system string file. We compare and discuss the coefficient effects to the formation of patterns and find out the combinations of coefficient which will form patterns meaningful. Finally, we combine the vein reconstructed by L-system strings and Turing system to do the research of pattern formation in butterfly wings with different coefficients and boundary condition described by reconstructed veins. When the boundary condition described above applied, we found that there’s almost no influence on pattern styles, however, influence on pattern distribution.This study can simulate concentric eyespot and wave stripe along the boundary of wing in Nymphalidae, besides, we can also simulate smaller and non-regular wave pattern which distributes over the wing. By this study, we figure out the patterns on the butterfly wings can be classified into two types, global patterns belong to the background, and local eyespot patterns according to the wing cell midline, furthermore, our study simulates these patterns by computer simulation.

並列關鍵字

L-system ； Turing system ； biological pattern formation

參考文獻

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