Butterfly wing color patterns are a representative model system for studying biological pattern formation, due to their two-dimensional simple structural and high inter- and intra-specific variabilities. Moreover, butterfly color patterns have demonstrated roles in mate choice, thermoregulation, and predator avoidance via disruptive coloration, attack deflection, aposematism, mimicry, and masquerade. Because of the importance of color patterns to many aspects of butterfly biology and their apparent tractability for study, color patterns have been the subjects of many attempts to model their development. Early attempts focused on generalized mechanisms of pattern formation such as reaction-diffusion, diffusion gradient, lateral inhibition, and threshold responses, without reference to any specific gene products. As candidate genes with expression patterns that resembled incipient color patterns were identified, genetic regulatory networks were proposed for color pattern formation based on gene functions inferred from other insects with wings, such as Drosophila. Particularly detailed networks incorporating the gene products, Distal-less (Dll), Engrailed (En), Hedgehog (Hh), Cubitus interruptus (Ci), Transforming growth factor-$\beta$ (TGF-$\beta$), and Wingless (Wg), have been proposed for butterfly border ocelli (eyespots) which helps the investigation of the formation of these patterns. Thus, in this work, we develop a mathematical model including the gene products En, Hh, Ci, TGF-$\beta$, and Wg to mimic and investigate the eyespot formation in butterflies. Our simulations show that the level of En has peaks in the inner and outer rings and the level of Ci has peaks in the inner and middle rings. The interactions among these peaks activate cells to produce white, black, and yellow pigments in the inner, middle, and outer rings, respectively, which captures the eyespot pattern of wild type Bicyclus anynana butterflies. Additionally, our simulations suggest that lack of En generates a single black spot and lack of Hh or Ci generates a single white spot, and a deficiency of TGF-$\beta$ or Wg will cause the loss of the outer yellow ring. These deficient patterns are similar to those observed in the eyespots of Vanessa atalanta, Vanessa altissima, and Chlosyne nycteis. Thus, our model also provides a hypothesis to explain the mechanism of generating the deficient patterns in these species.

蝴蝶翅膀颜色图案是研究生物图案形成的代表性模型系统，由于其二维简单的结构和高的种间和种内变异性。此外，蝴蝶颜色图案已通过破坏性着色、攻击偏转、分离、模仿和伪装在配偶选择、体温调节和避免捕食者方面发挥作用。由于颜色图案对蝴蝶生物学的许多方面的重要性及其明显的研究易处理性，颜色图案已成为许多尝试对其发展建模的主题。早期的尝试集中在模式形成的广义机制上，例如反应扩散、扩散梯度、侧向抑制和阈值反应，而不涉及任何特定的基因产物。果蝇。包含基因产物的特别详细的网络，Distal-less (Dll)、Engrailed (En)、Hedgehog (Hh)、Cubitus interruptus (Ci)、转化生长因子-$\beta$ (TGF-$\beta$) 和 Wingless (Wg) 已被提议用于蝴蝶边界 ocelli（眼点），这有助于研究这些模式的形成。因此，在这项工作中，我们开发了一个数学模型，包括基因产物 En、Hh、Ci、TGF-$\beta$, 和 Wg 来模拟和研究蝴蝶的眼斑形成。我们的模拟表明，En 的能级在内环和外环有峰值，Ci 的能级在内环和中环有峰值。这些峰之间的相互作用激活细胞在内环、中环和外环中分别产生白色、黑色和黄色色素，从而捕捉到野生型Bicyclus anynana蝴蝶的眼点图案。此外，我们的模拟表明，缺乏 En 会产生一个单一的黑点，缺乏 Hh 或 Ci 会产生一个单一的白点，缺乏 TGF-$\beta$或者wg会造成外黄环丢失。这些有缺陷的图案类似于在Vanessa atalanta、Vanessa altissima和Chlosyne nycteis的眼点中观察到的图案。因此，我们的模型还提供了一个假设来解释在这些物种中产生缺陷模式的机制。