Irregular spots on body surfaces of vertebrates induced by supercritical pitchfork bifurcations

The classical Turing mechanism containing a long-range inhibition and a short-range self-enhancement provides a type of explanation for the formation of patterns on body surfaces of some vertebrates, e.g., zebras, giraffes, and cheetahs. For other type of patterns (irregular spots) on body surfaces of some vertebrates, e.g., loaches, finless eels, and dalmatian dogs, the classical Turing mechanism no longer applies. Here, we propose a mechanism, i.e., the supercritical pitchfork bifurcation, which may explain the formation of this type of irregular spots, and present a method to quantify the similarity of such patterns. We assume that, under certain conditions, the only stable state of "morphogen" loses its stability and transitions to two newly generated stable states with the influence of external noise, thus producing such ruleless piebald patterns in space. The difference between the competitiveness of these two states may affect the resulting pattern. Moreover, we propose a mathematical model based on this conjecture and obtain this type of irregular patterns by numerical simulation. Furthermore, we also study the influence of parameters in the model on pattern structures and obtain the corresponding pattern structures of some vertebrates in nature, which verifies our conjecture.