A mathematical secret of lizard camouflage

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A multidisciplinary team at the University of Geneva has succeeded in explaining the complex distribution of scales in the ocellated lizard by means of a simple equation.

The shape-shifting clouds of starling birds, the organization of neural networks or the structure of an anthill: nature is full of complex systems whose behaviors can be modeled using mathematical tools. The same is true for the labyrinthine patterns formed by the green or black scales of the ocellated lizard.  A multidisciplinary team from the University of Geneva explains, thanks to a very simple mathematical equation, the complexity of the system that generates these patterns. This discovery contributes to a better understanding of the evolution of skin color patterns: the process allows for many different locations of green and black scales but always leads to an optimal pattern for the animal survival. These results are published in the journal Physical Review Letters.

A complex system is composed of several elements (sometimes only two) whose local interactions lead to global properties that are difficult to predict. The result of a complex system will not be the sum of these elements taken separately since the interactions between them will generate an unexpected behavior of the whole. The group of Michel Milinkovitch, Professor at the Department of Genetics and Evolution, and Stanislav Smirnov, Professor at the Section of Mathematics of the Faculty of Science of the University of Geneva, have been interested in the complexity of the distribution of colored scales on the skin of ocellated lizards.

Labyrinths of scales

The individual scales of the ocellated lizard (Timon lepidus) change color (from green to black, and vice versa) over the course of the animal’s life, gradually forming a complex labyrinthine pattern as it reaches adulthood. The researchers from the University of Geneva have previously shown that the labyrinths emerge on the skin surface because the network of scales constitutes a so-called ’cellular automaton’. "This is a computing system invented in 1948 by the mathematician John von Neumann in which each element changes its state according to the states of the neighboring elements," explains Stanislav Smirnov.

In the case of the ocellated lizard, the scales change state - green or black - depending on the colors of their neighbors according to a precise mathematical rule. Milinkovitch had demonstrated that this cellular automaton mechanism emerges from the superposition of, on one hand, the geometry of the skin (thick within scales and much thinner between scales) and, on the other hand, the interactions among the pigmentary cells of the skin.