Reaction-diffusion systems and pattern formation

Abstract

This bachelor’s degree final thesis deals with the problem of pattern formation in living forms, with special interest in animal skin. Mathematically, this phenomenon is caused due to perturbations on a system of differential equations initially in equilibrium, the so-called reaction-diffusion system, which is time independent but it becomes spatially unstable when is perturbed. This spatial instability is the one which causes the pattern. Therefore, we have started this project obtaining this mathematical model and then studying under which conditions the system becomes spatially unstable. Particularly, these conditions were given by the mathematician Alan Turing in The Chemical Basis of Morphogenesis in 1952. Furthermore, another important point developed in this undergraduate thesis has been to apply this theory on animal body, particularly, to see how the geometry of the body affects the design of the resulting pattern (striped or spotted). Finally, we have made some simulations of the system of differential equations in adapted domains with the characteristics we want to emphasize of animal body (scale, thickness and curvature). In order to do them, we have designed three programs in C which approximate the solution of the system using the explicit scheme of a numerical method, the so-called finite difference method.