A multigrid method for pattern formation problems in biology

Abstract

Mathematical models of many pattern formation problems in biology are reaction-diffusion systems. These systems are important for computer simulations of the patterns, parameter estimations as well as analysis of the biological properties. In order to solve reaction-diffusion systems efficiently, fast and stable numerical algorithms are essential for the pattern formation problems. In this paper, a fairly general reaction-diffusion system is considered. We propose a fully implicit discretization combined with a multigrid V-cycle solver for solving the reaction-diffusion system. Theorems about unconditional stability and convergence of the algorithm are given to show that the algorithm is highly stable and efficient. Numerical experiment results are given for two reaction-diffusion systems that can be used for generating animal coat markings. We also show the comparison results of the multigrid algorithm with other numerical algorithms.