Image analysis for patterns in solutions of reaction–diffusion equations

Abstract

We consider reaction–diffusion equations in form of a Turing system, i.e., a system of partial differential equations (PDEs) in time and two-dimensional space. The equations model the formation of patterns in animal coats and skins, for example. Stationary solutions represent a final pattern, which depends both on physical parameters and initial values. A spatial finite difference method yields a high-dimensional system of ordinary differential equations (ODEs). Stationary solutions of the ODEs are computed numerically to approximate steady-state solutions of the PDEs. We investigate properties of the patterns using image analysis. The two-dimensional stationary solutions are converted into binary images. This approach allows for the application of methods and algorithms on the binary image. The Euler characteristic yields the number of objects in this case. Properties of each object like area, Feret diameters, circularity, etc. can be computed. Moreover, we examine associated statistical quantities for sample sets of initial values in a Monte-Carlo simulation. Our aim is to analyse the dependence as well as the sensitivity of patterns on the physical parameters. We present results of numerical computations for different selections of those parameters.