Biological pattern formation in cell dynamics under cross-diffusion: An Isogeometric analysis perspective

Abstract

This note presents an efficient numerical method based on isogeometric analysis (IgA) and an operator splitting approach for solving nonlinear reaction–diffusion systems with cross-diffusion. Such problems are often used in mathematical modeling of developmental biology and are subject to highly rigid reactive and diffusive terms. Similarly, the interactions between substances produce complex morphologies (Roth, 2011) [1]. In this note we present two different types of solutions. Mainly, the Turing patterns and the traveling waves, which are a direct result of the presence of linear diffusion and/or cross-diffusion in the dynamical system. To deal with the multiphysical nature of the nonlinear system, we propose a time-splitting method. The spatial discretization is performed using IgA-based Non-Uniform Ratinal B-spline (NURBS) functions, where the semidiscrete problem is integrated using an implicit scheme. The nonlinear terms are treated by an adaptive fourth-order Runge–Kutta method. The well-known FitzHugh–Nagumo and Gray–Scott models are used to study the performance of the new method. The results obtained demonstrate the ability of our algorithm to accurately maintain the shape of the solution in the presence of complex patterns arising from biological cells on complex geometries. Furthermore, the energy dissipation in the Allen-Cahn equation is analyzed and the new method clarifies the effect of the geometry on the formed patterns and on the energy decay for the considered benchmarks.