Numerical simulations of Turing patterns in a reaction-diffusion model with the Chebyshev spectral method

Abstract

Multi-species models play an important role in both ecology and mathematical ecology due to their practical relevance and universal existence. Some phenomena include but are not limited to osculating solutions behavior, multiple steady states and spatial patterns formation. In this article we study the numerical approximation of Turing patterns corresponding to the steady state solutions of systems of reaction-diffusion equations subject to zero-flux boundary conditions. We apply Chebyshev spectral methods which proved to be numerical methods that can significantly speed up the computation of systems of reaction-diffusion equations in the spatial part, while the temporal part is discretized using the Euler scheme in one dimension. For the evaluation of Turing instabilities and bifurcation of the steady state problem, we used the eigenvalues of the Jacobian matrix. The proposed scheme is then extended to the two-dimensional problem. We found that our numerical scheme is in very good agreement with other schemes available in the literature.