Pattern formation in the diffusive Fisher equation†

In this paper, numerical simulations of nonlinear Fisher’s equation in oneand twodimensions have been considered. The derivatives and integrals are replaced by the necessary matrices, and the resulting algebraic system of equations was advanced by the popular fourth-order exponential time differencing Runge-Kutta (ETDRK4) schemes proposed by Cox and Matthew [Exponential time differencing for stiff systems, Journal of Computational Physics 176 (2002) 430-455], and later in a modified version modified by Kassam and Trefethen [Fourth-order time-stepping for stiff PDEs, SIAM Journal on Scientific Computing 26 (2005) 1214-1233]. Numerical results obtained in this paper have further granted an insight to the understanding of pattern formation in both oneand twodimensional systems. Computations are carried out on a large spatial domain size l to actually give enough room for waves propagation. Some initial data and parameter values were chosen to mimic some of the existing patterns. 2010 Mathematics Subject Classification: 92B05, 65L06, 65M20, 35Q80, 35C07.