Effect of spatial configuration of an extended nonlinear Kierstead–Slobodkin reaction-transport model with adaptive numerical scheme

In this paper, we consider the numerical simulations of an extended nonlinear form of Kierstead–Slobodkin reaction-transport system in one and two dimensions. We employ the popular fourth-order exponential time differencing Runge–Kutta (ETDRK4) schemes proposed by Cox and Matthew (J Comput Phys 176:430–455, 2002), that was modified by Kassam and Trefethen (SIAM J Sci Comput 26:1214–1233, 2005), for the time integration of spatially discretized partial differential equations. We demonstrate the supremacy of ETDRK4 over the existing exponential time differencing integrators that are of standard approaches and provide timings and error comparison. Numerical results obtained in this paper have granted further insight to the question ‘What is the minimal size of the spatial domain so that the population persists?’ posed by Kierstead and Slobodkin (J Mar Res 12:141–147, 1953), with a conclusive remark that the population size increases with the size of the domain. In attempt to examine the biological wave phenomena of the solutions, we present the numerical results in both one- and two-dimensional space, which have interesting ecological implications. Initial data and parameter values were chosen to mimic some existing patterns.