Summary: We have given an extension to the study of Kierstead, Slobodkin, and Skellam (KiSS) model. We present the theoretical results based on the survival and permanence of the species. To guarantee the long-term existence and permanence, the patch size denoted as \(L\) must be greater than the critical patch size \(L_c\). It was also observed that the reaction-diffusion problem can be split into two parts: the linear and nonlinear terms. Hence, the use of two classical methods in space and time is permitted. We use spectral method in the area of mathematical community to remove the stiffness associated with the linear or diffusive terms. The resulting system is advanced with a modified exponential time-differencing method whose formulation was based on the fourth-order Runge-Kutta scheme. With high-order method, this extends the one-dimensional work and presents experiments for two-dimensional problem. The complexity of the dynamical model is discussed theoretically and graphically simulated to demonstrate and compare the behavior of the time-dependent density function.