In this work, we present analysis of a scaled time-dependent reaction–diffusion system modeling three competitive species dynamics that is of Lotka–Volterra type for coexistence, permanence and stability. The linear analysis is based on the application of qualitative theory of ordinary differential equations and dynamical systems. We consider two notable spatial discretization methods in conjunction with an adaptive time stepping method to verify the biological wave phenomena of the solutions and present the numerical results in one dimensional space. Adequate numerical resulting are provided in one and two dimensions to justify theoretical investigations. In addition, efficiency of the proposed numerical schemes are justified.