On a conditionally stable nonlinear method to approximate some monotone and bounded solutions of a generalized population model

In this work, we design a numerical method to approximate some solutions of a generalization of a nonlinear diffusion-reaction model which appears in the context of population dynamics. The existence of traveling-wave solutions for the equation under consideration is a well-known fact. Some of such solutions are positive, bounded from above, and monotone in both space and time. Motivated by these facts, we propose an explicit, nonlinear, finite-difference methodology to approximate consistently the solutions of the model under investigation. In the linear regime, the method is consistent of first order in time and of second order in space. Under certain, flexible parameter conditions, the method is capable of preserving the positivity, the boundedness, and the spatial and the temporal monotonicity of the traveling-wave solutions. Moreover, we establish analytically and numerically that the nonlinear method is conditionally stable. A computational implementation of our technique shows that the method preserves in practice the mathematical features of interest of the exact solutions considered.