On a fully discrete finite-difference approximation of a nonlinear diffusion–reaction model in microbial ecology

Abstract

In this work, we introduce a numerical method to approximate the solutions of a multidimensional parabolic partial differential equation with nonlinear diffusion and reaction, subject to nonnegative initial data and homogeneous boundary conditions of the Neumann type. The equation considered is a model for both the growth of biological films and the propagation of mutant genes which are advantageous to a population. The initial-boundary-value problem under investigation is fully discretized temporally and spatially following a finite-difference methodology which results in a simple, linear, implicit scheme that is consistent with respect to the continuous problem. The method is a two-step technique that preserves the positivity and the boundedness of initial profiles. We provide some simulations on the growth of microbial colonies, and comparisons versus a standard approach.

Keywords:

• parabolic partial differential equation
• nonlinear diffusion and reaction
• finite-difference scheme
• microbial colonies
• positivity and boundedness
• inverse-positive matrices

2010 AMS Subject Classifications :

• 65M06
• 35K57
• 35K61

Acknowledgements

The corresponding author wishes to express his deepest gratitude to Dr F.J. Avelar-González, professor of the Department of Physiology and Pharmacology at the Universidad Autónoma de Aguascalientes, for all his invaluable help and support during the realization of this study. The authors also wish to thank the anonymous reviewers and the editor in charge of handling the present manuscript, for their constructive criticism and comments, all of which helped to improve the quality of this paper.