A positive finite-difference model in the computational simulation of complex biological film models

Abstract

In this work, we design a linear, two-step, finite-difference method to approximate the solutions of a biological system that describes the interaction between a microbial colony and a surrounding substrate. The model is a system of four partial differential equations with nonlinear diffusion and reaction, and the colony is formed by an active portion, an inert component and the contribution of extracellular polymeric substances. In this work, we extend the computational approach proposed by Eberl and Demaret [A finite difference scheme for a degenerated diffusion equation arising in microbial ecology, Electr. J. Differ. Equ. 15 (2007) pp. 77–95], in order to design a numerical technique to approximate the solutions of a more complicated model proposed in the literature. As we will see in this work, this approach guarantees that positive and bounded initial solutions will evolve uniquely into positive and bounded, new approximations. We provide numerical simulations to evince the preservation of the positive character of solutions.

Keywords::

• complex biological film
• extracellular polymeric substance biofilm model
• linear finite-difference scheme
• positivity-preserving
• boundedness-preserving
• M-matrices

Keywords::

• 65Q10
• 65M06
• 39A14
• 92D25

Acknowledgements

The author expresses his deepest gratitude to the anonymous reviewers for all their invaluable criticisms and suggestions, which contributed greatly to improve the quality of this manuscript.