![Sample our Mathematics & Statistics journals, sign in here to start your FREE access for 14 days]({"type":"image" , "src" : "/sda/5943/TOC-Subject-Sample-2021-Mathematics-Statistics.jpg"})
Abstract
Departing from a system of parabolic partial differential equations that describes the interaction of a microbial colony and a substrate of nutrients, we propose a finite-difference discretization to approximate the bounded and non-negative solutions of the model. The literature establishes the existence and uniqueness of bounded and non-negative solutions of the continuous problem under suitable, analytical conditions; however, the exact determination of such solutions for arbitrary initial-boundary-value problems is a difficult task, whence the need of designing numerical techniques to approximate them is pragmatically justified. The numerical properties of existence and uniqueness of non-negative and bounded solutions are established using the theory of M-matrices. We provide some illustrative simulations to evince the fact that the method preserves the properties of non-negativity and boundedness in the practice.
Acknowledgements
The corresponding main author wants to acknowledge enlightening conversations with Prof. I. E. Medina-Ramíreq of the Department of Chemistry at the Universidad Autónoma de Aguascalientes and Prof. F. J. Avelar-González of the Department of Physiology and Pharmacology at the same institution. The authors wish to thank the anonymous reviewers and the editor who handled this manuscript, for all their invaluable comments and criticism.