Abstract
The study of biofilm formation is undoubtedly important due to micro-organisms forming a protected mode from the host defense mechanism, which may result in alteration in the host gene transcription and growth rate. A mathematical model of the nonlinear advection–diffusion–reaction equation has been studied for biofilm formation. In this paper, we present two novel non-standard finite difference schemes to obtain an approximate solution to the mathematical model of biofilm formation. One explicit non-standard finite difference scheme is proposed for biomass density equation and one property-conserving scheme for a coupled substrate–biomass system of equations. The nonlinear term in the mathematical model has been handled efficiently. The proposed schemes maintain dynamical consistency (positivity, boundedness, merging of colonies, biofilm annihilation), which is revealed through experimental observation. In order to verify the accuracy and effectiveness of our proposed schemes, we compare our results with those obtained from standard finite difference schemes and earlier known results in the literature. The proposed schemes (NSFD1 and NSFD2) show good performance. The NSFD2 scheme reveals that the processes of biofilm formation and nutritive substrate growth are intricately linked.
Highlights
The word biofilm refers to aggregation of smaller organisms existing on a changing interface embedded in a polymeric matrix adhered to abiotic or biotic surfaces
Results of Numerical Experiment 2 All the three methods give almost the same profile at times 5, 10 and 20. One observation from these numerical simulations is that the biofilm colonies are annihilated as we progress in time, and this is due to rate of reaction being negative
We display the results of the numerical solution vs. x vs. y using NSFD1 and classical schemes in Figures 6 and 7
Summary
The word biofilm refers to aggregation of smaller organisms existing on a changing interface embedded in a polymeric matrix adhered to abiotic or biotic surfaces. Nilsson et al [5] probed the effect of changes in acylated homoserine lactone (AHL) concentration and investigated the possibility of biofilm formation using a mathematical model Their model consists of a set of coupled ordinary differential equations (ODEs) in which the bacterial growth is modelled using the well-known logistic equation: dηbc dt. Many known mathematical models in the literature are represented in terms of partial differential equations (PDEs) with a nonlinear density-dependent diffusion reaction describing biofilm growth. Eberl and Demaret [9] provided a non-classical finite difference scheme for the numerical solution to Equation (2) which preserves the condition of merging of two different colonies. The novelty of this work is the design of a novel structure-preserving (positivity, boundedness, merging of colonies, biofilm annihilation) finite-difference scheme to approximate the full and limiting cases of Equation (2), which describes the growth/decay of a microbial colony. We compare the performance of our scheme against earlier known work in the literature and with the classical finite difference scheme
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