<\/p>\n
In this work, we design a linear, two-step implicit finite difference method to approximate the solutions of a biological system that describes the interaction between a microbial colony and a surrounding substrate. Three separate models are analyzed, all of which can be described as systems of partial differential equations (PDEs) with nonlinear diffusion and reaction, where the biological colony grows and decays based on the substrate bioavailability. The systems under investigation are all complex models describing the dynamics of biological films. In view of the difficulties to calculate analytical solutions of the models, we design here a numerical technique to consistently approximate the system evolution dynamics, guaranteeing that nonnegative initial conditions will evolve uniquely into new, nonnegative approximations. This property of our technique is established using the theory of M-matrices, which are nonsingular matrices where all the entries of their inverses are positive numbers. We provide numerical simulations to evince the preservation of the nonnegative character of solutions under homogeneous Dirichlet and Neumann boundary conditions. The computational results suggest that the method proposed in this work is stable, and that it also preserves the bounded character of the discrete solutions.<\/p>\n
Finally, for \\( \\mathscr{E}_3 \\) and \\( \\mathscr{E}_4 \\) and just an expansion in spatial dimensions of \\( \\mathscr{E}_2 \\), we have for \\( \\mathscr{E}_4 \\):<\/p>\n
\\[
\n\\begin{aligned}
\n&\\alpha^{k,-}_{l,m,n;z}S^{k+1}_{l-1,m,n} + \\alpha^{k,-}_{l,m,n;x}S^{k+1}_{l,m-1,n} + \\alpha^{k,-}_{l,m,n;y}S^{k+1}_{l,m,n-1} + \\beta^{k,S}_{l,m,n}S^{k+1}_{l,m,n} + \\\\
\n&\\alpha^{k,+}_{l,m,n;z}S^{k+1}_{l+1,m,n} + \\alpha^{k,+}_{l,m,n;x}S^{k+1}_{l,m+1,n} + \\alpha^{k,+}_{l,m,n;y}S^{k+1}_{l,m,n+1} + \\zeta^{k,1}_{l,m,n}X^{k+1}_{l,m,n} + \\eta^k_{l,m,n}E^{k+1}_{l,m,n} = S^k_{l,m,n}
\n\\end{aligned}
\n\\]<\/p>\n
Similar relations hold for \\( X^{k+1}_{l,m,n} \\), \\( I^{k+1}_{l,m,n} \\), and \\( E^{k+1}_{l,m,n} \\) as described in the manuscript.<\/p>\n
Constants are defined as:<\/p>\n
\\[
\n\\begin{aligned}
\n\\alpha^{k,\\pm}_{l,m,n;w} &= -R^{k,Y}_w D(\\mu^{\\pm}_{w}U^k_{l,m,n}) \\\\
\n\\beta^{k,S}_{l,m,n} &= 1 – \\sum_{w\\in\\{x,y,z\\}}(\\alpha^{k,-}_{l,m,n;w} + \\alpha^{k,+}_{l,m,n;w}) + a_1\\frac{\\Delta t_k X^k_{l,m,n}}{\\kappa_S + S^k_{l,m,n}} \\\\
\n\\beta^{k,X}_{l,m,n} &= 1 – \\sum_{w\\in\\{x,y,z\\}}(\\alpha^{k,-}_{l,m,n;w} + \\alpha^{k,+}_{l,m,n;w}) + \\frac{a_4 \\Delta t_k}{\\kappa_S + S^k_{l,m,n}} \\\\
\n\\gamma^{k,1}_{l,m,n} &= -\\frac{\\mu \\Delta t_k X^k_{l,m,n}}{\\kappa_S + S^k_{l,m,n}}
\n\\end{aligned}
\n\\]<\/p>\n
By inspection:<\/p>\n
With these definitions, the discrete formulation for \\( \\mathscr{E}_1 \\) can be written compactly as
\n\\[
\n\\mathcal{M}^k_{\\mathscr{E}_1} \\mathbf{v}^{k+1}_{\\mathscr{E}_1} = \\mathbf{v}^{k}_{\\mathscr{E}_1}
\n\\]
\nwhere each block \\( \\mathcal{A}^k_{\\mathscr{E}_1} \\) and \\( \\mathcal{B}^k_{\\mathscr{E}_1} \\) is constructed as in the equations above.<\/p>\n<\/div>\n
<\/p>\n","protected":false},"excerpt":{"rendered":"
Abstract In this work, we design a linear, two-step implicit finite difference method to approximate the solutions of a biological system that<\/p>\n