Positive computational modelling of the dynamics of active and inert biomass with extracellular polymeric substances |
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Authors: | J. E. Macías-Díaz |
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Affiliation: | 1. Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, Aguascalientes, AGS 20131, Mexicojemacias@correo.uaa.mx |
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Abstract: | Departing from a complex system of nonlinear partial differential equations that models the growth dynamics of biological films, we provide a finite-difference model to approximate its solutions. The variables of interest are measured in absolute scales, whence the need of preserving the positivity of the solutions is a mathematical constraint that must be observed. In this work, we provide a numerical discretization of our mathematical model which is capable of preserving the non-negative character of approximations under suitable conditions on the model and computational parameters. As opposed to the nonlinear model which motivates this report, our numerical technique is a linear method which, under suitable circumstances, may be represented by an M-matrix. The fact that our method is a positivity-preserving scheme is established using the inverse-positive properties of these matrices. Computer simulations corroborate the validity of the theoretical findings. |
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Keywords: | two-dimensional diffusion–reaction system growth dynamics of complex biological films extracellular polymeric substances positivity linear finite-difference model |
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