Positive computational modelling of the dynamics of active and inert biomass with extracellular polymeric substances

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.

Keywords::

• two-dimensional diffusion–reaction system
• growth dynamics of complex biological films
• extracellular polymeric substances
• positivity
• linear finite-difference model

AMS Subject Classification::

• 22E46
• 53C35
• 57S20

Acknowledgements

The author would like to thank the anonymous reviewers and the anonymous associate editor who handled this manuscript, for all their criticisms and comments. Indeed, their observations and suggestions helped to improve the quality of this work.

Disclosure statement

No potential conflict of interest was reported by the author.