# Volume 28 - 2018 - 2019 - Mathematics for Biology and the Environment

Selected papers from CARI 2016 on the topics of mathematics for Biology and the Environment Editor : Nabil Gmati

### 1. Modèles mathématiques de digestion anaérobie: effet de l’hydrolyse sur la production du biogaz

We investigate, in this work, the effects of hydrolysis on the behavior of the anaero- bic digestion process and the production of biogas (namely, the methane and the hydrogen). Two modelisations of the hydrolysis process are involved. We consider, first, that the microbial enzymatic activity is constant, then we take into consideration an explicit hydrolytic microbial compartiment for the substrate biodegradation. The considered models include the inhibition of acetoclastic and hy- drogenotrophic methanogens. To examine the effects of these inhibitions in presence of a hydrolysis step, we first study an inhibition-free model. We determine the steady states and give sufficient and necessary conditions for their stability. The existence and stability of the steady states are illustrated by operating diagrams. We prove that modelling the hydrolysis phase by a constant enzymatic activity affects the production of methane and hydrogen. Furthermore, introducing the hydrolytic microbial compartment makes appear new steady states and affects the stability regions. We prove that the biogas production occurs at only one of the steady states according to the operating parameters and state variables and we determine the maximal rate of biogas produced, in each case.

### 2. Stochastic modeling for biotechnologies Anaerobic model AM2b

Le modèle AM2b est classiquement représenté par un système d'équations différentielles. Toutefois ce modèle n'est valide qu'en grande population et notre objectif est d'établir plusieurs mo-dèles stochastiques à différentes échelles. À l'échelle microscopique, on propose un modèle sto-chastique de saut pur que l'on peut simuler de fa con exacte. Mais dans la plupart des situations ce genre de simulation n'est pas réaliste, et nous proposons des méthodes de simulation approchées de type poissonnien ou de type diffusif. La méthode de simulation de type diffusif peut être vue comme une discrétisation d'une équation différentielle stochastique. Nous présentons enfin de fa con infor-melle un résultat de type loi des grands nombres/théorème central limite fonctionnelle qui démontre la convergence de ses modèles stochastiques vers le modèles déterministe initial.

### 3. Efficient high order schemes for stiff ODEs in cardiac electrophysiology

In this work we analyze the resort to high order exponential solvers for stiff ODEs in the context of cardiac electrophysiology modeling. The exponential Adams-Bashforth and the Rush-Larsen schemes will be considered up to order 4. These methods are explicit multistep schemes.The accuracy and the cost of these methods are numerically analyzed in this paper and benchmarked with several classical explicit and implicit schemes at various orders. This analysis has been led considering data of high particular interest in cardiac electrophysiology : the activation time ($t_a$ ), the recovery time ($t_r$) and the action potential duration ($APD$). The Beeler Reuter ionic model, especially designed for cardiac ventricular cells, has been used for this study. It is shown that, in spite of the stiffness of the considered model, exponential solvers allow computation at large time steps, as large as for implicit methods. Moreover, in terms of cost for a given accuracy, a significant gain is achieved with exponential solvers. We conclude that accurate computations at large time step are possible with explicit high order methods. This is a quite important feature when considering stiff non linear ODEs.