### Mathematical study of an anaerobic digestion model, part 1

#### Abstract

We will start by studying in great detail the simplest possible model that we will call ”minimal model”. It is minimal in the sense that if we tried to simplify it still a little, nothing would remain of what characterizes a ”real” Chemostat. For this minimal model we assume that s⇒(s) is a function of the substrate only and that the yield y(.)=Y is constant. We will engage in a very precise mathematical study of this model. It is not very difficult but it is imperative to understand all the details well because all the later studies, more complex, are based on the properties of the minimal model. In particular, the notion of ”growth threshold” is fundamental. In a first section we establish the mathematical properties that we interpret in the following section and finally we produce some simulation. In a last section we propose four possible extensions of the minimal model. The mathematical treatment will be faster either because it does not present any difficulty for the reader who has assimilated the above or, on the contrary, because it is more delicate and falls outside the scope of this work.

**Published:**2022-10-31

**How to Cite this Article:**Mohammed Hanaki, Hassan Sfouli, Mathematical study of an anaerobic digestion model, part 1, Commun. Math. Biol. Neurosci., 2022 (2022), Article ID 112 Copyright © 2022 Mohammed Hanaki, Hassan Sfouli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Commun. Math. Biol. Neurosci.

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