Section: New Results

Macroscopic models

About species coexistence

Participants : Fabien Campillo, Jérôme Harmand, Claude Lobry, Alain Rapaport, Tewfik Sari.

The so called “Principle of Competitive Exclusion” states that in the chemostat model, in presence of $p$ substrates only $p$ species can coexist. By contrast, in a bioreactor used for decontamination, hundreds to thousand different species are observed in presence of just very few substrates. Actually the classical chemostat models rely on assumptions: perfect mixing, substrate-dependent growth rate, constant environment, only asymptotic results are considered, deterministic continuous models...

A long term objective since Mere Inria project-team is to revisit the chemostat model in the absence of one or more of these hypotheses having in view the question of coexistence. In our “major publications” we proved coexistence in absence of the second hypothesis [6] or during long transient [9] . In [57] , we consider the case where the environment (in some sense) is periodic in time. Our results concerning non continuous and/or stochastic models (see Section 6.2.3 ) are also a first step in avoiding the fifth hypothesis.

Modeling and analysis of bioprocesses

Participants : Boumédiène Benyahia, Radhouane Fekih-Salem, Jérôme Harmand, Claude Lobry, Guilherme Pimentel, Alain Rapaport, Tewfik Sari.

Within the supervision of the PhD thesis of R. Fekih-Salem, we have studied a chemostat model where the species are present in two forms, isolated and aggregated individuals, such as attached bacteria in biofilm or bacteria in flocks [22] . We show that our general model contains a lot of models that were previously considered in the literature [77] . Assuming that flocculation and deflocculation dynamics are fast compared to the growth of the species, we construct and analyse a reduced chemostat-like model in which both the growth functions and the apparent dilution rate depend on the density of the species.

Within the framework of the PhD thesis of B. Benyahia, we have included the fouling dynamics of membranes into the AM2 (or AMOCO) model and we have analyzed the resulting model (called the AM2b) [15] . In particular, we have integrated into this model the production and the degradation of Soluble Microbial Products (SMP), which are known to play an important role in the membrane fouling phenomenon. We show that under some general assumptions, the AM2b model has the same number of equilibria as the AM2 model and can exhibit bi-stability. However, under certain operating conditions or if biological parameters values are slightly modified, the AM2b model exhibits equilibria bifurcations and multi-stability properties.

The available anaerobic digestion models used for control purposes do usually only consider soluble matter. In fact, part of the pollutants are not soluble but are under a particulate form. In order to establish whether adding the dynamics of such matter into the models is important for the system behavior or not, we have studied new anaerobic models and established that depending on the kinetics of this additional reaction step, the qualitative behavior of the process may be significantly modified [44] .

This year, G. Pimentel as started a PhD co-supervised with the University of Mons, about modeling of the membrane fouling in bioreactors in view of control. The objective is to represent cake formation and air cross-flow as a manipulated variable in the models, in view of future studies of control strategies for improving the efficiency of MBR processes [35] , [47] , [52] , [53] .

Ecosystem functioning in heterogeneous environments

Participants : Céline Casenave, Jérôme Harmand, Alain Rapaport.

This year, we have carried out a study of particular spatial interconnections such as “buffered” configurations, and its ecological impacts in terms of setup of a species in environments that are unfavorable when perfectly mixed. We have extended our previous results about the design of configurations for obtaining a global stability [28] . New conditions have been obtained for a species to setup when it is impossible in a perfectly mixed environment. At the opposite, we have characterized configurations that could destabilize bioprocesses.

With UMR Géosciences (Univ. of Rennes 1), we have carried on our analysis of the equivalence of two soil fracture models in terms of transfer functions [19] : the MINC (Multiple INteractiong Continua) and MRMT (Multiple Rate Mass Transfer) models that are quite popular in soil hydrodynamics. We have shown that a strict equivalence can be obtained if one considers different volumes in the discretization of the MINC model. For the moment, this study concerns the transfer of abiotic substances only.

In soil ecosystems, it often happens that several functional groups can be detected to operate concomitantly. We have investigated the mathematical properties of a relatively simple model that has been proposed by the UREP lab (Inra Clermont), that distinguishes explicitly two functional groups of micro-organisms: the decomposers of SOM (soil organic mater) and the producers of SOM, and compared it with a single microbial compartment model in terms of prediction of the so-called “priming effect” [27] .

Together with agronomists of the UMR Eco & Sols (Cirad, Inra, IRD, Montpellier SupAgro) and the supervision of the MSc thesis of C. Droin, we have proposed and started to study a new model of consumer/resource for soil microbial ecosystems, in which we explicitly distinguish available and recalcitrant resources [71] .