Section: Research Program
Modeling and simulating microbial ecosystems
The chemostat model is quite popular in microbiology and bioprocess engineering [58] , [60] . Although the wording “chemostat” refers to the experimental apparatus dedicated to continuous culture, invented in the fifties by Monod and Novick & Szilard, the chemostat model often serves as a mathematical representation of biotic/abiotic interactions in more general (industrial or natural) frameworks of microbial ecology. The team carries a significant activity about generalizations and extensions of the classical model (see Equation (1 ) and Section 3.1.1 ) which assumes that the sizes of the populations are large and that the biomass can be faithfully represented as a set of deterministic continuous variables.
However recent observations tools based notably on molecular biology (e.g. molecular fingerprints) allow to distinguish much more precisely than in the past the internal composition of biomass. In particular, it has been reported by biologists that minority species could play an important role during transients (in the initialization phase of bioprocesses or when the ecosystem is recovering from disturbances), that cannot be satisfactorily explained by the above deterministic models because the size of those populations could be too small for these models to be valid.
Therefore, we are studying extension of the classical model that could integrate stochastic/continuous macroscopic aspects, or microscopic/discrete aspects (in terms of population size or even with explicit individually based representation of the bacteria), as well as hybrid representations. One important question is the links between these chemostat models (see Section 3.1.2 ).
About the chemostat model
The classical mathematical chemostat model:
$\begin{array}{ccc}\dot{s}\hfill & =\hfill & {\displaystyle \sum _{j=1}^{n}\frac{1}{{y}_{j}}{\mu}_{j}\left(s\right)\phantom{\rule{0.166667em}{0ex}}{x}_{j}+D\phantom{\rule{0.166667em}{0ex}}({s}_{in}s)}\hfill \\ {\dot{x}}_{i}\hfill & =\hfill & {\mu}_{i}\left(s\right)\phantom{\rule{0.166667em}{0ex}}{x}_{i}D\phantom{\rule{0.166667em}{0ex}}{x}_{i}\phantom{\rule{2.em}{0ex}}(i=1\cdots n)\hfill \end{array}$  (1) 
for $n$ species in concentrations ${x}_{i}$ competing for a substrate in concentration $s$, leads to the socalled “Competitive Exclusion Principle”, that states that generically no more species than limiting resources can survive on a long term [59] . Apart some very precise laboratory experiments that have validated this principle, such an exclusion is rarely observed in practice.
Several possible improvements of the model (1 ) need to be investigated, related to biologists' knowledge and observations, in order to provide better interpretations and predictive tools. Various extensions have already been studied in the literature (e.g. crowding effect, interspecific interactions, predating, spatialization, timevarying inputs...) to which the team has also contributed. This is always an active research topic in biomathematics and theoretical ecology, and several questions remains open or unclear, although numerical simulations guide the results to be proven.
Thanks to the proximity with biologists, the team is in position to propose new extensions relevant for experiments or processes conducted among the application partners. Among them, we can mention: intra and interspecific interactions terms between microbial species; distinction between planktonic and attached biomass; effects of interconnected vessels; consideration of maintenance or variable yield in the growth reactions; coupling with membrane fouling mechanisms.
Our philosophy is to study how complex or not very well known mechanisms could be represented satisfactorily by simple models. It often happens that these mechanisms have different time scales (for instance the flocculation of bacteria is expected to be much faster than the biomass growth), and we typically use singular perturbations
techniques to produce reduced models.
Stochastic and multiscale models
Comparatively to deterministic differential equations models, quite few stochastic models of microbial growth have been worked out in the literature. Nonetheless, numerous problems could benefit from such an approach (dynamics with small population sizes, persistence and extinction, random environments...).
For example, the need to clarify the role of minority species conducts to revisit thoroughly the chemostat model at a microscopic level, with birth and death or pure jump processes, and to investigate which kind of continuous models it raises at a macroscopic scale. For this purpose, we consider the general framework of Markov processes [57] .
It also happens that minority species cohabit with other populations of much larger size, or fluctuate with time between small and large sizes. There is consequently a need to build new “hybrid” models, that have individualbased and deterministic continuous parts at the same time. The persistence (temporarily or not) of minority species on the long term is quite a new questioning spread in several applications domains at the Inra Institute.
Continuous cultures of microorganisms often face random abiotic environments, that could be considered as random switching between favorable or unfavorable environments. This feature could lead to nonintuitive behaviors in long run, concerning persistence or extinction of populations. We consider here the framework of piecewise deterministic Markov processes [55] .
Computer simulation
The simulation of dynamical models of microbial ecosystems with the features described in Section 3.1.2 raises specific and original algorithmic problems:

simultaneous presence in the same algorithms of both continuous variables (concentration of chemicals or very large populations) and discrete (when the population has a very small number of individuals),

simultaneous presence in the same algorithms of stochastic aspects (for demographic and environmental noises) and deterministic ones (when the previous noises are negligible at macroscopic scales)

use of individualbased models (IBM) (usually for small population sizes).
We believe that these questions must be addressed in a rigorous mathematical framework and that their solutions as efficient algorithms are a formidable scientific challenge.