Section: New Results

Theoretical results

Models resource/consumer

The team maintains a significant activity about the theory of the chemostat model, proposing and studying extensions of the classical models.

Theory of competition and coexistence

Participants : Jérôme Harmand, Claude Lobry, Tewfik Sari.

In the papers [41] , [50] we consider deterministic models of competition. We study the persistence of species. In [25] we study a syntrophic relation between microbial species. In [26] , we give a global asymptotic stability result for a mathematical model of competition between several species in a chemostat, by using a new Lyapunov function. The model includes both monotone and non-monotone response functions, distinct removal rates for the species and variable yields, depending on the concentration of substrate.

Study of interconnected chemostats

Participants : Jérôme Harmand, Alain Rapaport.

We have shown how a particular spatial structure with a buffer globally stabilizes the chemostat dynamics with non-monotonic response function, while this is not possible with single, serial or parallel chemostats of the same total volume and input flow. We give a characterization of the set of such configurations that enjoy this property, as well as the configuration that ensures the best nutrient conversion. Furthermore, we characterize the minimal buffer volume to add to a single chemostat for obtaining the global stability. These results are illustrated with the Haldane function that models inhibition in micro-organisms growth [67] .

In industrial applications, the attraction of the wash-out equilibrium is undesired because it presents a risk that may ruin the culture in case of disturbance, temporarily pump breakdown or presence of toxic material that could drive the state in the attracting basin of the wash-out equilibrium. This approach has led to a patent deposit by INRA [59] during the PhD of H. Haidar, a former PhD student of the team [80] .

Aggregation models in the chemostat

Participants : Radhouene Fekih-Salem, Jérôme Harmand, Claude Lobry, Alain Rapaport, Tewfik Sari.

We have studied a model of the chemostat where the species are present in two forms, isolated and aggregated individuals, such as attached bacteria in biofilm or bacteria in flocks. We show that our general model contains a lot of models that were previously considered in the literature. Assuming that flocculation and deflocculation dynamics are fast compared to the growth of the species, we construct a reduced chemostat-like model in which both the growth functions and the apparent dilution rate depend on the density of the species. We also show that such a model involving monotonic growth rates may exhibit bi-stability, while it may occur in the classical chemostat model, but when the growth rate is non monotonic [21] , [54] . This work is part of the PhD of R. Fekih-Salem co-supervised by A. Rapaport and T. Sari.

This research subject has been mainly motivated by the DISCO project (see Section 7.3 ).

Overyielding in continuous bioprocesses

Participants : Denis Dochain, Alain Rapaport.

We have shown that for certain configurations of two chemostats fed in parallel, the presence of two different species in each tank can improve the yield of the whole process, compared to the same configuration having the same species in each volume. This leads to a (so-called) “transgressive over-yielding” due to spatialization [35] .

This work has been achieved during the stay of Prof. P. de Leenheer (Univ. Florida).

Measuring taxonomic diversity of microbial communities

Participant : Bart Haegeman.

Diversity is considered to be a main determinant of the behavior of microbial communities. However, measuring microbial diversity is challenging. Although metagenomic techniques allow us to sample microbial communities at unprecedented depths, the disparity between community (e.g., ${10}^{15}$ organisms) and sample (e.g., ${10}^5$ organisms) remains large. We have studied what the diversity observed in a sample tells us about the real diversity of the community.

For a given empirical sample the aim is to construct the community from which this sample was taken. It turns out that a large set of community structures are consistent with the sample data. Some diversity metrics vary widely over this set of consistent communities, and are therefore difficult to infer from the sample data. Other diversity metrics are approximately constant over the set of consistent communities, and are therefore much easier to infer from the sample data.

The analysis of the set of consistent communities has yielded the following insights. First, it is impossible to robustly estimate the number of species from sample data. This is easy to understand. Microbial communities typically contain a large number of rare species, and these rare species are unlikely to be present in the sample. Hence, sample data are lacking crucial information to estimate species richness. Second, other diversity metrics, in particular Shannon and Simpson diversity, can be robustly estimated from sample data. We have constructed lower and upper estimates for a general class of diversity metrics, and we have shown that the difference between the extremal estimators, that is, the estimation uncertainty, is small for Shannon and Simpson diversity.

A theory of genetic diversity within bacterial species

Participant : Bart Haegeman.

With the wide availability of DNA sequencing, microbiologists are now able to rapidly sequence entire bacterial genomes. Comparison of these genomes has revealed a large genetic diversity within bacterial species. For example, one genome of the bacteria E. coli has about 4000 different genes, but a set of 10 genomes of E. coli has typically over 10000 different genes. Some of these genes are shared by all or almost all of the genomes, but many other genes are only present in one or a few of the genomes. This observation has important implications for the definition of bacterial species and for the description of the functional characteristics of bacteria.

In [23] we propose a theory for the frequency distribution of genes within a set of genomes. The model describes the genetic diversity as a balance between two forces. Demographic processes such as division and death tend to reduce the genetic diversity; horizontal gene transfer from other species can increase the genetic diversity. Our model predicts that the gene frequency distribution is U-shaped, meaning that there are a large number of genes present in only a few genomes, a small number of genes present in about half of the genomes, and a large number of genes present in almost all genomes. This prediction is consistent with the gene frequency distributions of six bacterial species we have analyzed (about 100 sequenced genomes in total). Importantly, the model does not assume any functional difference between the genes, that is, genes are considered to be selectively neutral. By showing that empirical gene frequency distributions can be reproduced by a neutral genome model, we contend that the frequency of a gene should not be interpreted as an indication of its essentiality or importance.

Individual-based modelling for bacterial ecosystems

Participants : Fabien Campillo, Chloé Deygout, Coralie Fritsch, Jérôme Harmand, Marc Joannides, Claude Lobry.

In terms of computational modelling of ecosystems, individual-based models (IBMs) are an interesting path to explore. We can outline two types of IBMs. On the one hand “detailed IBM” attempt to integrate in an ad-hoc way all the knowledge available about an ecosystem. On the other hand, “simplified IBM” are limited to one or several mechanisms to simplify the analysis. The former may be more realistic but are often difficult to analyze. Although the latter are too simplistic in realistic situations they lend themselves to the analysis and numerical analysis. We focus on the latter.

The IBMs offer an interdisciplinary language between biologists, biotechnologists, mathematicians, and computer scientists, to develop models in the form of relatively simple rules. In the case of simplified IBMs it is possible to translate these rules in the form of a branching Markov process with values in a space of measures. Using scaling methods, the IBMs can be approximated by integro-differential equations; using model simplification methods IBMs can be reduced to stochastic or ordinary differential equations. The mathematical interpretation of the IBMs and their analysis is relatively recent and still very few studies exist [78] . The numerical analysis of these models is yet to be built. Under certain conditions, IBMs themselves can be simulated through adapted Monte Carlo procedures.

The MODEMIC project-team develops many studies in the field of IBMs. The first is part of the ANR MODECOL on the modelling of clonal plant growth (see Section 7.4 ); the second is part of the ANR DISCO on modelling of biofilms (see Section 7.3 ); the third is also part of the ANR DISCO is dedicated to the modelling of biofilms in plug-flow reactors (see Section 6.2.2 ); the last one is the ongoing thesis of Coralie Fritsch at the École Doctorale I2E of the University of Montpellier 2, under the supervision of Fabien Campillo and Jérôme Harmand. The thesis aims at developing and analyzing “simple” individual-based microbial ecosystems models.

In all cases, we aim at developing the Monte Carlo simulation of the IBM as well as analyzing their links with integro-differential models. We also seek to make connections with non-IBM models proposed in Section 6.1.5 .

Stochastic/discrete and stochastic/continuous modelling for biotechnology and population dynamics

Participants : Fabien Campillo, Marc Joannides, Claude Lobry.

In [14] , we consider a stochastic model of the two-dimensional chemostat as a diffusion process for the concentration of substrate and the concentration of biomass. The model allows for the washout phenomenon: the disappearance of the biomass inside the chemostat. We establish the Fokker-Planck equation associated with this diffusion process, in particular we describe the boundary conditions that modelize the washout. We propose an adapted finite difference scheme for the approximation of the solution of the Fokker-Planck equation.

In [15] , we consider a hybrid version of the classical predator-prey differential equation model. The proposed model is hybrid: continuous/discrete and deterministic/stochastic. It contains a parameter $\omega$ which represents the number of individuals for one unit of prey – if $x$ denotes the quantity of prey in the differential equation model $x=1$ means that there are $\omega$ individuals in the discrete model – is derived from the classical birth and death process. It is shown by the mean of simulations and explained by a mathematical analysis based on results in singular perturbation theory (the so called theory of Canards) that qualitative properties of the model like persistence or extinction are dramatically sensitive to $\omega$. This means that we must be very cautious when we use continuous variables in place of jump processes in dynamic population.

Optimal control of continuous bioprocesses

Participants : Walid Bouhafs, Amel Ghouali, Jérôme Harmand, Alain Rapaport.

In continuous bioprocesses, a usual objective is to stabilize the output of the bioreactors about a desired steady state (in wastewater industry, this value is typically chosen under the norm of authorized discharge). It happens more and more frequently that transient trajectories are expected also to maximize a product of interest.

We have begun to study the maximization of the gaseous production of methane in anaerobic processes over a given period of time on specific problems. For the moment we have proven that the optimal trajectory consists in approaching a unique singular arc as fast as possible when only one limiting substrate has to be converted, but the problem is still open when involving several substrates [39] . Another problem arises for alternating aerobic-anoxic systems. Revisiting the results obtained several years ago within the framework of Djalel Mazouni's thesis, we aims at proposing optimal time control policies for sequencing batch reactors in which simultaneous nitrification and denitrification take place. The solution for the original problem is a difficult task but we have already proposed a solution for a slightly modified model [30] . These last results have been obtained within the framework of the PhD thesis of W. Bouhafs.

Reference points in batch processes can be mimicked by a series of continuously stirred bioreactors in series at steady state (see applications 6.2.4 and 7.1 ). We study the minimal time problem to drive the nutrients concentrations of a cascade of chemostats. The control variable is the dilution rates of each tank, under the constraint that each dilution rate is bounded by the one of the previous tank, that makes the system not locally controllable. For the particular case of two tanks with total mass at steady state, the planar feedback synthesis has been found but the problem is still under investigation for the general case.

One important issue in bioprocesses is to find optimal feedback control laws in order to steer a system describing a perfectly-mixed bioreactor to a given target value in a minimal amount of time. Finding adequate feeding strategies can significantly improve the performance of the system. A typical target (for fed-batch bioreactors) is to consider the substrate concentration to be lower than a given reference value at the end of the process. Other criterium can be studied such as the maximization of the production of biomass in a given time period. Singular strategies (in reference to the theory of singular arcs in optimal control theory) can be defined in this context and are natural due to the constraints on the system. One objective of our research is to characterize optimal feedback control laws using mathematical tools from optimal control laws, and also to develop numerical methods that can handle problems with a large number of parameters.

Modelling the functioning of soil ecosystems

Participants : Céline Casenave, Jérôme Harmand, Alain Rapaport.

In ecology, one of the important challenges is the understanding of the biodiversity observed in the natural ecosystems. Several models have been proposed to explain this biodiversity, and in particular the coexistence of different species. In these models, it is often assumed that, when they die, the micro-organisms are directly converted in an assimilable resource, that is a resource that alive organisms can consume. However, we know that it is not the case in reality. Indeed, the organic matter stemmed from the dead organisms has to be transformed in assimilable resource before it can be consumed. This transformation is performed by some micro-organisms which have this special ability.

We have proposed a new model of soil ecosystems, of chemostat type. This model is rather simple, but also original because it takes into account several mechanisms:

• the growth, the mortality and the respiration,

• the ability of some organisms to transform the non assimilable resources in assimilable ones,

• the advantage that an organism can gain from this ability of transformation.

For the moment, we have considered the case where only one or two types of organisms are present. The model is finally composed of 3 (or 4) nonlinear ordinary differential equations: one per type of organisms, one for the assimilable resource and one for the non assimilable one. The study of the equilibrium points has first shown the possibility of coexistence, at equilibrium, of the two organisms. Then, in numerical simulations, we have observed the possible existence of limit cycles, which can also explain the coexistence of organisms observed in the nature.

This problem is still under study; a working group (in particular with researchers of the UMR Eco & Sols, conducted by B. Jaillard) meet regularly to discuss about the problems of modelling in ecology.

We have also investigated the sampling effect that occurs when micro-biologists select randomly species in a natural ecosystem for studying reconstituted ecosystems in a controlled environment. We have proposed a very simple probabilistic model, that shows that observing average increases or decreases on the performances of these reconstitute ecosystems when modifying the size of the sampling, allows to infer kinds and proportions of the interactions among species present in the original ecosystem [65] . This research is conducted with the UMR Eco & Sols, Montpellier, (B. Jaillard) and the UMR BIOEMCO, Grignon (N. Nunan).

Equivalence between models of fractured porous media

Participants : Céline Casenave, Jérôme Harmand, Alain Rapaport, Alejandro Rojas-Palma.

In geosciences, models of fractured porous media are often described as a mobile zone driven by advection, and one or several immobile zones directly or indirectly connected to the mobile zone by diffusion terms. We believe that these models are also relevant to describe flows in soil or in porous media such as biofilms. They are very close from the spatial representations used in Section 6.2.6 . We have shown, using a transfer function approach, that two schemes often used in the literature: the MINC (Multiple INteractive Continua) where diffusive compartments are connected in series, and the MRMT (Multiple Rate Mass Transfer) where diffusive compartments are connected in star around the mobile zone, are equivalent input-output representations [56] , and providing formulas (up to three compartments) to pass from one representation to another. This result means that one can simply choose the most convenient approach when dealing with control or optimization without any loss of generality. We are currently working on the general case of $n$ compartments with $n$ larger than three, and looking for equivalent classes of configurations that could be half way between MINC and MRMT and fit better the spatial representations of real world.

This work is performed with the UMR GéoSciences Rennes (J.R. de Dreuzy), and has led to the internship of a Chilean MsC student (A. Rojas-Palma).

Non-linear filtering for the chemostat

Participants : Boumediene Benyahia, Amine Boutoub, Fabien Campillo, Jérôme Harmand.

Monte Carlo-based inference methods like particle filtering are bound to develop in the context of biotechnology. In contrast with the classical observer approach, inference through Monte Carlo methods can handle measurements in discrete time in low frequency and with low signal-to- noise ratio. Based on the stochastic modeling of the chemostat, these approaches may also be used for model selection and hypothesis testing.

In a preliminary work [28] we consider the bootstrap particle filter applied to a 2-dimensional chemostat model. The internship of Amine Boutoub dedicated to the study of particle filtering for more realistic chemostat models has recently started.

Functional identification of growth functions in the chemostat

Participant : Alain Rapaport.

We have proposed an adaptive control law that allows one to identify unstable steady states of the open-loop system in the single-species chemostat model without the knowledge of the growth function. We have then shown how to use a continuation technique to reconstruct the whole graph of the growth function, providing a new method for identifying non-monotonic growths [42] , [27] . Two variants, in continuous and discrete time, have been studied. An analysis of the case of two species in competition has shown the ability of the method to detect a mixed culture for which dominance depends on the dilution rate, due to a property of stability loss in slow-fast dynamics. This method leaves open future extensions for extremum seeking problems.

This work has been conducted in cooperation with Universities of Exeter (J. Sieber) and Plymouth (S. Rodrigues), and the EPI SISYPHE (M. Desroches).

Diffusive representation of integro-differential models

Participant : Céline Casenave.

This work is done in collaboration with Emmanuel Montseny (LAAS/CNRS), Gérard Montseny (LAAS/CNRS), and Christophe Prieur (LIAFA/CNRS).

In lots of dynamic systems of Physics or others scientific fields such as Biology (Volterra models), dynamic integral operators, often of convolution type, are involved. Problems relating to integro-differential models are often difficult to solve, especially because these models are not time-local. In this context, the methodology called “diffusive representation” presents some interests: an integral operator is represented by its gamma-symbol, directly deduced from its transfer function. It can be formulated by means of a state realization whose dimension is numerically reasonable whatever the size of the system may be. In addition to this interesting practical side, the diffusive representation offers a unified mathematical framework, well adapted to analysis of integral convolution operators.

Several dynamic problems can be tackled in an original and quite simple way by using the diffusive representation. In fact, all the operatorial problems of modeling, simulation, control, model identification, model reduction, etc. can be formulated in such a way that the object of the problem is the gamma-symbol of the operator solution.

Several problems are under study:

• the identification of integro-differential models [66] ,

• the controllability of some SISO Volterra models [63] ,

• the simulation and the analysis of a model of porous media[64] .

These works follow up on the ones developed during the PhD thesis of Céline Casenave, which deals with the problem of the operator inversion for the application to non local dynamic problems.