Section: New Results

Theoretical results

Theory of competition for one limited resource

Participants : Claude Lobry, Tewfik Sari, Radhouene Fekih-Salem.

In the paper [24] , 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.

In the paper [14] , we consider the mathematical model of two species microbial competition on a single food resource in a chemostat, when one takes into account species interactions between the two populations of microorganisms and intraspecific interactions between individuals themselves, using strictly monotonic growth functions and distinct dilution rates.

Study of input/output maps of interconnected chemostats

Participants : Alain Rapaport, Ihab Haidar.

Patch or island models are popular in ecology, and are a convenient way to study the influence of a spatial structure of a geography on the distribution of the abundance of resources. Coupling such a structure with abiotic/biotic models and studying its input-output properties has been very rarely tackled in the literature. In biotechnology engineering, dead-zones models, that distinguish two sub-domains (a “living” and a “dead” one) are often used for approximating non perfectly mixed tanks. No more sophisticated representation, apart continuous space models (systems of partial differential equations), have been investigated.

We consider an hydric capacity and an nutrient flow that are fixed, and analyze the influence of different structures, having the same total hydric volume, on the output concentrations at steady-state, Three configurations are compared, under the assumption of a monotonic growth rate: perfectly-mixed, serial and parallel with diffusion rate. In each case, we show the uniqueness of a steady-state different to the washout equilibrium and its global asymptotic stability in the positive orthant. We prove the existence of a threshold on the input concentration of nutrient for which the benefits of the serial and parallel configurations over the perfectly-mixed one are reversed. In addition, we show that the dependency of the output concentrations on the diffusion rate can be non-monotonic, and give precise conditions for the diffusion effect to be advantageous [19] , [33] . The study encompasses the dead-zone models.

The possibly non-monotonic influence of the diffusion parameter on the output steady state is not intuitive, and leaves further investigations open for understanding or taking benefit of this property for natural ecosystems (such as saturated soils or wetlands) as well as for bioprocesses (such as wastewater treatments). This result can be also of interest for reverse engineering when deciding which among serial or parallel configurations is a better fit for the modelling of chemostat-like ecosystems, providing that one has an estimation of the hydric capacity of the system.

This work is part of the material thesis of I. Haidar [11] .

Aggregation models in the chemostat

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

Bacteria aggregation often occurs in bioprocesses, creating flocks or biofilms (the latter being attached to the tank walls). At a macroscopic level with large populations of aggregated and non-aggregated individuals, a simple way of modelling this phenomenon in the chemostat is to distinguish explicitly two populations: planktonic or free bacteria and attached ones. The main differences between flocks and biofilms rely in the attachment/detachment terms and the effective dilution rate (assumed to be zero or very small for biofilms). Typically, the specific growth rate of free bacteria is expected to be larger than the attached one (that have in average a restricted access to nutrient and use part of their energy to glue together).

Based on former works of the team and the main assumption that attachment and detachment dynamics are much faster than the biological one, we have shown that a significant difference between the specific dilution rates of the free and attached populations can surprisingly lead to bi-stability, even for a single species and monotonic growth rates [30] , [42] , [43] .

A work in progress addresses the case of two species, one of them having a non-monotonic growth rate (due to substrate inhibition) and the ability to form flocks. Without flock, the Competitive Exclusion Principle extended by G. Wolkowicz and her co-authors [47] shows the possibility to have one of the two species winning the competition depending on their initial repartition. Here, the presence of flocks may lead on the contrary to a single winner.

Neutral community models for microbial ecology

Participant : Bart Haegeman.

Hubbell's neutral model [50] describes the dynamics of an ecological community in terms of random birth, death, immigration and speciation events, attributing equivalent characteristics to all species. Despite the absurd simplicity of these assumptions, remarkable agreement between neutral model predictions (e.g., the distribution of the abundance of the species present in the community) and empirical observations has been reported for some, mostly rather diverse, ecological communities.

There is some evidence that also certain aspects of microbial communities can be well described by the neutral model. Highly diverse microbial communities have been difficult to deal with using more traditional modelling approaches from community ecology. The neutrality assumption could lead to an effective global description, without requiring quantitative species data (growth characteristics, interaction strengths, etc). We are actively participating in the development of neutral community models, with a focus on microbial systems.

(1) Effect of speciation process

It has been argued that the neutral model predictions are rather insensitive to its assumptions. However, we have found that the details of the way new species appear in the community (i.e., the speciation process) do matter, and can drastically change the model predictions. In particular, we have studied the neutral community model with random fission speciation. This speciation model is quite different from the point mutation model usually considered in neutral community model, and is generally believed to be more realistic.

Using a technique from theoretical physics, we have obtained the stationary distribution of species abundances for the random fission model. We have compared our solution with the well-known stationary distribution of species abundances for the point mutation model on empirical data (tree communities in tropical forests) [13] . Surprisingly, we found that the point mutation model fits the data better than the random fission model, although the latter is believed to be more realistic.

(2) Comparison with niche models

Neutral community models challenge more traditional, niche-based models in community ecology. Niche theory states that species can coexist only if they differ sufficiently in their characteristics (for example, their use of available substrates). Neutral theory assumes that all species have approximately equal characteristics. Hence, the two theories describe species coexistence in fundamentally different ways.

We have tried to narrow the gap between the two theories. We have proposed a mathematical model that combines essential features of niche-based and neutral community models [17] . It integrates species niches, described as Lotka-Volterra interactions, in the standard neutral community model. The analysis of this model indicates that the addition of species interactions has a limited effect on the species abundance distribution. We have further clarified this result using a slightly different model that also combines niche and neutral features [16] . For the latter model we have proved that the niche structure does not affect at all the species abundance distribution.

Quantifying genetic diversity of bacteria

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.

We have been dealing with the problem of how to quantify this observed genetic diversity. Microbiologists have introduced notions like the pan genome of a bacterial species (that is, the set of genes that are present in at least one of the organisms from the species) and the core genome of a bacterial species (that is, the set of genes that are present in every organisms from the species). However, we have argued that both the pan and core genome are difficult to estimate, and should not be used for quantitative purposes [21] . Instead, we have proposed a measure of genetic diversity that has much better estimation properties. It is based on the average number of genes shared by a pair of genomes when sequencing two randomly sampled organisms from the species under consideration. We have applied our estimator on six bacterial species (about 100 sequenced genomes in total). Software for our robust estimation procedure of genetic diversity is freely available, see http://ecotheory.biology.gatech.edu/downloads/genomic-fluidity-scripts .

Individual-based modelling

Participants : Fabien Campillo, Chloé Deygout, Coralie Fritsch, 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 [48] . 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 three studies in the field of IBMs. The first is part of the ANR MODECOL on the modelling of clonal plant growth (see Section 7.5 ); the second is part of the ANR DISCO on modelling of biofilms (see Section 7.4 ), the last one is a starting thesis.

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 6.1.8 .

In October 2011, Coralie Fritsch started a thesis at the École Doctorale I2E of the University of Montpellier 2, under the supervision of Fabien Campillo, Jérôme Harmand and Marc Joannides. This thesis is supported by a grant of the MESR and a grant of INRA from the MEM Meta-program (Méta-omiques des écosystèmes microbiens). The thesis aims at developing and analyzing individual-based microbial ecosystems models that capture both the spatial, biodiversity and function of these ecosystems. The thesis received the Agreenium label in December.

Hybrid modelling of biofilms in plug-flow reactors

Participants : Fabien Campillo, Chloé Deygout, Annick Lesne, Alain Rapaport.

Within the DISCO project of the SYSCOMM program founded by the ANR, we have proposed a multi-scaled modelling that combines three scales: a microscopic one for the individual bacteria, a mesoscopic or “coarse-grained” one that homogenises at an intermediate scale the quantities relevant to the attachment/detachment process, and a macroscopic one in terms of substrate concentration (see the Section 7.4 ).

Such an “hybrid” approach allows for modelling and understanding in plug-flow reactors [41] the interplay between

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    the formation of the biofilm at a microscopic scale, that starts from a small number of bacteria (thus a stochastic individual based description),

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    the limitation of the biofilm, due the carrying capacity of the wall attachment, at a mesoscopic scale,

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    the consumption of nutrient along the flow at a macroscopic level, as a solution of a coupled transport-reaction partial differential equation.

The numerical computation of such a model requires a software architecture that allows the simultaneous simulation of stochastic events at the bacteria scale and the continuous evolution (in space an time) of the substrate density.

Experiments on real tubular plug-flow reactors are currently driven at Cemagref HBAN with the perspective of comparison with numerical simulations. After spending one year at Montpellier for deriving and simulating the theoretical model, our post-doctoral fellow C. Deygout is presently participating to the real experiments at Cemagref Antony.

The multi-species case with different bacteria specialized in different environments (poor or rich in nutrient) is a work in progress.

Stochastic modelling for biotechnology

Participants : Fabien Campillo, Marc Joannides, Claude Lobry.

This year we continue to study stochastic models for the chemostat [12] , [39] , [27] . Starting from the well-known ordinary differential equation systems, we propose first a pure jump process model at the microscopic scale that leads to a stochastic differential equation at the intermediate scale and to an ordinary differential equation at the macroscopic level (fluid limit model). After developing the model, we establish the Fokker-Planck partial differential equation for the diffusion model. This PDE integrates a specific washing-out term. We proposed an ad hoc numerical integration scheme for the simulation of this PDE [39] .

In [40] , we consider a stochastic version of the basic predator-prey differential equation model. The model, which contains a parameter ω 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 ω individuals in the discontinuous one – 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 ω. For instance, in our example, if ω=10 7 we have extinction and if ω=10 8 we have persistence. This means that we must be very cautious when we use continuous variables in place of jump processes in dynamic population modelling even when we use stochastic differential equations in place of deterministic ones.

Minimal time control of batch bioprocesses

Participants : Denis Dochain, Alain Rapaport.

Minimal time control problems often occur in biotechnology when one has to fill tanks. Typically, the objective to be reached is to have the tank full with a prescribed value of substrate or product concentrations, the tank being filled with a high concentration of nutrient.

When a single reaction occurs, the optimal solution is already known and has been rigorously proved by. J. Moreno in 1999 [51] using the Green's theorem: it consists in a “bang-bang” strategy (fill as fast as possible or do not fill) and possibly a singular arc when the growth function presents an inhibition (i.e. a maximum growth for a precise concentration of nutrient). When impulse controls in addition to regular control are allowed, an extension of this result has been recently proposed with a different technique that do not use the Green's theorem [4] . This technique has also allowed to solve partially the problems when several species compete for the degradation of the substrate, but when all of them have a monotonic growth.

In the presence of complex non monotonic kinetics, typically characterized by the combination of two non-monotonic growth functions, aimed at emphasizing the presence of two parallel metabolic pathways to transform the limiting substrate into the biomass, the candidate singular arcs are multiple and determining which singular arc is eventually optimal is clearly a crucial issue. The local optimality conditions based on the Pontryagin Maximum Principle allow to characterize the geometric structure of the extremal trajectories, in which there may be singular arcs, but these necessary conditions are not always sufficient for determining which extremals are (globally) optimal. Then one has to compute the cost of each extremal or use global optimization methods such as dynamic programming or Hamilton-Jacobi- Bellman equation. The extremals are traditionally determined numerically by considering shooting methods, but for bang-bang control, it is well known that one may face numerical troubles because the shooting function is in general not smooth. For this problem, we have used an approximation technique first proposed and studied by C. Lobry and his students [53] and later by C. Silva and E. Trélat [52] , that consist in adding an artificial control. In [22] , we have proposed a new proof of convergence based on differential inclusions arguments that allows to relax the assumption of the uniqueness of the optimal solution for the convergence of the optimal paths of [52] . Then we have shown how to apply numerically this approximation procedure for analyzing the field of extremals on the whole state space. This technique appears to be quite effective for the practical determination of optimal synthesis in the planar case even in presence of multiple singular arcs.

Optimal control of continuous bioprocesses

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

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 proved 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. These works are part of the PhD work of A. Ghouali and W. Bouhafs.

Reference points in batch processes can be mimicked by a series of continuously stirred bioreactors in series at steady state (see applications 6.2.2 and 7.2 ). 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.

Minimal time bioremediation of natural resources

Participants : Jérôme Harmand, Alain Rapaport, Antoine Rousseau.

In biological wastewater treatment (batch or continuous bioprocesses), one has always to separate biomass from the purified liquid phase at the output of the tanks, that is not possible when tanks are rather natural reservoirs such as lakes or water tables.

We have proposed a new operation strategy that consists in treating with the help of a bioreactor aside. No bacteria are introduced in the reservoir but water is pumped and treated by microorganisms in a smaller tank, and treated water returns to the reservoir after being separated from the biomass. Consequently, there is no need of a separation operation for the reservoir.

The minimal time control problem consists in controlling the flow rate for having the substrate concentration of the whole reservoir below a given reference value as fast as possible.

Last year, we have determined analytical expressions of optimal feedback strategies for a general class of growth functions under the assumptions that the volume of the bioreactor is much smaller than the reservoir one, and that the spatial repartition of the concentration of the pollutant in the reservoir can be modelled by simple spatial representations: either perfectly mixed or discrete one directional gradient [44] , [15] . This year, we have studied more realistic spatial motifs:

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    dead-zones: we have shown that the optimal synthesis is identical to the perfectly mixed case, even though the time to reach the target is larger [31] ,

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    two parallel zones, allowing to control the repartition of the flow rate between the two zones. Without diffusion between the zones, the optimal solution is almost straightforward and under investigation in presence of lateral diffusion.

This work is mainly achieved in cooperation with Chilean researchers and PhD students within the associated team DYMECOS.