Section: New Results

Mathematical models for microbial ecology

Differential equations models

Participants : Céline Casenave, Jérôme Harmand, Claude Lobry, Alain Rapaport, Alejandro Maximiliano Rojas.

Anaerobic digestion refers to the transformation of biodegradable material by micro-organisms in absence of oxygen (it can be found in waste-water treatments or industrial fermentation, and occurs naturally in soils). It receives an increasing consideration due to recent technological advances, but also because it is a source of renewable energy (bio-gas, fuel...). The anaerobic digestion is a complex set of bio-processes, for which there is a strong expectation of tractable models. We have proposed and studied new mathematical models that takes into account the following features:

• 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 models and established that depending on the kinetics of this additional reaction step, the qualitative behavior of the process may be significantly modified [26] .

• Microbial food chains are present in anaerobic digestion where the different reaction steps can be seen as such: the waste products of the organisms at one trophic level (i.e. one reaction step) are consumed by organisms at the next trophic level (i.e. the next reaction step). In [55] we study a model of a two-tiered microbial `food chain' with feedback inhibition, which was recently presented [63] as a reduced and simplified version of the anaerobic digestion model ADM1 of the International Water Association (IWA). It is known that in the absence of maintenance (or decay) the microbial `food chain' is stable. In [63] , using a purely numerical approach and ADM1 consensus parameter values, it was shown that the model remains stable when decay terms are added. In [55] we prove that introducing decay in the model preserves stability whatever its parameters values are and for a wide range of kinetics.

For the study of spatial heterogeneity in the models, we have carried on mathematical analyses of the properties of interconnected chemostats, in particular when growth rates present a substrate inhibition. In addition to the stabilizability properties discovered last year on “buffered” interconnections [33] , we have studied this year yielding performances at steady state for the stabilizing configurations and characterized the set of the most efficient ones. For such configurations, we have shown that under certain circumstances, a “by-pass” of the main tank could be the best solution to ensure a global stabilization.

We have also analyzed two kinds of models, suited to specific characteristics of the microbial activity in soils:

• In [30] , we have studied analytically and numerically a piece-wise linear model of carbon mineralization by two functional groups of micro-organisms in view of predicting the “priming effect” in soil ecosystems. The conclusion is that under a climate change, the augmentation of $C{0}^2$ will not affect to primary production and carbon storage when the plants are limited by nitrogen, but surprisingly a higher carbon input in soil should lead to a deplete of sequestered carbon and the increase of nitrogen release.

• In collaboration with Géosciences Rennes (Jean-Raynald de Dreuzy, Tristan Babey) and in the scope of the co-supervision of the PhD of Alejandro Rojas (also in the collaboration within the associated team with Chile), we investigate the equivalence between networks that represent interconnections of mobile/immobile zones in mass transfer models for soil ecosystems. For Structured INteracting Continua (SINC) models, that are described as the combination of a finite number of diffusion-dominated interconnected immobile zones exchanging with an advection-dominated mobile domain, we have proved an equivalence with Multi-Rate Mass Transfer (MRMT) and proposed a method for the identification of the equivalent MRMT model [14] . Moreover, we have shown the role of the controllability properties of a sub-system, in addition to the irreducibility of the network graph, for the input/output equivalence between several representations (work in preparation).

Stochastic and hybrid discrete-continuous dynamical models

Participants : Fabien Campillo, Bertrand Cloez, Coralie Fritsch.

Hybrid mass-structured chemostat models

Within the context of Coralie Fritsch thesis [12] , we adopt a new modeling approach where instead on focusing on one type of model we propose different models and their interconnections, on the numerical viewpoint as well as the analytical one. Namely we propose an hybrid model of the chemostat where the population of bacteria is individually-based, each individual being described by its mass, and the subtract concentration is represented as a classic differential equation. We proved the convergence of this model in high population size toward an integro-differential system [20] . We proposed specific numerical schemes for the two approaches (see 5.3 ) [27] .

Evolutionary invasion analysis and simulation for the chemostat

Still in the context of Coralie Fritsch thesis [12] and following her last year stay at the University of Helsinki in Otso Ovaskainen's Research Group, we consider an hybrid mass-structured mass-structured chemostat models with trait. The trait could for example be the factor of mass dissymmetry in the binary fission of a bacteria. In this context we proved an equivalence between invasion fitness for the hybrid-IBM model and the integro-differential system. We also numerically exhibit an evolutionarily singular strategy: with this given trait a monomorphic resident population cannot be invaded by a mutated population; the result is true for the two models.

PDE and stochastic models

In collaboration with M. Joannides and I. Larramendy-Valverde (I3M, University of Montpellier) we consider a stochastic growth model for which extinction eventually occurs almost surely. The associated complete Fokker–Planck equation describing the law of the process is established and studied. In dimension one, e.g. for the stochastic logistic model this equation combines a PDE and an ODE (paper under revision); in dimension two, e.g. for the stochastic chemostat model this equation combines a 2D PDE and a 1D PDE [22] . We then design a finite differences numerical scheme under a probabilistic viewpoint.

Other modeling approachs

Participants : Anne Bisson, Jérôme Harmand, Alain Rapaport.

A collaboration with the UMR Eco & Sols has led to the development of a (static) probabilistic model for inferring nature and number of interactions in communities assembly [29] . This model has brought new insights on a data set from reconstituted soil ecosystems. Because of the curse of dimensionality, we have begun this year to extend this approach to “assembling motifs” instead of considering all the possible assemblages (paper in preparation).

In [28] , ecological trade-offs between species are studied to explain species coexistence in ecological communities. In our model, plant species compete for sites where each site has a fixed stress condition. Species differ both in stress tolerance and competitive ability. We derive the deterministic discrete-time dynamical system for the species abundances. We prove the conditions under which plant species can coexist in an stable equilibrium. We compare our model with a recently proposed, continuous-time dynamical system for a tolerance-fecundity trade-off in plant communities, and we show that this model is a special case of the continuous-time version of our model.