Section: New Results

Mathematical models for microbial ecology

Differential equations models

Participants : Jérôme Harmand, Claude Lobry, Alain Rapaport, Yessmine Daoud, Sonia Hassam, Zeyneb Khedim, 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:

• 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 [54] we study a model of a two-tiered microbial ‘food chain’ with feedback inhibition, which was recently presented as a reduced and simplified version of the anaerobic digestion model ADM1 of the International Water Association (IWA) (cf. [61] ). It is known that in the absence of maintenance (or decay) the microbial ‘food chain’ is stable. In [61] , using a purely numerical approach and ADM1 consensus parameter values, it was shown that the model remains stable when decay terms are added. In [54] we prove that introducing decay in the model preserves stability whatever its parameters values are and for a wide range of kinetics.

• In the thesis by Sonia Hassam [13] , we have proposed a new procedure to easily and systematically obtain a simple model useful for control purposes of any process for which an ADM1 is available. The simplified model has two major characteristics : its states keep their physical meaning and it remains nonlinear. The technique is based on the state association technique proposed in [26] .

• Zeyneb Khedim (University of Tlemcen, Algeria) has began her PhD in 2014. She is working on the modeling and control of anaerobic digestors. In particular, she works on the reduction of models using the state association approach proposed by Sonia Hassam but for substrates highly loaded in nitrogen such as algae. She has published this year a survey with Sonia Hassam [36] .

• Yessmine Daoud (ENIT-LAMSIN, Tunis, Tunisia) continues her work on the analysis of a model of the literature to optimize anaerobic processes [35] . She is preparing a journal paper which should be submitted during 2016.

Formerly, the team has studied chemostat models where the bacterial compartment is split into “planktonic” and “attached” bacteria (such as in flocculation or biofilms formation), under the hypothesis that attachment and detachment are fast phenomena. Under certain mixing conditions, this last condition is no longer satisfied. We have studied on the non-reduced model the competition between a species that presents growth inhibition in planktonic form with a species that does not attach. This consideration leads to multiple positive equilibria but surprisingly it can also conduct to limit cycles [53] (paper under revision for Applied Math. Model.).

Spatial heterogeneity is often observed in non perfectly mixed bioprocesses or in populations in natural environments. The representation o spatial heterogeneity in population models with patches or interconnected models, rather than p.d.e., is one of the specialties of the team, that allows us to characterize non intuitive effects of spatialization :

• The very basic Rosenzweig-MacArthur model is subject to the "atto-fox" problem [2] when considered for homogeneous populations. Is it still true in case of heterogeneous populations? The idea is: the resource population being not small at the same time in different places is it possible that, thanks to dispersal, it will not disappear? One possible idealization of heterogeneous populations is to use reaction-diffusion equations. We do not take this direction for two reasons

• (i) Due to the presence of a limit cycle in the homogeneous system mathematics of such reaction diffusion are difficult.

• (ii) Idealization through reaction-diffusion is not the best one; patch-systems (or lattice differential equations in mathematical terms) are better in many cases.

Our ultimate objective is to provide mathematical results for systems with a large number of patches but, as a first step, in the paper [27] we consider two patches. It is proved that for some migration rates, stable periodic solutions avoiding "atto-fox" exist.

• The standard model for the dynamics of a fragmented density-dependent population is built from several local logistic models coupled by migrations.

First introduced in the 1970s and used in innumerable articles, this standard model applied to a two-patch situation has never been completely analyzed. The motivation for studying this problem came out from discussions at the Bernoulli semester organized in 2014 and 2015 by the team at the EPFL (see the 2014 activity report and Section 9.3.3.1 ). It addresses very fundamental issues in theoretical ecology. In the paper [15] written in collaboration with R. Arditi (U. Fribourg) an T. Sari (IRSTEA Montpellier), we complete this analysis and we delineate the conditions under which fragmentation associated to dispersal is either beneficial or detrimental to total population abundance. Therefore, this is a contribution to the SLOSS question. Importantly, we also show that, depending on the underlying mechanism, there is no unique way to generalize the logistic model to a patchy situation. In many cases, the standard model is not the correct generalization. We analyze several alternative models and compare their predictions. Finally, we emphasize the shortcomings of the logistic model when written in the r-K parameterization and we explain why Verhulst's original polynomial expression is to be preferred.

• We have carried on our former work on the role of particular interconnections patterns on the global stability of chemostat model with inhibition. While we focused formerly on the conditions for which a spatial structure ensures the global stability when the chemostat model is bi-stable in homogeneous environment, we have shown that at the opposite a spatial structure can make unstable the dynamics of the chemostat model with inhibition when it is stable in a homogeneous environment [30] .

• 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 have carried on the complete equivalence between several models used in Geosciences to characterize soil fractures : MINC (Multiple INteracting Continua), MRMT (Multi-Rate Mass Transfer) and SINC (Structured INteracting Continua). We have shown that the irreducibility of the network graph is not sufficient to obtain equivalence : a controllability assumption has also to be fulfilled [42] . Moreover, this kind of models has been used to fit experimental data of reconstituted soils at Inra Grignon and has shown the role of convection in the acquisition of pesticides by micro-organisms [46] (paper in preparation). This work will be continued in the framework of the new ANR project Soil$\mu 3D$ (see Section 9.2.1 ).

In resources/consumers models, heterogeneity can be also due to time varying inputs of resources (e.g. light in micro-algae populations). While, most of the literature studies periodic inputs, we have begun investigations of more general time varying inputs in chemostat like models, having in mind to characterize “pull-back attractors” (rather than forward attractors) [43] .

Stochastic and hybrid discrete-continuous dynamical models

Participants : Bertrand Cloez, Claude Lobry.

Approximation of quasi-stationary distributions

The study of the long-time behavior of a stochastic process is one of the main questions of interest for modeling. In a standard Markov setting, this leads to the study of the convergence towards the invariant distribution. However, in many applications such as population dynamics for instance, the stochastic dynamics is killed in a finite (random) time so that the standard asymptotic regime is trivial. In this case, it can be interesting to focus on the behavior of the process conditionally to its non-extinction before a given time $t$ Under appropriate assumptions, one can exhibit a convergence of this conditional distribution towards a law called Quasi-Stationary Distribution. Properties of this law is then fundamental. In [21] , we study an algorithm to approximate this distribution and we provide proof of convergence as well as precise rates for convergence. This one is based on a reinforced random walk.

Lotka Volterra in fluctuating environment

In the paper [49] , we consider two dimensional Lotka-Volterra systems in a fluctuating environment. Relying on recent results on stochastic persistence and piecewise deterministic Markov processes, we show that random switching between two environments that are both favorable to the same species can lead to the extinction of this species or coexistence of the two competing species. This work has been accepted in Journal of applied probabilities, provided major revisions. We submitted a new version with the new title: Lotka Volterra with randomly fluctuating environments or "how switching between beneficial environments can make survival harder".