Section: New Results

Applications

Modelling and control of Anaerobic Digestion processes

Participants : Amine Charfi, Radhouene Fekih-Salem, Jérôme Harmand, Boumediene Benyahia, Tewfik Sari.

We consider the AM2 or AMOCO model developed in [46] and extend both the model in itself and its analysis to the following cases:

• Depending on the AM2 model parameters, the steady states were analytically characterized and their stability were analyzed. Following this study, it was shown that the overloading tolerance, a parameter proposed in [49] to on-line monitoring anaerobic processes, may be not adapted under certain operating conditions and even lead to bad operating decisions.

• Within the framework of the PhD theses of Amine Charfi and Boumediene Benyahia, we have included the fouling dynamics of membranes into the AM2 and we have analyzed the resulting model (called the AM2b).

• We actually work towards two directions: (i) we are extending these results in including into the AM2 an additional process, i. e. the hydrolysis step in order to study bioprocesses treating solid waste (the resulting model being called the AM3); (ii) we try to find links between complex models such as the ADM1 model and simple models such as the AM2b or the AM3.

Modelling and control of cascade biosystems to mimic batch wine making processes

Participants : Jérôme Harmand, Alain Rapaport, José Fernandez.

An experimental setup of four tanks connected in series has been designed by the research unit SPO (Montpellier) for studying four physiological stages of yeast as steady state. The manipulated variables are the flow rates $Q_i$ of each tank with the constraint $Q_i\ge Q_{i-1}\ge 0$, and the objective is to reach simultaneously four set-points in the four tanks. We are studying two kinds of control strategies:

• a linearizing feedback law that drives exponentially the dynamics to the target. This is not the fastest strategy but is has good robustness properties. Nevertheless, the inputs constraint imposes to use saturation functions that provide satisfactory convergence in simulations but that is hard to prove mathematically.

• a minimal time feedback. Due to lack of local controllability imposed by the constraint on the inputs, the optimal synthesis is not smooth with the presence of “barriers” (see Section 6.1.10 ).

Those feedback laws will be implemented and tested on real pilot plant at SPO lab in the scope of the European project CAFE described in Section 7.2 .

Modelling and simulating terrestrial plant ecological dynamics

Participants : Fabien Campillo, Ihab Haidar.

This study is part of the ANR Syscomm MODECOL that is done in collaboration particularly with the University of Rennes I, the University of La Rochelle and INRIA. This is the second year of the three years program. We propose a stochastic individual-based model for clonal plant dynamics in continuous time and space, focusing on the effects of the network structure of the plants on the reproductive strategy of ramets. This model is coupled with an explicit advection-diffusion dynamics for resources. We develop a partially exact simulation scheme of the model ; the capacity of the model to reproduce specific features of clonal plants, such as their efficiency to forage resources over the field, is numerically studied. Next, we propose a large population approximation of the model for phalanx-type populations, taking the form of an advection-diffusion PDE for population densities, where the influence of the local graph structure of the plant takes the form of a nonlinear dependence in the gradient of resources. This year we improved the simulation code that was proposed last year and made three communications in international conferences [26] [36] [35] .

Modelling and inferring agricultural dynamics

Participants : Fabien Campillo, Angelo Raherinirina.

The International Laboratory LIRMA supports this work that is done in collaboration with the University of Fianarantsoa in Madagascar and with Dominique Hervé (IRD, Fianarantsoa, Madagascar). The aim is to study the dynamics of agricultural plots on the edge of primary forest. In [38] we propose a Markov chain model where the transition matrix is estimated both by maximum likelihood and Bayesian approaches. We also test if the Markov chain model is adapted to this problem. In an ongoing work we develop semi-Markov models for an extended data set.

Modelling and simulating microbial ecosystems in soils

Participants : Ihab Haidar, Jérôme Harmand, Alain Rapaport.

The team studies simple representations of the spatial inhomogeneity for bioprocesses, in terms of networks of interconnected compartments. Each compartment is modelled as a perfectly mixed bioreactor. Simulation of such networks is performed with the software developed by the VITELBIO project (see 5.1 ). Comparisons of simple structures (cascade of bioreactors) with numerical simulators based on REV (Representative Elementary Volumes) have been performed. The objective is to understand the role of the topology of the network on the biological functions of the overall system. For the moment simple configurations with two to four nodes and one single species have been investigated.

In addition, we study in simulation how software implementations of transport reactions models such as MIN3P, can mimic such simple configurations. Transport reactions models are often used in the community of soil functioning modelling by simulation. Here we compare their numerical computation with the exact solutions that can be approximated with a a good accuracy using classical o.d.e. solvers. We found that one has to be careful when the trajectories are close to a non-hyperbolic equilibrium [45] , [18] .

Numerical optimisation in non perfectly mixed tanks

Participants : Jérôme Harmand, Alain Rapaport, Antoine Rousseau.

The optimisation of series of bioreactors in terms of minimizing the total residence time have been already investigated in the literature. In such models, the space has a one-dimensional representation. There is comparatively much less work on 2D or 3D space models. We consider different shapes of reactors of the same volume, and simulate with the help of multi-physics numerical software the coupling of the hydrodynamics laws in 3D or 2D (under cylindrical symmetry), solved numerically by the Navier-Stokes equations, with the system of differential equations of biotic/abiotic concentrations [34] .

We are studying the influence of the shape of the domain and possible obstacles on the output concentration at steady state.

With A. Rousseau (EPI MOISE, INRIA Rhône-Alpes) we study with simulation of 2D Navier-Stokes equations the benefits of having several pumping points and how to control the pumps speed to minimize the treatment duration for the bioremediation of natural reservoirs presented in Section 6.1.11 . We evaluate the feedback strategies that are derived to be optimal for simple models of o.d.e. in the more realistic framework of hydrodynamics simulation.

Individual-based models for the bacterial degradation of the cellulose

Participants : Fabien Campillo, Chloé Deygout, Marc Joannides.

We propose an individual-based model for the degradation of one cellulose bead (dozens of micrometers in diameter) by cellulolytic bacteria. Our aim is to determine the macroscopic degradation behavior. The initial stages of the degradation process may involve a very limited number of bacteria that cannot be properly modelled by classical models based on deterministic equations. In the present work we only consider a two-dimensional model for the degradation of a cellulose disc.

Non-linear filtering for the chemostat

Participants : Boumediene Benyahia, Fabien Campillo, Jérôme Harmand.

We propose numerical non-linear filtering approaches for the identification of non-observed components of dynamical systems in the context of the chemostat. This recently started study relies on the work of the project-team in the stochastic modelling of the chemostat. In a preliminary work we consider the bootstrap particle filter.