Section: New Results

Methodology

Mathematical study of models

Participants : Jean-Luc Gouzé, Olivier Bernard, Frédéric Grognard, Madalena Chaves, Pierre Bernhard, Pierre Masci, Andrei Akhmetzhanov.

Study of structured models of cell growth

We have developed a structured model representing the development of microalgal cells through three main phases of the cell cycle: G1, G2 and M. The model is made of three interdependent Droop models. The model was validated through extensive comparison with experimental results in both condition of periodic light forcing and nitrogen limitation. The model turns out to accurately reproduce the experimental observations [97] .

Mathematical study of models of anaerobic plants.

We have studied an unstable biological model of an anaerobic wastewater treatment plant [87] , [88] . This ecosystem can have two locally stable equilibria and an unstable one. The risk of destabilization associated to a control policy has first been evaluated on a static basis by estimating the size of the attraction basin of the working point. A dynamical risk has been defined based on the sequence of transitions, and zones of the state-space have been classified according to their dangerousness [24] . The proposed approach relies on the model structure; it does not depend on the parameter values and is thus very robust.

Mathematical study of models of competing species

When several species are in competition for a single substrate in a chemostat, and when the growth rates of the different species only depend on the substrate, it is known that the generic equilibrium state for a given dilution rate consists in the survival of only one of the species. The species that requires the smallest substrate concentration in order to have a growth rate equivalent to the dilution rate wins the competition; the most efficient species at this rate survives. This observation has been validated through laboratory experiments [86] . This competitive exclusion principle was already demonstrated in the Monod and the Droop model with n species. By a qualitative study of the microorganisms'internal quota, we demonstrated that it still holds when species modelled by Droop and Monod models compete [95] .

Moreover, it is in some cases theoretically possible to change the result of competition by changing the dilution rate. We are currently validating this theoretical result experimentally, in collaboration with the LOV (Laboratoire d'Oceanographie de Villefranche-sur-mer).

Bistability in biological systems

The following notion of bistability was proposed in  [80] : a system is bistable if its state space contains two disconnected invariant sets. In [18] this notion of bistability is used to analyze a small model of a caspase cascade at the core of apoptosis, or programed cell death. Conditions on the parameters are given that characterize different biological scenarios (e.g., healthy or malfunctioning cells). This is work in collaboration with T. Eissing and F. Allgöwer from the University of Stuttgart, Germany.

Towards an algorithmic reduction of models using orders of magnitude

We consider large biological models, described by an ordinary differential equation, with different scales with respect to time and space; moreover, the parameters have different order of magnitude. We use orders of magnitudes of these variables and parameters to obtain a partition of the state space in boxes (hyper-rectangles). From the fast system in each box, we derive rules of transition, and obtain a transition graph. This graph can be used for a qualitative simulation and validation of the system [51] .

Life history traits

In this work we study the evolution of a prey-predator system with seasonal character of the dynamics. We specify two main parts of the process. First, we consider the system during one season with a fixed length: the preys lay eggs continuously and the predators lay eggs or hunt the preys according to the solution of an optimal control problem [57] , [63] . Secondly, we study the long-scale discrete dynamics over seasons. We investigate the qualitative behavior of the dynamics with respect to the parameters of the problem and show that, depending on the parameters of the model, extinction or co-existence of the predators and preys can be evidenced [63] .

In a second work, we suppose that the evolution of the system during a season of fixed length is governed by optimal game dynamics with two players. On the one hand, the predator has the choice between foraging the food (eating preys) or reproducing for the next year (laying eggs at a rate proportional to its energy). On the other hand, the prey has a chance to hide from the predator but in this case the prey only has a mortality rate and its population can decrease faster than when foraged by the predator. The preys lay eggs at a constant rate whether they are hiding or eating. The aim of both is to maximize their population (the number of offsprings) for the next season. This non-zero sum dynamic game yields complex dynamics whose analysis gives rise to bang-bang control or bi-singular region depending on the initial condition [37] .

Model design, identification and validation

Participants : Olivier Bernard, Jean-Luc Gouzé, Madalena Chaves.

Model design and identification

One of the main families of biological systems that we have studied involves mass transfer between compartments, whether these compartments are microorganisms or substrates in a bioreactor, or species populations in an ecosystem. We have developed methods to estimate the models of such systems [68] . These systems can be represented by models having the general structure popularized by [65] , [76] , [77] , and based on an underlying reaction network:

We address two problems: the determination of the pseudo-stoichiometric matrix K and the modelling of the reaction rates r(, ) .

In order to identify K , a two-step procedure is presented. The first step is the identification of the minimum number of reactions to be taken into account to explain a set of data. If additional information on the process structure is available, we showed how to apply the second step: the estimation of the pseudo-stoichiometric matrix associated with the reaction network [67] . These methods lead to identifiability conditions on the pseudo-stoichiometric coefficients and provide a framework for their estimation. They have been improved in order to better filter noise using modulating functions [70] . This approach has been applied to various bioproduction processes, most recently on activated sludge processes [64] and anaerobic digestion [43] [106] .

These approaches have been combined with neural networks, in order to better constrain the network design and to ensure that it keeps a realistic behaviour even far from the training data set [25] .

Identifying operational interactions in genetic networks

For Boolean networks, we introduce the notion of operational interactions , corresponding to those interactions that are “active” in a certain region of the state space, and hence responsible for the dynamical behaviour in that region of the space. In [35] , we develop a method to identify operational interactions, in two steps. The first step consists in the decomposition of the asynchronous transition graph of the Boolean network into its strongly connected components (SCCs). The second step consists of choosing a desired region of this transition graph (for instance, a SCC), computing its reachable set and use an identification algorithm to reconstruct the interactions which are responsible for the dynamics in this set of states. This method was applied to a Boolean model of an apoptosis network [42] , and two distinct subsystems were found: one is responsible for generating oscillatory behaviour in the presence of an input, while the other subsystem is responsible for generating bistability in the absence of an input. To re-introduce the notion of time into this modelling approach, transition probabilities can be associated with the asynchronous transition graph, and the most frequent dynamics can be analysed [48] .

The feasible parameter space for biochemical networks

For biological models in general, the parameters can be identified up to a given set. The properties (namely, volume, geometry and topology) of the set of all biologically feasible parameters provide a measure of the robustness of the system. For example, a feasible parameter set which is composed of various disconnected components will, in general, be less robust than a simply connected set with the same volume. A method is proposed in [16] for writing a full description of the feasible parameter set, as a hierarchy of intervals. Application of this method to the segment polarity genes network (fruit fly) shows that the space of parameters is composed of five disconnected regions, which are connected by faces of lower dimension. The effect of random perturbations in the parameters is studied in [17] . This is work in collaboration with E. Sontag, A. Sengupta, and A. Dayarian from Rutgers University, USA.

Nonlinear observers

Participants : Jean-Luc Gouzé, Olivier Bernard.

Interval observers

We designed so-called bundles of observers [72] , [73] made of a set of interval observers. Each observer computes intervals in which the state lies, provided intervals for parameters and initial conditions (and more generally, all the uncertainties) are known. We then take the lower envelope of this set to improve the overall estimation.

We have extended the results of hybrid interval observers [84] , introducing an optimality criterion to compute an optimal gain, leading to the best interval estimates [98] ,[32] .

The combination of the observers has also been improved in the case where various types of interval observers are run in parallel [44] , and the approach has been applied to estimation of the microbial growth rate [50] .

In order to demonstrate the efficiency of the interval observer design, even with chaotic systems, a special application of the interval observer has been developed for Chua's chaotic system. The interval estimation of the state variables are performed considering parameters uncertainties of the system and biased output [101] . These techniques have been improved by introducing a linear, time-varying change of coordinates. For some class of systems, this method allows the design of interval observers when it was not possible in the original basis [46] , [30] . This approach was then extended to n -dimensional linear systems, leading to the design of interval observers in high dimensions [96] .

These interval observers have been extended to the case where only discrete time measurements are available [23] , [22] , and applied to experimental data of phytoplankton growth.

Finally, our approach was also extended to deal dynamical systems with time delay [47] .

Nonlinear control

Participants : Jean-Luc Gouzé, Olivier Bernard, Frédéric Grognard, Pierre Masci, Sapna Nundloll, Andrei Akhmetzhanov.

Global stabilization of partially known positive systems

We have constructed strict Lyapunov functions for general nonlinear systems satisfying Matrosov type conditions [31] . We illustrate the practical interest of our design using two globally asymptotically stable biological models.

Control of competition in the chemostat

We had designed a closed loop control procedure for microorganisms in the chemostat [93] . The objective is to select species with interesting characteristics in chosen environmental conditions. In particular, by controlling the dilution rate and the input substrate concentration, it is possible to select a species which maximises a criterion. This selection method was adapted to a model of anaerobic digestion during its start-up phase [45] . In such a model, we proposed controls which enable to select, among several hundred of species, the ones with maximum growth rate in the reactor steady state operating mode. A first experimental validation of this control strategy was done at the INRA-LBE Laboratory.

Mathematical study of impulsive biological control models

The global stability of the interconnection of a continuous prey-predator ODE model and periodic impulses has been studied. This work was motivated by the biological control of pests in a continuously grown greenhouse. The prey-predator dynamics are continuous and are augmented by discrete components representing the periodic release of predators. Our analysis consists in establishing the existence and global stability of a pest-free periodic solution of the system driven by the repeated predator releases. The latter is achieved using Floquet Theory and is explicitly formulated as a minimal bound on the number of predators to release per unit of time (the minimal rate of predator release).

The influence of various parameters on the stability conditions has been investigated. We first considered the general predator-prey model with density-dependent functional response, and showed that the minimal rate is independent of the release period, but that this period should be as small as possible in order to be able to optimally counter unexpected pest invasions [29] .

Then, the effects of two types of intrapredatory interferences have been analyzed: Beddington-DeAngelis and squabbling interferences. The first one represents interference for the access to the prey and the second represents squabbling between the predators through the addition of a quadratic term to the predators mortality (which is otherwise linear). In both cases, we show that the minimal rate that ensures stability is increasing with the period of release [33] . This result has also been shown for a generalised form of the Beddington-DeAngelis interference [104] .

This work has been done in collaboration with L. Mailleret (URIH, INRA Sophia-Antipolis).

Evolutionary games

Participants : Pierre Bernhard, Frédéric Grognard, Andrei Akhmetzhanov.

As an addition to our investigations in population dynamics and optimal control, we have embraced evolutionary games dynamics per se as one of our domain of investigation, adding Evolutionary Stable Strategies (ESS) and non-invadable strategies as equilibrium paradigms.

After characterizing ESS's as Wardrop equilibria (the so-called Nash property or “first ESS condition” of ESS's), we have provided a simple test for a matrix Wardrop equilibrium to actually be an ESS. In the same vein, we have also obtained a sufficient condition for the nonlinear theory. Because our main focus is on dynamic systems, we have developped this test in the infinite dimensional case. [53] , [36] , [52]

It turns out that the theory of evolutionary games, born in the investigation of biological populations, has found many applications in other domains. This has lead to some publications in networking theory, as joint work with our colleagues of the Maestro research project [12] .

As an outlet of earlier research in other applications of dynamical games, we obtained in a joint work with Guy Barles, of the university François Rabelais of Tours, and Naïma El Farouq, of the university Blaise Pascal of Clermont-Ferrand, rather technical results concerning the uniqueness of viscosity solutions of some quasi-variational inequalities associated with minimax impulse control problems [19] .