Section: New Results

Fields of application

Bioenergy

Modelling and optimization of microalgae production

Participants : Olivier Bernard, Antoine Sciandra, Frédéric Grognard, Philipp Hartmann, Rafael Munoz Tamayo, Andrei Akhmetzhanov, Nina Moelants, Hubert Bonnefond.

Experiments have been carried out to study the effects of nitrogen limitation on the lipid production in microalgae [74] and support model development. We have proposed a new model for lipid production by microalgae which describes the fate of the CO${}_2$ incorporated during photosynthesis [23] . This model describes the accumulation of neutral lipids (which can be turned into biofuel), carbohydrates and structural carbon. It has been calibrated and validated with experimental data. Experiments have also attempted to simultaneously represent the effect of an osmotic stress [55] . This model highlights and explains the phenomenon of hysteresis in lipid production which has been experimentally verified. It has been extended to account for light/dark cycles [36] .

On the other hand, a new dynamical model has been developed to describe microalgal growth in a photobioreactor under light and nitrogen limitations [13] . The strength of this model is that it takes into account the strong interactions between the biological phenomena (effects of light and nitrogen on growth, photoacclimation [34] , [48] ...) and the radiative transfer in the photobioreactor (light attenuation due to the microalgae).

Using these two approaches, we have developed a model which describes lipid production in a photobioreactor under light limitation. This model is used to predict lipid production in the perspective of large scale biofuel production. Simpler models have also been developed and have been used to provide optimization strategies: first, biomass production has been optimized in a constant light environment [79] , yielding results emphasizing the importance of the optical depth of the reactor. In a second work, we focused on the optimal operating conditions for the biomass productivity under day/night cycles using Pontryiagin's maximum principle (assuming a periodic working mode) [72] , [73] .

Another model has been developed to represent growth of microalgae colimited by nitrogen and phosphorus [69] . It has been shown, from qualitative analysis of the model that uptake of nitrogen and phosphorus are non symmetric.

Modelling the effect of light and temperature on microalgae

Participants : Olivier Bernard, Antoine Sciandra, Frédéric Grognard, Philipp Hartmann, Rafael Munoz Tamayo, Kerstin Ebert, Nina Moelants, Hubert Bonnefond.

The light distribution within a photobioreactor was estimated thanks to a multi photon Monte-Carlo simulation. From measurements of absorption and scattering properties, it was thus possible to extrapolate and validate the light distribution within a photobioreactor or a raceway.

The impact of the hydrodynamics on the light percept by a single cell was studied thanks to fluid dynamics simulations of raceway pond [48] . The light signals that a cell experiences at the Lagrangian scale, depending on the fluid velocity, were then estimated. A Droop-Han model was used to assess the impact of light fluctuation on photosynthesis [48] .

Finally, the effect of temperature on microalgae has been represented by adapting the CTMI model developped for bacteria [88] . The proposed model [59] , associated with a parameter identification procedure, was able to correctly represent the growth response to temperature for 12 different species [48] .

CO${}_2$ fixation by microalgae

Participants : Olivier Bernard, Antoine Sciandra, Philipp Hartmann, Nina Moelants.

We have run experiments to observe the response of a population of microalgal cells to various periodic light/dark or nitrate signals. The measurements show the synchronicity of the cells for some conditions. These experiments support the hypothesis that uptake of nitrogen stops during cell division [82] . On this basis, we have developed a structured model representing the development of microalgal cells through three main phases of their cell cycle: G1, G2 and M. The model is made of three interdependent Droop models [13] . 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 [81] . The effect of cell synchronization on lipid content were experimentally studied [18] and included into microalgae growth models [36] .

The effect of CO${}_2$ partial pressure increase on photosynthesis and calcification of the calcareous microalgae Emiliania huxleyi have been experimentally observed. It results that an increase of the coccolith size together with a decrease in the calcification rate has been observed.

Three models accounting for the possible coupling between photosynthesis and calcification were included in an ocean model, including settling and predation by grazers, and a bloom of coccolithophorids was simulated [67] , [68] . It was shown using Monte Carlo simulations that the uncertainty on the mechanisms driving calcification together with parametric uncertainties lead to uncertainties which are in the same range as the effect of an increase or the CO${}_2$ partial pressure.

Design of ecologically friendly plant production systems

Controlling plant pests

Participants : Frédéric Grognard, Ludovic Mailleret, Mickaël Teixeira-Alves.

The influence of an alternative prey in biological control programs

We have developed a model based on the classical Leslie-Gower predator prey model, that allows for the choice that a predator might have for its diet. In a biological control framework, this choice might be between a pest that we want to eradicate and another prey that could be fed to the predator in order to help the biological control efficiency or between the pest and an alternative prey that is present in the field and might keep the predator from acting as a natural enemy of the pest. We put the problem in a time-partitioning framework: the predator has to split its time between the two prey. We then compared two time-partitioning strategies: one where the predator always spends a fixed proportion of its time on each prey and one where the predator always chooses the prey that is instantaneously most profitable ( adaptive foraging). We then studied the effect of the presence of one prey on the other (indirect effect since it is mediated by the presence of the predator). We showed that, in the Leslie-Gower framework, one of the two prey always benefits from the presence of the other and that this effect is even stronger in the adaptive foraging framework, where the presence of the other prey is never detrimental to the one considered. That way, with very little assumptions, we showed the existence of apparent predation, commensalism and apparent mutualism, while most existing theoretical results tend to evidence apparent competition [51] . Such mechanisms may explain why generalist biological control agents are, in general, not as efficient as specialists in controlling crop pests.

The influence of plant dynamics on pest eradication

Pests-biocontrol agents models have been developed in order to build biological control strategies that can achieve pest eradication through augmentative biological control [85] . In the present work, we aim at introducing a plant compartment since its dynamics clearly have an influence on that of the pests and since the final objective of biological control is to maintain a sufficient plant yield. In a first step, we focused on plant-insect interactions and showed how the level and timing of the pest invasion could influence the final plant yield. Introducing pests control interventions and studying its timing, we showed how it eventually could have important effects on the growth pattern and the final biomass. As a reference, we consider the novel invasive pest Tuta absoluta on tomato plants [56] .

This work is done in collaboration with Yves Dumont (Cirad).

Controlling plant pathogens

Participants : Frédéric Grognard, Ludovic Mailleret.

Sustainable management of plant resistance

The introduction of plants strains that are resistant to one pathogen often leads to the appearance of virulent pathogenic strains that are capable of infecting the resistant plants. The resistance strain then becomes useless. It is therefore necessary to develop ways of introducing such resistance into crop production without jeopardizing its future efficiency. We did so by choosing the proportion of resistant plants that are mixed with the non-resistant ones. In this work, we studied a vector borne pathogen in a seasonal environment, with healthy crop being planted at the beginning of each season and cropped at its end, the pathogen surviving in the environment during the 'winter'. Two strategies have been proposed, one that aims at minimizing the cumulated damage over a 15 years horizon and one that aims at preventing the virulent strain outbreak. In the first case, both plant strains need to be mixed, but it results in the loss of the resistance at the end of the 15 year period; in the second case, the damage is higher and the maximal proportion of resistant plant is smaller, but the resistance is preserved [58] , [16] .

This work is done in collaboration with Frédéric Fabre (INRA Avignon).

Plant pathogen dynamics and cropping management practices

The coexistence of closely related plant parasites is widespread. Yet, understanding the ecological determinants of evolutionary divergence in plant parasites remains an issue. Niche differentiation through resource specialization has been widely researched, but it hardly explains the coexistence of parasites exploiting the same host plant. Agricultural systems are characterized by the cyclical presence and absence of the crop, due to cropping practices such as harvest and planting. We studied the influence that time-partitioning, i.e. the specialization of a parasite for the beginning or the end of crop presence, can have on co-existence. In modelling the epidemiology through a semi-discrete model we showed through an evolutionary invasion analysis that evolutionary divergence, and thus co-existence of different strains, of the parasite phenotype can occur [17] , [44] . Also, in a similar context, we underlined why modelling seasonal plant epidemiology did not necessarily lead to competitive exclusion; indeed, generating a compact model by rigorously isolating the slow dynamics from a large detailed model of plant epidemiology, we found out the possibility of coexistence [49] , [21] . Such a result contrasts with classical competitive exclusion principles found in compact models which rely on the arguable density independent nature of the pathogen infections occuring during the very beginning of the cropping seasons.

This work is done in collaboration with Frédéric Hamelin (Agrocampus Ouest).

Biological depollution

Coupling microalgae to anaerobic digestion

Participants : Olivier Bernard, Antoine Sciandra, Jean-Philippe Steyer, Frédéric Grognard, Philipp Hartmann.

The coupling between a microalgal pond and an anaerobic digester is a promising alternative for sustainable energy production and wastewater treatment by transforming carbon dioxide into methane using light energy. The ANR Symbiose project is aiming at evaluating the potential of this process [90] , [89] .

In a first stage, we developed models for anaerobic digestion of microalgae. Two approaches were used: First, a dynamic model has been developed trying to keep a low level of complexity so that it can be mathematically tractable for optimisation [37] , [32] , [22] . Considering three main reactions, this model fits adequately the experimental data of an anaerobic digester fed with Chlorella vulgaris (data from INRA LBE). On the other hand, we have tested the ability of ADM1 [91] (a reference model which considers 19 biochemical reactions) to represent the same dataset. This model, after modification of the hydrolysis step [24] , [38] , [41] has then been used to evaluate process performances (methane yield, productivity...) and stability though numerical simulations.

In a second stage, a model describing the coupling between anaerobic digestion process and microalgae culture (including the feeding of the algae with anaerobic digestion effluents) has been developed. The model is based on the three steps model for anaerobic digestion, and on the photoacclimation model for microalgae [13] . The model also includes the modelling of heterotrophs in the microalgae pond.

Life Cycle Assessment of microalgae production

Participants : Olivier Bernard, Jean-Philippe Steyer.

This work is the result of a collaboration with Laurent Lardon and Arnaud Helias of INRA-LBE through the co-supervision of Pierre Collet's PhD thesis.

An analysis of the potential environmental impacts of biodiesel production from microalgae has been carried out using the life cycle assessment (LCA) methodology [75] . This study has allowed to identify the obstacles and limitations which should receive specific research efforts to make this process environmentally sustainable.

This study has been updated and the effects of technological improvements (leading to higher productivities) have been compared to the source of electricity. It turns out that the overall environmental balance can much more easily be improved when renewable electricty is produced on the plant [47] , [46] . As a consequence, a new paradigm to transform solar energy (in the large) into transportation biofuel is proposed, including a simultaneous energy production stage.

A LCA has been carried out to assess the environmental impact of methane production by coupling microalgae and anaerobic digestion. The study highlights the limitation derived by the low biodegradability of the considered microalgae [15] which induces a large digester design and thus more energy to mix and heat it.

Models of ecosystems

Adaptive behaviour in seasonal consumer-resource dynamics

Participants : Frédéric Grognard, Ludovic Mailleret, Pierre Bernhard, Andrei Akhmetzhanov.

In this work we studied the evolution of a consumer-resource (or predator-prey) system with seasonal character of the dynamics. We specified two main parts of the process. First, we considered the system during one season with a fixed length: the prey lay eggs continuously and the predators lay eggs or hunt the preys (choose their behaviour) according to the solution of an optimal control problem [12] . Secondly, we studied the long-scale discrete dynamics over seasons. We investigated the qualitative behaviour of the dynamics with respect to the parameters of the problem and showed that, depending on the parameters of the model, extinction or co-existence of the predators and preys can be evidenced [12] .

We then examined how (resident) predators adopting this behaviour would fare when faced with a small population of selfish mutants that would be identical to the resident but would have the freedom to choose a different behaviour. We studied the resulting optimal control problem where the mutants maximize their own number of offspring using the knowledge of the resident's behaviour, and showed that, in most situations, mutants can take advantage of their low frequency and fare better than the residents. Over the course of a large number of seasons, the mutants replace the residents, only to find themselves applying the original resident behaviour [52] . We have then proposed a strategy for the predator in which it would prevent the invasion by the mutant instead of maximizing its number of offspring, which corresponds to the computation of evolutionarily stable life history strategies.

We have then considered that the resource itself could adapt its behaviour over time to limit the damage caused by the consumer, and maximize its own offspring. This problem requires the solution of a non zero-sum differential game, the consumer and the resource being the two players. We showed that the patterns of the strategies of the consumers and the resources are identical to the ones that can be obtained if the opposing player adopts a constant behaviour; the timing of the switchings varies however [42] .

Including phytoplankton photoadaptation into biogeochemical models

Participant : Olivier Bernard.

The complexity of the marine ecosystem models and the representation of biological processes, such as photoadaptation, remain open questions. We compared several marine ecosystem models with increasing complexity in the phytoplankton physiology representation in order to assess the consequences of photoadaptation model complexity in biogeochemical model predictions. Three models of increasing complexity were considered, and the models were calibrated to reproduce ocean data acquired at the Bermuda Atlantic Time-series Study (BATS) from in situ JGOFS data. It turns out that the more complex model are trickier to calibrate and that intermediate complexity models, with an adapted calibration procedure, have a better prediction capability [43] .

Growth models of zooplankton

Participants : Jean-Luc Gouzé, Jonathan Rault, Eric Benoît.

The model built to describe a zooplankton community is some variant of the McKendrick-Von Foerster Equation. The model includes cannibalism within zooplankton and predation on phytoplankton. Dynamic mass budget theory is used in order to describe individual behaviour and allows mass conservation. Also we have added phytoplankton dynamics, and we use environmental data as an input for the model. The aim is to compare simulations with data provided by the Laboratoire d'Océanographie de Villefranche. Since the model incorporates lots of parameters, which are not always known in the literature, we have to use optimization techniques to find them. Further, equilibria of such models and their local stability is studied in using strongly continuous semigroup approach [50]

We have also built a discrete size-structured model. Discrete models are less numerically demanding and so can be more easily incorporated into bigger models. Moreover the study of discrete models are often easier than that of continuous ones. We focus our study on the impact of cannibalism within zooplankton community and show that under some hypotheses, cannibalism can stabilize the equilibrium of the model [86] .