Section: New Results
Modelling and Applied Mathematics
Stochastic Dynamic Equations of Growth
The stochastic version of GreenLab (GL2) was developed by M.Z. Kang [8] . A stochastic formal language adapted to the botanical concepts underlying the GreenLab organogenesis model was introduced in [60] . It is based on stochastic L-systems (parallel rewriting grammars) and on multi-type branching processes: stochastic processes control bud productions and at each growth cycle, each new growth unit is the result of a random variable.
From 2008, these preliminray results were extensively revisited by a new PhD student, C. Loi. A new probabilistic framework for stochastic L-systems was introduced, and linked to multitype branching processes. The results were applied to the Greenlab organogenesis model in order to characterize the probability distributions and moments of the numbers of organs in plant structure. Plant organogenesis can be seen as the combination of two subprocesses driving the bud population dynamics, branching and differentiation. The generating function of the whole process is shown to be the composition of the generating function associated to the two subprocesses. The modelling of stochastic branching is classical but, to model differentiation, we introduced a new framework based on multivariate phase-type random vectors. By differentiating the generating function of the whole process, we were able to write the recursive relationships for the expectation and the variance of the number of metamers in the plant. These results were published [12] .
A new generalization of these results was also developed, in collaboration with Jean Françon (Univ. Strasbourg), based on the symbolic method of Flajolet [72] . Generating functions of any kinf od substructures (or patterns) can be derived for any stochastic 0L-Systems [37] .
Parameter identification of trees
The theoretical advances concerning the GL3 model (model of interaction between growth and development) [16] and the development of new tools allowed confronting the model to new plant species, with more complex architectures than the previous versions. Based on the PhD of V. Letort, there is an important work on the validation of the GreenLab model for trees. Important results were published in [10] and new works are carried on in China on pine trees (in collaboration with the Chinese Academy of Forestry), in Guyana on Cecropia (in collaboration with P. Heuret, INRA-AMAP) and in Africa on Accacia (in collaboration with UMR BIOEMCO). The last results were published in PMA09 conference [33] , [20] .
Sensitivity and Uncertainty Analysis
In order to make the model more robust and suitable for applications, it is necessary to master its uncertainty. For this purpose, mathematical studies on model structure and parameterization are carried out. It concerns Qiongli Wu's PhD and the post-doctoral work of Samis Trevezas.
It first involves a proper parameter sensitivity analysis of the model, in order to assess the importance of each parameter and their potential source of uncertainty [46] . Sobol's method is used. The complexity of the functional structural model and of the number of parameters involved led us to develop new powerful computing techniques.
Regarding uncertainty on parameter estimation, the current estimation process used in the Digiplante software, does not provide a precise quantification of the parameter uncertainty. The difficulty lies in the estimation of the covariance matrix of the observation error. A new statistical framework is currently developed by S. Trevezas and should hopefully lead to interesting results.
Modeling Inter-Individual Variability
Modeling heterogeneity in field crops is a key issue for a better characterization of field production. First works were achieved on sugar beet. Several sources of individual variability within plant populations are identified: namely, initial condition (seed biomass, emergence delay), genetic variability (including phyllochron) and environment (including spacing and competition). A mathematical framework is introduced to integrate the different sources of variability in plant growth models. It is based on the classical method of Taylor Series Expansion, which allows the propagation of uncertainty in the dynamic system of growth and the computation of the approximate means and standard deviations of the model outputs. The method was applied to the GreenLab model of plant growth and more specifically to sugar beet [27] . It opens perspectives in order to assess the different sources of variability in plant populations and estimate their parameters from experimental data.
A continuous version of the GreenLab model
To model plant-environment interactions, the synchronization of the plant growth model with biophysical models has revealed limitations of the discrete formulation of the GreenLab model at the growth cycle step. Therefore, a continuous approach to model functional-structural plant growth was developed by V. Le Chevalier and Z. Li, based on the discrete GreenLab model. The continuous dynamics is driven by a system of differential equations with respect to calendar time, with a continuous mechanism of senescence introducing delay terms. A numerical scheme for solving the system is studied. It was tested on the model of sugar beet growth, to compare different approximation methods including the classical discrete model. With a higher precision, the simulation based on the continuous approach reveals significant differences with the discrete model. Moreover, an approximation of the continuous model is derived with a daily time step, which makes it suitable for agronomy applications [34] .
Plant - environment interactions
So far, experiments for GreenLab calibration were conducted in unstressed conditions. In order to develop optimal control for agriculture, it is important to develop a good model of plant - soil interactions, especially for water and nitrogen.
A new PhD (Z. Li) has started in collaboration with CEMAGREF Montpellier (J.C. mailhol), with the objective of coupling the GreenLab model with a model of soil water budget. The first result is the combination of the GreenLab model of plant production to the PILOTE model [66] . A proper study of the model is in process and experiments on wheat and maize will help estimate the model parameters.
Moreover, the post-doc of Jessica Bertheloot helped develop a model integrating fully both Carbon and Nitrogen metabolisms [25] . The parameterization of the model is difficult and more work is necessary to validate the model.
Another study is to represent 3D landscapes from the outputs of the classical Process Based Models (PBM) used to compute yield of crops in agronomy and from the outputs of Empirical Forestry Models (EFM) used to assess wood quality and quantity in forestry. The data provided by these models are uncomplete, but it is possible to add a relevant botanical knowledge, coming from similar plants previously studied. For instance PBMs provide only dry biomass, LAI, harvest index, and EFMs provides only lengths, diameters and branch numbers on the trunk. Adding GreenLab knowledge about plant functioning and plant architecture allows to reconstruct 3D representations faithful to plants and thus to visualize a field, which is a first step towards "functional landscapes". PBMs and EFMs are numerous so it could be a chance for the GreenLab model to be widely used in Agronomy. This is the PhD subject of Feng Lu cosupervised by Digiplante and the Chinese University of Agriculture. The PBMs and EFMs used come from Cemagref (Pilote), Inra (Stics), Wageningen (Tomsim), CAF (Simtree).
Functionnal Landscape: coupling and synchronising models
A new formalism framework allowing model composability was defined in 2009. The new conceptual approach is close to P_DEVS approach (developped by Ziegler). Our approach allows to build simulation plateforms coupling various models sharing data in a synchronous way. The approach is based on Models components and Caches components to access Data.Data update is performed by a specific Model, the Manager at specific time steps, computed for model internal time requests. Models are responsible for their own evolution, and specify to the manager their time of Data update. The approach is modular, allows hierarchical nesting and parralel implementation.
This approach was implemented and two showcase developped: a specialized water competition example (set of pumps), and a specific GreenLab modelplant model. The model was revised, redesigned using a continuous formalism in order to fullfil approach requirement.
These results were submitted and recently accepted at PMA09 conference
This topic is confirmed beeing a growing concern in the team, involved in the coordination of several regional projects (RTRA-INRIA) dedicated to animate landscape integrative modelling and their plateform at regional level, in aim to build a scienti?c plateform with its partners (CIRAD-GREEN,INRA LISAH, etc). In the frame of this network, those partners lauch a new event dedicated to Integrative modelling and simulation plateform at landscape and ecosystems levels: LANDMOD2010, to be held on february 2010 in Montpellier, with the support of RTRA (Computation plants and Eco-systems Agropolis Fundation-INRIA Call)
Floral biology and fertility models
An important component of the yield that was not taken in account until now in GreenLab model is the production of seeds. Cirad working on tropical trees (Palm tree, Cocoa tree, Coffee tree) has developed previously such model to simulate the seed production from the ovules distribution in the flowers, the distribution of pollen seeds, and the abortion laws of seeds and pods. Few parameters calibrated from the measurements on flowers and fruits control quite well the seeds distribution in the pods. This allows separating the environment effect (weak pollination) from genetic effects (ovule fertility). Such issue occurs also on temperate crops such as rapeseed. A PhD Wang Xiujuan coming from CAU and cosupervised by Digiplante, Inra EGC, and CAU, works to improve the Cirad model in its mathematical shape and applied it successfully to the rapeseed in Grignon. The model is generic and can be used in many crops.
Yield optimization and plant architecture
The harvest index is the ratio between the yield and the total biomass produced by the field. Depending on the type of crop, the yield concerns seeds (corn), root (beetroot), fruits (tomato), wood (tree), and even leaves (salad, tea plants ...) that are source organs! The yield depends on the source and sink balance through plant growth and development. The strategy to follow for yield optimization is complex and must be adapted to plant architecture. Yield also depends on the interaction between plants and their environment. A first study was conducted in Digiplante, in cooperation with EPI Idopt (Le Dimet), to control irrigation. Another important problem is the interactions between plant growth and pest (fungus, insect). The thermal time drives both insect and plant developments and the leaf surface is the spot of their interaction. Insects need to eat leaves and plant need leaf area. Such issue has been previously studied by Cirad. In Digiplante, the Phd of Qi Rui (joint ECP - Liama) studies mathematically the three-antagonists model obtained, and its control.
Towards functional sinks
Formulations of the sources and sinks functions in GreenLab are constantly improved, with a more mechanistic approach. In 2006, the source function based on the hydraulic plant architecture was replaced by the light interception at the canopy level (Beer-Lambert Law) and adapted to individual plant using an optimized projection surface computed by inverse method. It allows the passage from Individual plant to plant population. Symmetrically studies on sink function are carried out between Digiplante, Liama and CAU. Sinks are relative, time duration of organ expansion is complicated to assess directly. Although the empirical sink functions used in Greenlab are very efficient and can be assessed by inverse method they are only descriptive, they depend on the thermal time and have no physiological meaning. A new formalism based on ecophysiological considerations (number and expansion of cells inside an organ) is proposed. Fitting the source and sink balance on crops with this new formulation has given promising results. Advantages are that the sinks can be absolute and it is no more required to assess the expansion time of organs. Once this study is completed in 2010, it could be possible to swap the sink function in 2010 as we did for the source function. The assumptions underlying the GreenLab model would thus be entirely functional.
