### From biological concepts to equations

This step corresponds to the writing of the model equations based on the agronomic / biological knowledge. It is done in close collaboration with the partner institutions. Besides the GreenLab model, other families of models are also considered (STICS INRA-Avignon, NEMA INRA-Grignon, SUNFLO-CORNFLO-SOYFLO Syngenta, etc.), with three speciﬁc priorities:

- better integration of the environment (speciﬁcally water and Nitrogen). This is necessary for model applications;
- modelling plant populations from the individual-based model, by studying competition between plants and the inter-individual variability (More: modelling the growth of plant populations, by Yuting Chen);
- modelling the genetic determinism of parameters. In a perfect model, we would write dX/dt = F (X(t), P, E(t)) where X(t) are the state variables (masses of plant components), E(t) represent the environmental factors (radiation, temperature, soil water content, etc.) and P are variety-speciﬁc parameters from genetic origins: P = H(G) where G represents plant genetics. Several methods are possible, including metabolic networks or methods derived from quantitative genetics. (More: Simulation of QTL Detection for Model Parameters and Application to Potential Yield Optimization by V. Letort et al., 2008.)

### Mathematical formalism

- Formal grammars and combinatorics: the former formalism in GreenLab (dual-scale automaton) has been linked to the theory of formal grammars (L-System). In the stochastic case, the link with branching processes was also studied, which allowed the computation of moments and generating functions for the numbers of elements in plants. In collaboration with J. Françon (Univ. Strasbourg), symbolic methods derived from the combinatorics approach of Flajolet allowed the computation of the generating functions of the occurrences of patterns in plants. Such results lead to the deﬁnition of new methods to estimate the parameters of stochastic models of plant organogenesis.
- Continuous models of plant growth, time-delay systems: traditional models of plant architectural growth adopt a discrete formalism (based on the discrete steps in the automaton or grammar theory deﬁning architectural growth cycles). It proves limiting when considering plant-environment interactions. Therefore, a continuous version of the GreenLab model has been derived, at least for the functional parts. It raises interesting numerical issues (discretization schemes and optimal control for time-delay systems). Current studies are carried out to write the structural part in a continuous way.

### Mathematical and statistical analysis of model structures

When model equations are written, a fundamental step is their mathematical analysis: limit and stability analysis, identiﬁability, sensitivity and uncertainty analysis. A few important results have been produced by Digiplante on this aspects (conditions for the generation of rhythms [Mathieu et al., 2008], designing speciﬁc methodology for the global sensitivity analysis of functional-structural plant models [Wu et al., 2011]). One of the key points to explore concerns the study of complex systems: plant integrative modelling (especially functional-structural plant modelling) implies diﬀerent scales of biophysical processes, some of them are particularly well-known, but rarely the interactions between these processes when considering more global phenomena at plant or ﬁeld scale. Global sensitivity analysis oﬀers very interesting perspectives to study such integrative models (as well as some linked methodologies: model reduction / meta-modelling).