Section: Scientific Foundations

Model Design and Analysis

From biological concepts to equations

This step corresponds to the writing of the model equations based on the agronomic / biological knowledge. It will be done in close collaboration with the partner institutions. We will continue working on the GreenLab model, and will also consider other family of models (STICS INRA-Avignon, NEMA INRA-Grignon, SUNFLO-CORNFLO-SOYFLO Syngenta ...). 3 specific points are now our priorities:

• better integration of the environment (specifically water and Nitrogen). This is still poorly taken into account in the GreenLab model, and is quite restrictive for model applications.

• modeling plant populations from the individual-based model, by studying competition between plants and the inter-individual variability,

• modeling the genetic determinism of parameters. In a perfect model, we would write $\frac{dX}{dt}=F\left(X\left(t\right),P,E\left(t\right)\right)$ where $X\left(t\right)$ are the state variables (masses of plant components), $E\left(t\right)$ represent the environmental factors (radiation, temperature, soil water content ...) and $P$ are variety-specific parameters from genetic origins. What would be very interesting is to write $P=H\left(G\right)$ where $G$ represents plant genetics. Several methods are possible, including metabolic networks, but in a first step we will consider methods derived from quantitative genetics.

Mathematical formalism

• Formal grammars and combinatorics: in the last two years, in the PhD thesis of Cedric Loi [Loi et al. 08,10,11] some very precious results have been obtained, linking the former formalism in GreenLab (dual-scale automaton) 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 also allowed the computation of the generating functions of the occurences of patterns in plants. Such results led to the definition of new methods to estimate the parameters of stochastic models of plant organogenesis. This promising approach still needs to be explored: extend the cases in which the distribution of patterns can be derived, comprehensive study of the estimation methods.

• Continuous models of plant growth, time-delay systems: traditional models of plant architectural growth (like GreenLab) 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 [Li et al., 2009], at least for the functional parts. It raises interesting numerical issues (discretization schemes and optimal control for time-delay systems). Moreover, the structural part is not yet written in a continuous way. Current studies are carried out.

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 modeling (especially functional-structural plant modeling) 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-modeling). A collaboration with one of the major group in the world (Saltelli, Tarantola in the Joint Research Centre (JRC) of the European Commission, Ispra Italy) is starting about the results of Qiongli Wu's PhD.