1 Mathematical Models of Plant Growth for Applications in Agriculture, Forestry and Ecology Paul-Henry Cournède Applied Maths and Systems, Ecole Centrale Paris

2 « DigiPlante » Modelling, Simulation and Visualization of Plant Growth a joint team between associated to agronomy research institutes with very strong links with (Institute of Automation, China Academy of Sciences) (China Agricultural University) All these institutions collaborating to develop a joint model:

3 Plant Growth Modelling: a multidisciplinary subject Bioclimatology Soil Sciences Botany Plant Architecture Agronomy: Ecophysiology Applied Mathematics Stochastic Processes Dynamical Systems Optimization, Control Computer sciences Simulation visualization Plant Growth Simulation

4 Outline 1.Modelling plant growth and development 2.Model identification from experimental data 3.Exemples of applications

5 Outline 1.Modelling plant growth and development 2.Model identification from experimental data 3.Exemples of applications

6 Environment of the plant Genetic model Ecophysiological model Endogenous parameters for: development sources, sinks Plant architecture Plant growth Mathematical growth model An integrative model of plant growth For applications, a « mesoscale » approach: individual-based,at the level of organs Genome of the plant QTL

7 A model combining two approaches Biomass water Nutriments CO 2 Organogenesis + empirical Geometry = Plant Architecture. Plant development coming from meristem trajectory (organogenesis) Biomass acquisition (Photosynthesis, root nutriment uptake) + biomass partitioning (organ expansion) Compartment level Morphological models => simulation of 3D development Process-based models =>yield prediction as a function of environmental conditions Functional-structural models

8 Sim HP De Reffye 1980 Amap Jaeger 1988 Amapsim I Barczi 1993 Amaphydro Blaise 1998 Organogenesis + Geometry Botany Functional growth Simulation AMAPGREENLAB GL1 &2 Kang 2003 Mathematical Models GL3 Cournède 2004 Interaction Photosynthesis x organogenesis prototype Dynamic model AfricaFranceChina A familly of Functional-Structural Models of plant growth initiated by P. de Reffye France

9 Guo and Ma (2006) A « Growth Cycle » based on plant organogenesis Development of new architectural units: continuous : agronomic plants or tropical trees rhythmic : trees in temperate regions. Organogenesis Cycle = Growth Cycle, time discretization for the model Phytomer = botanical elementary unit, spatial discretization step For continuous growth, the number of phytomers depends linearly on the sum of daily temperatures.

10 Flowchart for plant growth and plant development seed photosynthesis H2OH2O Pool of biomass leaves roots Organogenesis + organs expansion transpiration CHO fruits branches GreenLab Plant

11 A formal grammar for plant development (L-system) Alphabet = {metamers, buds} (according to their physiological ages = morphogenetic characteristics) Production Rules : at each growth cycle, each bud in the structure gives a new architectural growth unit. Factorization of the growth grammar factorization of the plant into « substructures » Computation time proportional to plant chronological age and not to the number of organs !

12 A generic equation to describe sources-sinks dynamics along plant growth No stress W. stress Energetic efficiency Environment Light Interception Development Sinks Function

13 Plant architectural plasticity.  Same set of model parameters  Different light conditions Age L-System production rules for the development depend on plant trophic state = function of the ratio of available biomass to demand (Q / D)

14 Inter-tree competition at stand level Isolated tree Central tree Tree on the edge Trees in competition for light Method of intersections of projected areas

15 Outline 1.Modelling plant growth and development 2.Model identification from experimental data 3.Exemples of applications

16 Plant = Dynamic System State variables = vector of biomass production Input Variables = environnement (light, temperature, soil water content) Parameters Observations Trace back organogenesis dynamics from static data collected on plant architecture (numbers of organs produced, modelling of bud functioning) Trace back source-sink dynamics (biomass production and allocation) from static data on organ masses. Estimate : Estimation of model parameters from experimental data

17 Exemple of parameter estimation on Maize Water use efficiency Light interception plant Environment Sources Fonctions Sinks Fonctions Architecture and climate Data Fitting architecture Estimation results endogenous Parameters Parameters of climatic functions source-sink functions Internodes Blades cob time Root mass as a function of time Blades Internodes Cob

18 Guo et al., 2006 Simulation and visualization of Maize growth

19 Arabidopsis (Christophe et al., 2008)Young pine (Guo et al., 2008) Genericity of the growth model

20 Genericity of the growth model

21 Outline 1.Modelling plant growth and development 2.Model identification from experimental data 3.Exemples of applications

22 For (a), sunflower height is 105.9, fruit weight is , total weight is For (b), sunflower height is 98.3, fruit weight is , total weight is For (c), sunflower height is 112.1, fruit weight is , total weight is For (a), sunflower height is 105.9, fruit weight is , total weight is For (b), sunflower height is 98.3, fruit weight is , total weight is For (c), sunflower height is 112.1, fruit weight is , total weight is Optimal control of irrigation for Sunflower Wu et al., 2005

23 Biomechanics and computation of constraints in trees Mechanical constraints Tree growth in a constrained environment Fourcaud et al., 2003

Functional landscapes in ecology Simulation and visualization of ecological process dynamics at landscape scale (from 100 to km²). E.g.: water ressource allocation. prototype : C++ and OpenGL (glut), multi-platform Visualization of dynamics and image post-treatments Le Chevalier et al., 2007

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