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Mathematical Models of Plant Growth for Applications in Agriculture, Forestry and Ecology
Paul-Henry Cournède
Applied Maths and Systems, Ecole Centrale Paris
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« 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:
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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 visualizationPlant Growth Simulation
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Outline
1. Modelling plant growth and development
2. Model identification from experimental data
3. Exemples of applications
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Outline
1. Modelling plant growth and development
2. Model identification from experimental data
3. Exemples of applications
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A model combining two approaches
Biomass
water Nutriments
CO2
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
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Sim HPDe Reffye
1980
AmapJaeger 1988
Amapsim IBarczi1993
AmaphydroBlaise1998
Organogenesis + GeometryBotany Functional growth
Simulation
AMAP GREENLAB
GL1 &2Kang2003
Mathematical Models
GL3Cournède
2004
Interaction
Photosynthesis x organogenesis
prototype
Dynamic model
Africa France China
A familly of Functional-Structural Models of plant growthinitiated by P. de Reffye
France
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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.
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Flowchart for plant growth and plant development
seed
photosynthesis
H2O
Pool of biomass
leaves
roots
Organogenesis + organs
expansion
transpiration
CHO
fruits
branches
GreenLab Plant
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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 !
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n
ija
a
n
tni jD
jQijpiN
Spe
kSpnEnQ
a)(
)1()1()(
.exp1)()(
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A generic equation to describe sources-sinks dynamics along plant growth
No stressW. stress
Energetic efficiency
EnvironmentLight
Interception
variation puits limbe
0
0.05
0.1
0.15
0.2
1 2 3 4 5 6 7 8 9 10 11 12
age organe
forc
e de
pu
its
Development Sinks Function
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Plant architectural plasticity.
Same set of model parameters Different light conditions
Age 15 15 15
L-System production rules for the development depend on plant trophic state = function of the ratio of available biomass to demand (Q / D)
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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
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Outline
1. Modelling plant growth and development
2. Model identification from experimental data
3. Exemples of applications
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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
nnn UPXFX ,,1
nXnU
P PXGY N ,
P ArgMin modèlealexpériment PYYP
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Exemple of parameter estimation on Maize
variationpuits limbe
0
0.05
0.1
0.15
0.2
1 2 3 4 5 6 7 8 9 10 11 12
organ age
sin
k st
ren
gh
t
Water use efficiency
Light interception
plant Environment
Sources Fonctions
Sinks Fonctions
Architecture and climate Data
Fitting architectureEstimation results
endogenous Parameters
Parameters of climatic functions
production biomasse S/SP
0
5
10
15
20
1 3 5 7 9 11 13 15
S/Sp
bio
ma
ss
pro
du
cti
on
source-sink functions
Internodes
Blades
cob
timeRoot mass as a function of time
Blades
Internodes
Cob
18Guo et al., 2006
Simulation and visualization of Maize growth
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Arabidopsis (Christophe et al., 2008) Young pine (Guo et al., 2008)
Genericity of the growth model
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Genericity of the growth model
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Outline
1. Modelling plant growth and development
2. Model identification from experimental data
3. Exemples of applications
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For (a), sunflower height is 105.9, fruit weight is 623.41, total weight is 1586.72
For (b), sunflower height is 98.3, fruit weight is 656.11, total weight is 1509.78.
For (c), sunflower height is 112.1, fruit weight is 881.52, total weight is 1998.97
For (a), sunflower height is 105.9, fruit weight is 623.41, total weight is 1586.72
For (b), sunflower height is 98.3, fruit weight is 656.11, total weight is 1509.78.
For (c), sunflower height is 112.1, fruit weight is 881.52, total weight is 1998.97
Optimal control of irrigation for Sunflower
Wu et al., 2005
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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 10000 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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