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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)()(

1

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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