COURSES

-> About this Resource
Scope *______
Map *____

-> Preliminary Courses
Contents & Objectives *__________________
Map *____
-> Botany
Contents & Objectives *__________________
Map *____
-> Axis Typology Patterns
Typology basis *___________
Pictograms *_________
Sexuality & development *___________________
Growth *______
Branching rhythms *______________
Branching delays *_____________
Branching positional *________________
Branching arrangement *__________________
Axis orientation *_____________
Architectural models *________________
-> Architectural Unit
About Arc. Models *______________
Models limitations *______________
Architectural Units *______________
Reiteration *_________
Sequence of development *___________________
Morphogenetic gradients *___________________
Physiological age *_____________
-> An Example
Wild Cherry (young) *_______________
Wild Cherry (adult) *______________
Wild Cherry (mature) *________________
Quiz *____
Case study Quiz *_____________
Supplementary resources *____________________

-> Eco-Physiology
Contents & Objectives *__________________
Map *____
-> Growth Factors
Factors affecting Growth *___________________
Endogenous Processes *_________________
Environmental Factors *_________________
Thermal Time *___________
-> Light interaction
P.A.R. *_____
Light absorption *_____________
Photosynthesis *___________
Respiration *_________
Maintenance respiration *__________________
L.U.E. Model *__________
Density effect *___________
Density effect on crop *__________________
-> Biomass
Biomass Pool *__________
Biomass Partitioning *_______________
Crop models *__________
A Crop model example *__________________
Quiz *____
Supplementary resources *___________________

-> Applied Mathematics
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Map *____
-> Probabilities
Section contents *____________
Discrete Random Variable *___________________
Expected value, Variance *___________________
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-> Useful Laws
Bernoulli Trials *___________
Binomial Law *__________
Geometric Law *____________
Negative Binomial Law *_________________
-> Dynamic systems
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Useful functions *____________
Beta density *__________
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Negative Exponential *________________
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Discrete dynamic systems *___________________
Parameter Identification *__________________
Parameter estimation *________________
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-> GreenLab courses
GreenLab presentation *__________________
-> Overview
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Biomass production *________________
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-> Principles
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-> About modelling
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-> Development
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Map *____
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-> Stochastic modelling
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-> Organogenesis equations
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-> Structure construction
Construction modes *_______________
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Implicit construction *________________
Explicit construction *________________
3D construction *____________
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-> Production-Expansion
Presentation & Objectives *____________________
Map *____
-> EcoPhysiology reminders
Relevant concepts *______________
Temperature *__________
Light interception *______________
Photosynthesis *___________
Biomass common pool *_________________
Density *______
-> Principals
Growth cycle *__________
Refining PbMs *___________
Organ cohorts *___________
GreenLab vs PbM & FSPM *___________________
-> GreenLab's equations
Summary *_______
Production equation *_______________
Plant demand *__________
Organ dimensions *______________
A dynamic system view *__________________
Equation terms *____________
Full Model *________
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-> Applications
Presentation & Objectives *____________________
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-> Measurements
Agronomic traits *_____________
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-> Fitting structure
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-> Development
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Damped growth *____________
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Rhythmic growth samples *___________________
Mortality *_______
Branching *________
-> Crown analysis
Analysis principles *______________
Equations *________
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-> Case study
Plant Architecture *______________
Development simulation *__________________
Introducing Biomass *_______________
Biomass partitioning *_______________
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-> Tools (software)
Presentation & Objectives *_____________________
Map *____
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Online tools *__________

Preliminary Course

Applied Mathematics

Usual Parametric functions


Negative exponential function
      In many plant growth models, (e.g. Pilote, Ceres, STICS, GreenLab), the biomass production at time t, Q(t), is expressed as a parametric function of the leaf area index, the LAI (the one-sided green leaf area per unit ground area, expressed in m2.m-2, so dimensionless). μ stands for a positive proportional coefficient.
      Q(t) = μ . ( 1 - e -k LAI(t))

      This expression assumes that biomass production is determined by the amount of radiation that is intercepted by the plant/field and can be derived from the Beer-Lambert law, an empirical relationship between the absorption of a monochromatic light wave in an isotropic and homogeneous medium to its concentration.
      The parameter k drives the curve variation. μ is a real positive value standing for a biological efficiency factor.

      Note that more than 95% of the incoming radiation is intercepted when LAI ≥ -ln(0.05)/k, i.e. when LAI ≥ ≈ 3 if k = 1.
      Above that value, increasing the LAI has nearly no effect on the production (phenomenon of 'LAI saturation').

      Negative Exponential
      Biomass production as a function of LAI (graph V. Letort- Le Chevalier, ECOLE CENTRALE PARIS)
      y stands for the production Q(t)

Exercise
    Let the function f be defined as f(x) = μ . (1 - e-k.x) with μ = 3.

    What is the value of k given that:

      the derivative of f at x = 0 is equal to 2.25 ?
           k = 1.00      k = 0.95      k = 0.75      k = 2.25      k = 6.75      k = 1.33
          detailled answer here

      f(x) reaches 97% of its supremum (least upper bound) at x = 6 ?
           k = 0      0.35 ≤ k ≤ 0.45      0.45 ≤ k ≤ 0.55
           k = 1      0.55 ≤ k ≤ 0.65      0.65 ≤ k ≤ 0.75
          detailled answer here

Bibliography
Pilote:

Mailhol, J. and Gonzalez, J. 1993. Furrow Irrigation Model for Real-Time Applications on Cracking Soils. Journal of Irrigation and Drainage Engineering, 1993, 119:5, pp. 768-783

CERES:

Jones, C.A., and J.R. Kiniry. 1986. CERES-Maize : a simulation model of maize growth and development. Edited by C.A. Jones and J.R. Kiniry. College Station : Texas A&M University Press, 1986. 194 p.

Stics:

Brisson, N.,Gary, C., Justes, E., Roche, R., Mary, B., Ripoche, D., Zimmer, D., Sierra, J., Bertuzzi, P., Burger, P., Bussière, F. Cabidoche, Y.M., Cellier, P., Debaeke, P., Gaudillère, J.P., Hénault, C., Maraux, F., Seguin, B., Sinoquet, H. 2003. An overview of the crop model stics, European Journal of Agronomy, Volume 18, Issues 3-4, January 2003, Pages 309-332, ISSN 1161-0301

GreenLab:

Kang, M.G., Cournède, P.H., Mathieu, A., Letort, V., & Qi, R. 2009. A Functional-Structural Plant Model: Theories and its Applications in Agronomy. Cao, W. and White, J. and Wang, E. Crop Modeling and Decision Support, Springer, pp. 148-160, 2009, 978-3-642-01131-3