-> 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 *_________________
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-> 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 *____
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-> Applied Mathematics
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Map *____
-> Probabilities
Section contents *____________
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-> Useful Laws
Bernoulli Trials *___________
Binomial Law *__________
Geometric Law *____________
Negative Binomial Law *_________________
-> Dynamic systems
Section contents *_____________
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Negative Exponential *________________
Systems functions *______________
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Parameter estimation *________________
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-> GreenLab courses
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-> Overview
Presentation & Objectives *____________________
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Biomass production *________________
Modelling - FSPM *______________
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-> Principles
Presentation & Objectives *____________________
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-> About modelling
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Organs: tree components *___________________
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Inside the growth cycle *___________________
Implementations *______________
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-> Development
Presentation & Objectives *____________________
Map *____
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-> Stochastic modelling
Principles *_______
-> Development
Growth Rhythm *____________
Damped growth *____________
Viability *______
Rhythmic axis *___________
Branching *________
Stochastic automaton *_________________
-> Organogenesis equations
Principles *_______
Organ cohorts *___________
Organ numbering *_____________
Substructure factorization *____________________
Stochastic case *____________
-> Structure construction
Construction modes *_______________
Construction basis *______________
Axis of development *________________
Stochastic reconstruction *___________________
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 *__________
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A dynamic system view *__________________
Equation terms *____________
Full Model *________
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-> Applications
Presentation & Objectives *____________________
Map *____
-> Measurements
Agronomic traits *_____________
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-> Fitting structure
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-> Development
Simple development *_______________
Damped growth *____________
Rhythmic growth *_____________
Rhythmic growth samples *___________________
Mortality *_______
Branching *________
-> Crown analysis
Analysis principles *______________
Equations *________
Example / Exercise *_______________
-> Case study
Plant Architecture *______________
Development simulation *__________________
Introducing Biomass *_______________
Biomass partitioning *_______________
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-> Tools (software)
Presentation & Objectives *_____________________
Map *____
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Simulation *_________
Online tools *__________

GreenLab Course


Stochastic development modelling

While some agronomic plants show a deterministic architecture (maize, wheat, etc.) which can be fully grasped by an automaton, most plants show significant variability in their structures, even considered at plant level, such as on two similar axes derived from the same phytomer: they may show a different number of phytomers.

Theoretical framework

    Since plant structure is established from discrete elements (phytomers), such modelling relies on statistical distribution modelling.

    The proposed process to be modelled concerns:
      - axis development: this means the probability of a new phytomer appearing (i.e. of repeating the current micro-state or of moving to the next one in terms of the automaton)
      - axis mortality: this means the probability of a metamer dying
      - axillary bud branching: this means the probability of a new axis developing

    The renewal theory is the branch of probability theory that generalizes Poisson processes for arbitrary holding times. In probability theory, a Poisson process is a stochastic process that counts the number of events and the time points at which those events occur in a given time interval. Such stochastic models are derived from the classic Renewal theory.
    According to Feller (1968), a renewal process is a stochastic model for events that occur randomly in time.
    The basic mathematical assumption is that the periods between successive arrivals, called renewal time, are independent and identically distributed. Given the mean and variance of renewal time, the number of events during time T (whose distribution is known as the counting law), is asymptotically normally distributed. In our discrete case, because of the convergence of the counting law towards a normal law, we can approximate this distribution by a binomial law, and the time T can then be replaced by a virtual discrete time N.

Axis development

    Modelling axis development: periodic and random rests in phytomer occurrences.

      In quantitative terms, we aim here to define the law characterizing the appearance of a new phytomer.

      Such a process can be modelled by a Bernoulli process, characterized by a probability p of success (occurrence of a new phytomer) and thus a probability q = 1- p of failure (a rest).

      Applying this process at successive time steps thus defines a list of Bernoulli trials, leading to a list of success (refered to here by the number "1") and failure (refered to by "0") sequences, where the number of successes ("1") stands for the number of phytomers.
      With this encoding, the number of phytomers is simply given by the sum of the numbers refering to the status of each step.

      In practice, the time step is chosen as an interval of thermal time, usually defined from the smallest time between appearance of two phytomers on the plant's main axis (i.e. on an axis of physiological age 1). This time step is defined as the plant growth cycle.

      In more detail, modelling the development of a given axis means characterizing two process:

        - The rhythm ratio wφ of the axis (of physiological age φ) defined as the ratio of potential new phytomers occurrence on this axis of physiological age φ to the potential occurrence on an axis of physiological age 1, standing as a reference.

        - Then, on the axis, definition of the probability of a success, or more generally, definition of the number of phytomers n generated in a given period N.

      In other words, the rhythm ratio involves periodic rests in phytomer production allowing different axis development rates in the plant structure, as opposed to random rests (failures) in phytomer occurrence.


Feller, W. 1968. An introduction to probability theory and its applications, vol. 1, 3rd edn. Wiley, New York