GreenLab Course
Development
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 microstate 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 w_{φ} 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.
Bibliography
Feller, W. 1968. An introduction to probability theory and its applications, vol. 1, 3rd edn. Wiley, New York