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Section: Application Domains

Behaviour of GreenLab model

A mathematical model needs several steps to fulfil the common requirement:

  1. The equations of the model must be a relevant translation of the reality.

  2. The behaviour of the model must be studied.

  3. The calibration of the model has to be undertaken on real data, and the model could be modified if necessary.

  4. The model is used for various applications using optimization and control.

The GreenLab model has been improved gradually, through the successive Phd subjects. Starting from the GreenLab-Liama Team, the research has been extended to the new Digiplante team born in Inria at the end of 2004.

The deterministic case GL1

Participant : H.P. Yan (GreenLab- Liama associated team to Digiplante).

First the deterministic case named GL1 has been studied. The plant development is monitored by the DSA as in Figure 1 .

Figure 3. Flowchart for the GreenLab model.
IMG/flowchart

Im10 $\mfenced o={ c=. \mtable{...}$(8)

Im11 $\mfenced o={ c=. \mtable{...}$(9)

At the step n of growth the number of organs Xn + 1 to create is computed thank to a function F deduced from the DSA shape (8 ). The Biomass production is computed from another equation (for example the Beer Law (5 )), that is represented by equation (9 ). Equation (9 ) is obtained by replacing the leaf surface by its explicit formulation coming from the source and sink formalism. The recurrence shape of the system is obvious. Sets U and V contains respectively the rules of the DSA and the sinks and sources parameters, the system starting from the seed. At this step a simple retroaction occurs between the biomass production and the plant development at the level of the organs geometry.

According to the functioning durations of the different organ types (bud, leaf, internode, fruit, layer, root), and the environmental conditions it is possible to compute the plant growth and to determine the system stability thank to sinks and sources parameters. It is thus possible to build pure virtual plants, whose organs expansions are exactly controlled during plant growth. Such a plant is described Figure 2 . Branches duration is t2=10 cycles. All organs (leaves, internodes, fruits) have ta=5 cycles for expansion (with constant sinks pa, pe, pf) and the leaves have 5 cycles of functioning.

Figure 4. GL1 case: Behaviour of a virtual plant during the growth process.
IMG/GL1-1IMG/GL1-2

The generic recurrence equation (8 ) is available for all the plants built with the GL1 system. Parameters A and B, are a combination of sources and sink parameters coming from equations (3 ) and (4 ).

Im12 ${Q_n=E·\#8721 _{i=1}^t_a\mfrac \mstyle {N_{n-i+1}^a\#8721 _{j=1}^i\mfrac \mstyle {p_j^aQ_{n-(i-j)-1}}D_{n-(-j)}}\mstyle {A+B\#8721 _{i=1}^i\mfrac \mstyle {p_j^aQ_{n-(i-j)-1}}D_{n-(-j)}}.}$(10)

The system will stabilize its biomass production Ql/ cycle according to the solution of equation (9 )

Im13 ${1=\#8721 _{i=1}^{ta}\mfrac {E·i(t2+1)}{{A·ta(t2+1)(pa+pe+pf)+iB}Q_l}.}$(11)

The sizes of organs depend explicitly of the environment E and of the sources and sinks parameters.

The stochastic case GL2

Participant : MZ Kang (GreenLab- Liama associated team to Digiplante).

The dual scale automaton can be easily adapted to the stochastic case named GL2 (Liama, Kang MZ Phd). Here we still consider that there are no interaction between the growth and the development schedule of the plant that is now stochastic. In the equation7 the U set contains also a set of probabilities.

The bud functioning is controlled thank to growth probability bk , reliability ck , and branching threshold ak , that monitor the macrostates creation and also the law of repetitions of the microstates inside macrostates. The means and the variances of both organs and biomass productions have been explicitly computed from the stochastic DSA parameters, using covariance formulations and differential statistic properties. This avoids performing heavy MonteCarlo simulations to get the shapes of the distributions.

Figure 5. Stochastic plants simulated by GL2 Case.
IMG/stochastic

Even substructure method is used here to shorten the simulation duration. For each chronological age and physiological age a set of limited repetitions is built, and then the accuracy of the simulation depends only of the number of repetitions. The time duration to build a stochastic tree is the same than for the deterministic case, once the substructure collection has been built for the first tree simulation.

The convergence toward Normal laws of the automaton production makes often the use of the computed means and variances sufficient to predict the organs and the biomass distributions.

Figure 6. Comparison between theoretical distributions and montecarlo simulations (250 trials) for stochastic trees generated by model GL2.
IMG/4-1IMG/4-2

The interactions between plant development and plant growth: GL3 Case

Participant : A. Mathieu.

Thank to the results obtained by the associated team in Liama for levels one and two of GreenLab model, the Digiplante team was ready to contemplate the integration of the feedbacks between the Growth and the Development at a third level named GL3. This was the main subject of Amelie Mathieu's Phd from ECP. Locations of the feedback relie in a plant mainly on the buds functioning behaviour. Under different external conditions a same bud can produce more or less metamers and set in place various numbers of axillary branches. As a result of this variation the same tree can be 15 cm or 15 m at 15th years old according to shadow or sunny conditions. The matter of such a plasticity was supposed coming from the ratio Q/D of the biomass supply Q coming from the photosynthesis and the plant demand D, that is the scalar product between the organs and their sinks. The main Botanical improvement from GL3 is considering the bud as an organ with a sink, mean while in GL1 and GL2 the demand relies only on the plant organs (leaves, internodes, ...).

The more Q/D is big the more the Growth Unit born from the bud will be developed. A simple linear relationship is assumed between the functioning thresholds and Q/D.

First only the deterministic case is considered and three main thresholds are identified:

This introduces a full retroaction between Development and Growth equations. Equations (8 ) and (9 ) becomes:

Im17 $\mfenced o={ c=. \mtable{...}$(12)

Im11 $\mfenced o={ c=. \mtable{...}$(13)

Figure 7. Schedule for the functioning during a growth cycle, for the buds.
IMG/schedule

The behaviour of the system made of equation (12 ) and (13 ) was successfully studied by Amelie Mathieu. The main results are to determine the conditions of the growth stabilisation according to the parameters, to retrieve the plant plasticity at every stages of growth, to control the conditions of the phenomena apparition into the plant architecture and to generate a periodical functioning that is often observed during growth of trees.

Figure 8. Plasticity of the GreenLab model, to simulate different plant architectures upon various climate conditions.
IMG/plasticity
Figure 9. Example of retroaction between the size of the growth unit in number of metamers and the biomass production in the case of a monoculm plant (Corner model).
IMG/example

In the simple case of a monocaulus plant, the retroaction between growth and development relies on a variable number of metamers/GU. Under explicit numerical conditions the system will stabilize or not its growth.

Figure 10. Rythmic growth for fruiting and branching in alternation depending of the retroaction between plant production and plant development.
IMG/rythmic

In the complex case the effect of the retroaction between plant production and plant development will generate cyclic phenomena at several levels. Biomass production, fruiting and branching alternation, number of internodes/GU etc ...Very simple rules linking thresholds for development with a linear function Q/D depending, are sufficient to retrieve classical phenomena observed in growth of plants.


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