Conditions for the Generation of Rhythms in a Discrete Dynamic System. Case of a Functional Structural Plant Growth Model Am´elie Mathieu Ecole Centrale de Paris, Laboratoire MAS Chatenay Malabry, France [email protected] Daniel Barth´el´emy CIRAD, AMAP Boulevard de la Lironde, 34398 Montpellier Cedex 5, France [email protected]

Abstract To model plasticity of plants in their environment, a new version of the functional-structural model GreenLab has been developed where a single variable representative of the plant sink and source balance controls different events in plant architecture, among which the numbers of fruits and of branches. For some sets of well-chosen parameters, the dynamic evolution of this variable induces variations in fruit and branch production. Simulations revealed emerging properties of this model like the automatic generation of rhythms. It corresponds to cyclic patterns observed in real plants. The model is based on the theory of discrete dynamic system. The writing of the equations helps us explain periodic patterns, control the behaviour of the system and understand some hidden mechanisms of tree growth. Conditions for the appearance of such rhythms are determined as functions of the plant endogenous parameters and the environmental conditions. Experiments are foreseen to calibrate the model.

1. Introduction The architecture of a plant is the expression of an equilibrium between endogenous processes and exogenous constraints exerted by the environment [4]. Several authors assessed that a better introduction of effects of growth on architecture would increase reliability of plant growth models ([15],[16]) and improve the modelling of interactions between a plant and its environment. Some models have been

Paul-Henry Courn`ede Ecole Centrale de Paris, Laboratoire MAS Chatenay Malabry, France [email protected] Philippe de Reffye CIRAD, AMAP et INRIA, Digiplante B.P. 105, 78153 Le Chesnay Cedex, France Philippe.de [email protected]

developed to reproduce plant plasticity, implementing either the spatial interactions ([22],[5]) or the dependence of architecture on functional mechanisms ([6],[23][1]) but they were mostly limited to the branching system. The work we present here is based on the GreenLab model ([24],[27]) that derived of AMAP models [3]. To take into account the interactions between organogenesis and photosynthesis, the plant architecture is built with rules that are controlled by a key variable representative of the source and sink balance at different levels of the plant [21]. In given conditions, the dynamic trend of this variable induces rhythms over successive growth cycles in fructification without being forced by an external user. There are numerous applications for such a model. In [25], the author is interested in the growth of temperate fruit trees like apple trees and notices that ”heavy fruiting, which reduces shoot growth, will also reduce the formation of new flowers and results in poor fruit production in the following year”. In perennial plants, these phenomena of ”mast seedings”, i.e. the succession of ”high and low years in reproduction whereas environmental conditions are not bimodal”, are well known ([13],[14]). In cucumber plants, the proportional matter distribution between vegetative parts and fruits showed a cyclic pattern [18] and a simulation model has been implemented to reproduce it [19]. The same authors lead studies on the sweet pepper and found a ”linear relationship between abortion and source-sink ratio” ([20],[12]). In our model, similar mechanisms of balance between sources and sinks combined with an influence of the photosynthesis on organogenesis provoke the apparition of rhythms in branch formation. Such phenomena were ob-

served in trees as for example the periodicity of whorl formation in Araucaria araucana[17]. The authors suggested that it was linked to the whole-plant carbon balance. It seems that the plant needs to accumulate enough reserves to be able to initiate the construction of a new branch whorl. The GreenLab model can be written under the form of a discrete dynamic system. The state vector describes the plant at a given moment. Emerging properties of the model like the generation of rhythms can be explained by the study of the equations. Among others, conditions for their apparition can be determined. After a presentation of the equations of the GreenLab model, we give illustrations of the generation of rhythms in both branch and fruit formation. All the simulations showed in this paper were obtained with the Digiplante software developed in Ecole Centrale de Paris [7].

2. Materials and Methods 2.1. Rhythms in a dynamic system A dynamic system consists of an initial state and a dynamic law which describes the rate of change of the variables of the system. We discretize here the growth of a plant and hence we will focus on discrete systems, that is to say that we express the state of the system at t + 1 as a function of the state at t. We consider a causal system in which the output may be expressed in terms of the past inputs and outputs of the system. We write a model of the form [9]: xt+1 = a(xt , ut , p)

(1)

xt = [xt1 , ..., xtm ] in the state vector of the system, ut is the input vector and p is the parameter vector of the system. We say that rhythms appear in the system if ∃p ∈ IN, ∀t, xt+p = xt

(2)

The smallest integer value satisfying equation 2 is the period of oscillations. Rhythms may appear after a transient stage, where the behaviour of the system tends to become periodic. In our dynamic system, we also consider rhythms even if only some coordinates of the vector are periodic, which can be written: ∃p ∈ IN, i ∈ {1, ..., m}, ∀t, xit+p = xti The system is said to be pseudo-rhythmic if there seems to be rhythm and we do not exactly have the relation ∀t, xt+p = xt but rather ∃p, ∀t, xt+p ≈ xt or ∃p, ∀t, xt+p+∆p = xt .

2.2. Equations of a model of plant growth A plant can be represented by a state vector X and its growth can be summed up by the equation X t+1 = F (X t ).

In the framework of the GreenLab model [24], [27], the plant growth is discretized with a time step, called as the Growth Cycle, that depends on the plant development. Its duration can vary from a few days for herbaceous plants to one year for temperate trees. At the beginning of Growth Cycle t, a set of organs called a growth unit appear on a growing axis according to organogenetic rules. Each axis is characterized by a physiological age p ∈ P = {1, ..., Pm } that corresponds to a degree of differentiation [2]. There are at most Pm different kind of axes in a tree. In the plant at cycle t, there are vpt (n) growth units of physiological age p and chronological age n, that is the number of growth cycle since an organ was built. The growth unit bears bpq (t − n + 1) lateral axes of physiological age q and chronological age n−1 and cp (n−t+1) fruits of chronological age n. It is composed of upq (t − n + 1), p ≤ q ≤ Pm metamers that are differentiated by the physiological age q of their axillary bud. This metamer bears mbpq potential buds and mopq organs of type o where o is replaced by a for leaves, e for internodes, f for fruits. Those are botanical data of the plant. All the leaves of the metamer have the same surface noted stpq (n). The state vector we chose to represent the plant is given in equation 3, see [21] for more details:     t spq (1) p∈P,p≤q≤P m   v t (1)   p∈P (3) Xt =  p   [upq (t)]p∈P,p≤q≤Pm  [cp (t)]p∈P In the following, we suppose to simplify that ∀p, q, t, upq (t) = u and we detail the equations that allow us to write X t+1 = F (X t ). The number of organs that will be created at cycle t + 1 is determined at the end of cycle t. In the model defined in [21], the coefficients upq (t), bpq (t) and cp (t) depends on the ratio of available biomass to plant demand that is also seen as a variable of particular interest and called the rate of growth demand satisfaction in [8].

2.3. Surface of leaves If we do not consider polycyclism, the architecture does not change during one growth cycle, and the amount of biomass produced by the leaves is computed with a photosynthetic model that can be chosen among several ones without changing the basic concepts of GreenLab. We suppose that a leaf is active during ta cycles and produces an amount of matter that is a function q of its surface. E is the average potential biomass production during one Growth Cycle and the total biomass Q(t) produced by the plant at cycle t is the sum of the production of all the leaves: Q(t) = E

ta XX p∈P n=1

vpt (n)

X q≥p

upq (t − n + 1)q(stpq (n)) (4)

The biomass so produced is distributed between the different organs according to a proportional allocation model [26]. Each organ o of chronological age n has a strength to attract biomass called sink po (n) and receives an amount of biomass proportional to this sink multiplied by the total biomass and divided by the sum of all the sinks that is called the demand. The one is the sum of the organ demand, the demand of the root system DR (t) and the demand of secondary growth DL (t) that can be computed by several ways that we will not detail here. Finally we have:

D(t) =

ta XX

vpt (n)dgu,t (n) + DR (t) + DL (t) p

The dynamic evolution of the rate of growth demand satisfaction creates variations in fruit productions along cycles and even alternations between cycles of low and high fruit production. Indeed, a sufficient value of the rate of growth demand satisfaction leads to fruit production. The fruit sinks will compete with the leaf ones, and those sources will receive less biomass than previously. Consequently, the increase in demand and decrease in biomass production due to smaller leaf areas will entail a lower value for the rate of growth demand satisfaction. Fruit production may be stopped. More biomass will be allocated to leaves till the rate of growth demand satisfaction increases enough, and so on. Rhythms are automatically generated.

p∈P n=1

With dgu,t (n) =X cp (t − n + 1)pf (n)X p + upq (t − n + 1) po (n)mopq

3. Results

o∈a,e

q≥p

(5)

2.4. Number of growth units If axes grow indefinitely, we can determine the number of new growth units of the plant by the equation: X ∀p ∈ {1, Pm }, vpt+1 (1) = vpt (1) + vlt (1)blp (t) (6) l≤p

For the growth units of one-cycle-old, we determine the number of active buds bpq (t) among the mbpq upq (t) potential ones:   Q(t) (7) bpq (t) = b1 + b2 D(t) bxc is the integer part of x ∈ IR. The biomass that has been allocated to dormant buds with the proportional allocation model is redistributed between all the active buds. That’s why we distinguish the demand of potential buds (Dpb (t)) from the one of active buds (Df b (t)). Each active bud gives birth to a growth unit of same physiological age. According to the value of parameters b1 , b2 , different behaviours are observed. If b2 is high enough, the ramification can be continuous. When b2 decreases, the ramification becomes sparser. Simulations of the model show that the ramification becomes periodic for some sets of parameters (see section 3.2).

2.5. Number of fruits Likewise, the number of fruits is determined at the level of the growth unit as a function of the rate of growth demand satisfaction, with the relation:   Dpb (t) Q(t) cp (t) = b1 + b2 f b (8) D (t) D(t)

In the following, we present a study of the generation of rhythm in two simple examples: the branching system of a Leeuwenberg architectural model [11] and the fruit production in a tree. The dynamic system is written in both cases, and the influence of parameters on the plant behaviour is examined.

3.1. Hypotheses and notations In the examples we present after that, we make hypotheses to simplify the writing of the system. Without loss of generality, we suppose that all the axes have the same characteristics, that is to say that there is only one physiological age and from now on we do not write the corresponding indices. We suppose that a growth unit of chronological age n is made of u similar metamers mt (n) with ma leaves each. We suppose that the amount of biomass produced by one leaf is a hyperbolic function of its surface s with parameters r1 and r2 : 1 (9) q(s) = r1 + r2 s The model of biomass allocation is simplified, the biomass allocated to root system is neglected, hence all the biomass is first allocated to buds and then used for the corresponding growth units. The expansion or organs is very short, compared to the duration of the cycle, and we consider organs immediately reach their final sizes. Hence we have po (1) = po and po (d) = 0, d ≥ 2. We note pm = ma pa + pe , pb fot the sink of a bud and e the thickness of a leaf. Leaves are supposed to be active during only one growth cycle (ta = 1). And min(x, y) represents the smallest value of reals x and y. We recall the state variables of the plant at growth cycle t: st (1) is the surface of the new leaves, v t (1) is the number of new growth units, u(t) and c(t) are respectively the number of metamers and fruits in the new growth unit.

3.2. Rhythms in a branching system In this section, we consider the branching system of a tree of a Leeuwenberg architectural model [11]. We neglect the secondary growth. There is one metamer per growth unit. At the end of Growth Cycle t, the terminal bud of the growth unit dies and is relieved by b(t) active buds. If we have b(t) = 0, the plant dies. The state vector of the plant is written:  a   t  p Q(t) s (1)  m    v t (1)   p e D(tt−1 t    (10) = b(t − 1)v (1)  X =  u(t)    1  c(t) 0 We deduce the demand D(t) and the total biomass producr1 epm tion Q(t) from equations 4, 5 and 9 and note A = pa and B = r2 . Thus we have:   Q(t − 1) b(t − 1) = min( a1 + a2 a t−1 , ma ) m v (1) D(t) = ma pb v t (1) Ema v t (1) Q(t) = Ev t (1)ma q(st (1)) = Av t (1) +B Q(t − 1)

(11)

We define a sequence α proportional to the rate of growth demand satisfaction: Q(n − 1) Q(n) = f ( n−1 ) v n (1) v (1) + with f : IR → IR (12) Ema k x→ A j x min( a1 + a2 a , ma ) + B x m The study of the sequence α presented in appendix A gives indications of the behaviour of the system. We suppose that ma E > A, otherwise the biomass production will decrease at least exponentially till the plant death (see equation 11). The simulated ramification is either periodic or continuous. It is continuous when the same number of potential buds becomes active at each cycle. We distinguish two cases that are represented on figure 1: ∀n, α(n) =

ma − a1 (E − A) > , ma buds become actives at B a2 each cycle on a growth unit. The number of new leaves increases exponentially and so will the biomass.

• if

• if ∃k ∈ {0; ...; ma − 1} such as ma

k − a1 a2

B a2 then limn→+∞ α(n) = ma

E−A . B

• or ∃k ∈ {0; ...; ma − 1}, k − a1 Ema − Ak k + 1 − a1 ma < < ma a2 B a2 Ema − Ak then limn→+∞ α(n) = . B Otherwise, the sequence oscillates around the value xk defined by the kth discontinuity of f and such as

A. Study of a recurrent sequence

limx→xk ,xxk f (x) We define a recurrent sequence α by ∀n, α(n) = f (α(n − 1)) With f : IR+ → IR+ Ema k x→ A j x min( a1 + a2 a , ma ) + B x m

(18)

The study of the function f on different intervals gives information on the behaviour of α. • If x ≤

ma − a1 a m , we define a2 f (x) = fma (x) =

Ema Ama +B x

• Else for k ∈ {0; ...; ma − 1}, weconsider the interval k − a1 a k + 1 − a1 a x∈ m ; m and define a2 a2 f (x) = fk (x) =

Ema A k+B x

On each interval, we can iterate the function fk and obtain the relation: fkp (x) = 

Ema

j p−1  X Ak Ak +B x Ema j=0 a = p−1  Em − Ak  Ak Ak Ak Ak (1 − )−B +B Ema x Ema Ema (19) Ak Ema

p−1

As we know f p on both intervals, it is possible to determine the number of iterations of f necessary between two points of discontinuity of f . This gives an indication of the period p of the sequence such as ∀n ∈ IN, α(n + p) = α(n) which corresponds to f p (α(n)) = α(n)

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