A Generalized Poisson Model to Estimate Inter-Plant Competition for Light Paul-Henry Courn`ede Ecole Centrale Paris Laboratory of Applied Mathematics 92295 Chˆatenay Malabry - FRANCE [email protected]

Abstract A new method is proposed to compute the functional effects of competition for light in plant populations. It is based on the estimation of exposed green leaf area for each individual. For this purpose, each individual in the stand is assigned a disk-shaped zone S p representing its assumed crown projection and competitive neighbours are defined as plants having overlapping disks. We combine the foliage distribution functions of neighbours to compute the probabilities of dominance and thus the exposed leaf area for each plant. The computation is fully detailed in the case of uniform distributions (Poisson model). In functionalstructural models of plant growth, the exposed leaf area thus obtained can be used to compute biomass production. The method is applied to the GREENLAB model and we present simulation results illustrating the effects of density.

Philippe de Reffye CIRAD - AMAP and INRIA Futurs - DigiPlante 34398 Montpellier - FRANCE philippe.de [email protected]

cepts developed in this model can be useful at the level of functional structural models for which biomass acquisition is deduced from light interception, (see reviews about functional structural models in [18], [17]). In this context, we propose a new method to compute competition for light in plant populations based on the estimation of exposed green leaf area for each individual. This paper starts with a theoretical presentation of the method. It is based on the combination of Poisson models of leaf distribution for all the neighbouring plants whose crown projection surfaces overlap. Then, the model of competition is applied to the GreenLab model of plant growth (see [12]) at stand level to determine resource acquisition of each indivdual in the stand. A simulation test case is also presented. It is important to note that this paper mostly focuses on the mathematical derivation of the model. It complements another paper of the authors [8] which is dedicated to present the validation of the model for applications in agronomy and forestry.

2. Description of the Method 1. Introduction Interactions among individual plants is a subject of interest in forest science or ecology, in order to model ecosystem dynamics, in terms of functioning or population. The general approach is a study of geometrical interactions between plants. The literature is abundant on the topic since Bella [3] or Mitchell [14]. For empirical or process-based models of forestry, the determination of spatial interaction helps define competition indices to characterize the growth condition of each tree. We refer for example to Biging and Dobbertin [4] and to Corral Rivas et al. [6], who present the most usual indices and compare them. Sorrensen-Cotthern et al. [20] present a process-based model integrating competition for light and plasticity in crown development, based on the Beer-Lambert law for resource acquisition and on a relative growth demand of tree ”sectors” for biomass allocation. Even though architecture is not described in detail and leaf area does not result from biomass allocation, the con-

2.1. Beer-Lambert’s extinction law and the concept of individual LAI At canopy level, the light interception ratio (LIR), i.e. the proportion of incoming irradiation intercepted by the canopy, is generally computed using the turbid medium analogy which supposes a uniform distribution of leaf area in the canopy. This analogy has been proved to be very robust since the Beer-Lambert equation is fairly insensitive to violations of the uniform assumption, see for example [15] or [16]. The light interception ratio is thus classicaly given by Equation (1) LIR = 1 − exp (−k LAI)

(1)

where LAI denotes the leaf area index whose original definition is the total one-sided area of photosynthetic tissue per unit ground surface area and k is the extinction coefficient

for the Beer Law, related to leaf orientation. In order to assess the effects of competition for light on the growth of individuals in a population, we need to compute the light interception for each plant. For this purpose, we extrapolate and define an individual LAI. As done in [3], [11] or [5] in the context of dendrometric forestry models, or more recently in [2] in ecological modelling and in [1] for computer graphics, we assign each individual in the stand a disk-shaped zone S p representing its assumed crown projection and competitive neighbours are defined as two plants having overlapping disks. However, unlike in dendrometric forestry models, we do not compute competition indices but estimate the exposed green leaf area for each plant with Beer-Lambert’s extinction law, so as to evaluate its individual biomass production. In the following, Li denotes the individual leaf area index for the plant Pi , that is to say the ratio Ai /Sip , with Ai the plant leaf area.

2.2. Partition of the crown projection surfaces Let(Pi )i∈[1;N ] denote the set of plants in the stand and Γ(i) denote the neighbourhood of the plant Pi , defined by : Γ(i) = {k ∈ [1; N ], k 6= i / Sip ∩ Skp 6= ∅}

(2)

Let Π (Γ(i)) be the set of all the subsets of Γ(i) and let η ∈ Π (Γ(i)). We define Si (η) by: \ p\ p S¯k (3) Sj Si (η) = Sip j∈η

Figure 1. Simple example of partition. We consider the plant Pi and its two neighbours Pj and Pk . The p partition of Sp i is given by Si = S1 ∪ S2 ∪ S3 ∪ S4 . S1 p p is the part of Si overlapped neither by Sp j nor by Sk , S2 p the one only overlapped by Sk , S3 the one overlapped p by both Sp j and Sk , and S4 the one only overlapped by p Sj .

generally be expressed as a decreasing exponential function of the LAI, even when the random turbid medium assumptions generally associated with the Poisson model are not satisfied. We deduce that the exposed foliage area Ali of the plant Pi is given by: Ali =

k∈η /

where S¯jp is the complementary set of Sjp . All the Sjp being disk-shaped surfaces, the exact formula giving Si (η) can be computed theoretically. However, the formula becomes quite tedious in the case of heterogeneous stands when Card(η) ≥ 3 and it may be necessary to resort to numerical computation. Finally: [ Sip = Si (η) (4) η∈Π(Γ(i))

and {Si (η)}η∈Π(Γ(i)) is a partition of Sip . A simple example is given in Figure 1.

2.3. Computing the exposed foliage area of the plant Pi The computation of the exposed photosynthetic foliage area is based on the assumption of vertical irradiation, which is justified by the time unit considered to compute photosynthesis (at least several days) and uses the so-called Poisson model: Nilson [15] demonstrated, citing both theoretical and empirical evidences, that the gap fraction can

X

Si (η) Pid (η)

(5)

η∈Π(Γ(i))

where Pid (η) is the probability that δS, an infinitesimal surface element of Sip , is covered by the foliage of the plant Pi and that moreover Pi dominates its neighbours in η over δS, that is to say that the upper foliage element covering δS belongs to Pi . Let dS denote the surface area of δS. For the sake of simplicity and without loss of generality, we consider in the following that the extinction coefficient k = 1, which would correspond to leaves always orientated towards the source of irradiation. In the general case, we would simply need to replace the leaf area Ai by ki Ai , ki denoting the specific extinction coefficient for each plant. If we assume that the vertical projection onto Sip of all the elementary surface elements of area dS that belong to Ai follows a uniform distribution, we deduce that the distribution law of the number of elementary foliage surface elp ements covering an elementary  element δS of Si  surface follows a binomial law: B dS/Sip

Ai dS dS , Sip

→ +∞ and Ai dS = Li dS Sip

. When dS → 0,

(6)

the height of the crown basis and the total height of Pj , we may take:  0 if z ≤ bj    (z − b ) j if bj ≤ z ≤ hj (11) Fj (z) =  (hj − bj )   1 if hj ≤ z

As a consequence, if Pic denotes the probability that δS, an infinitesimal surface element of Sip , is covered by the foliage of the plant Pi , we have:   Ai c Pi = 1 − P (Ni = 0) = 1 − exp − p (7) Si

With such hypotheses, integral 10 can thus be evaluated formally. Taking more complex foliage distribution would not enduce critical difficulties, we may simply need to resort to numerical computation of this integral. We finally have: X Pid (η) = Pid (η/N = K) P (N = K) (12) ∗ n K∈IN ×IN

The classical GREENLAB equation of resource acquisition for the individual plant is deduced from this relationship, see [12], as the exposed leaf area for the indivisual plant is directly given by:    Ai p c p l (8) Ai = Si Pi = Si 1 − exp − p Si

with

remains constant. Hence, the approximation of the binomial law by the Poisson law is justified and if Ni denotes the number of foliage elements of Pi covering a given surface element of the horizontal plane: k

P (Ni = k) =

exp (−Li ) (Li ) k!

It is important to note that in GreenLab, Sip is estimated from experimental data. Basically, it corresponds to using a theoretical distribution of leaf area whose vertical projection onto Sip is uniform and which is equivalent in terms of global light interception to the real distribution of leaf area in the plant crown. In the following, we consider η ∈ Π (Γ(i)) and we let n = Card(η), that is to say the number of elements in η. In the case of homogeneous stands, all the trees are supposed to have the same individual LAI and it is easy to see that Pid (η) =

1 (1 − exp (−(n + 1)Li )) . n+1

(9)

For heterogeneous stands, we can combine the Poisson models of the competitors to derive the exact formula giving Pid (η).  

Let K ∈ IN∗ × INn denote ki , (kj )j∈η and N denote   Ni , (Nj )j∈η (if we suppose the indices ordinated in η).

We first need to compute the probability Pid (η) knowing that Ni = ki and Nj = kj for all j ∈ η. It will be denoted by Pid (η/N = K). If Fj denotes the probability distributions of the vertical position of foliage area and fj the density functions, we can prove that: Pid

Z (η/N = K) = 0

+∞ Y 

 k Fj j (z) ki Fiki −1 (z)fi (z)dz

P (N = K)

= P (Ni = ki )

P (Nj = kj )

j∈η

=

exp (Li ) Lki i Y ki ! j∈η

k

exp (Lj ) Lj j kj !

!

(13)

2.4. Example of computation: Card(η) = 1 In this section, we detail the computation on Si (η) when Card(η) = 1, that is to say for a surface on which Pi has only one neighbour that is denoted by Pj . The vertical foliage distributions are supposed uniform and let α = min (hi , max (bi , bj )) and β = max (bi , min (hi , hj )) (14) Integral 10 can be rewritten: k j  ki −1 Z β z − bj z − bi ki dz Pid (η/N = K) = hj − b j h − bi hi − b i α ki −1i Z hi  z − bi ki dz + hi − b i hi − b i β (15) After integrating by parts iteratively, we finally have: # " ki k +l k −l β X (z − bj ) j (z − bi ) i l d Pi (η/N = K) = γi,j k k (hi − bi ) i (hj − bj ) j α l=1  k i hi − β + h i − bi (16) with ki !kj ! l γi,j = (−1)l−1 (17) (ki − l)! (kj + l)! Finally, Equation 12 rewrites:

j∈η

(10) If we suppose that the vertical distribution of the foliage in the crown is uniform, and if bj and hj denote respectively

Y

Pid

−Li−Lj

(η) = e

+∞ X +∞ k k X (Li) i (Lj ) j d P (η/N = K) (ki )! (kj )! i

ki =1 kj =0

(18)

It involves infinite series and only depends on the LAI of all the competitors, their total heights and the heights of their crown bases. Practically, we can see that only a few terms in the sum are not negligible.

3 presents the view from above of a clump of even-aged trees. Conditions of competition vary from one tree to another, which induces differences in resource acquisition at each growth cycle and strong architectural differenciation owing to plant plasticity.

Figure 2. Truncated series for the approximation of Pd i (η). We compare the truncated development of the series in Equation (18), to the theoretical value (for the infinite sum). Here, we consider the homogeneous case (so that the theoretical result is obtained explicitly from Equation (9)), for two different LAI, Li = Lj = 1 and Li = Lj = 2. We see that the series converges rapidly and it is sufficient to consider the sums to rank 6 or 7.

3. Application of the Method to the GreenLab Model In this section, we present simulation results of our model of competition applied to GREENLAB at stand scale, using Digiplant software [7]. The exposed leaf surface area (Equation 5) is computed for each plant in the stand with respect to its neighbours. We thus deduce resource acquisition (with the concept of light use efficiency) and resource allocation as it is classically done in the GREENLAB model [12]. We use the last version of this model which takes into account interactions between functional growth and architectural development [13]. Figure

Figure 3. Simulation of tree growth in heterogeneous conditions: view from above and detailed architectures of the individuals. In the upper right corner of the figure, Tree 1 grows in open-field like conditions. In the lower left corner, Tree 3 surrounded by its four neighbours (including Tree 2) severely suffers from competition. The variation of exposed leaf area according to tree age is also given for Tree 1, Tree 2, Tree 3

4. Discussion In this paper is presented a mathematical framework to compute the exposed leaf area of each individual in plant populations: the Poisson models of leaf repartition of competing plants are combined. The exposed leaf area is then used to compute biomass acquisition for each individual plant. The model seems very general compared to most

dendrometric methods based on competition indices (see [6] and [4]), since they usually resort to the computation of statistical coefficients that are specific to given conditions and are not easily extrapolated. From a theoretical point of view, the proposed method is very general and can be applied to both even-aged and uneven-aged plant populations, in both spatially homogeneous and heterogeneous conditions, with varying environmental conditions. However, its validation is still at its early stages and we refer to [8] for a detailed discussion. Besides, the method can not be applied directly to mixed crops and multi-species plant communities since different plant species have different strategies to obtain their necessary share of the resources in order to grow and reproduce [9]. Interactions between plant species have been investigated in ecology, often by performing two-species competition experiments, since de Wit [10]. The empirical concept of interspecific competitive forces was introduced. However, the way to make the link with our method to compute competition in the context of functional-structural growth model is still to study. The hypothesis of uniform distribution may seem too restrictive for application to real plant populations. In [19], it is shown that departure from randomness is mainly due to the spatial variations in leaf area density within the canopy volume. The method presented in this paper can be generalized to take into account spatial variations in leaf area density. It would simply concern the computation of Pid (η/N = K) in Equation (10), more complex distribution functions should be used. However, going into such details may not be necessary. As recalled above, in GREENLAB, Sip is estimated from experimental data, which corresponds to using a theoretical distribution with the adequate hypothesis of uniform distribution and which is equivalent in terms of global light interception to the real distribution of leaf area in the plant crown. So far, such an equivalent uniform distribution has always been found for all the plants on which the GREENLAB model has been validated. We are confident that the extrapolation at stand level should work, at least for mono-species populations. Preliminary results on field crops (maize, tomato and sugar beet) are very encouraging and an experimental study is currently carried out on natural tree communities (Cecropia) for a comprehensive test of the model.

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