Understanding the dynamics of an undisturbed ... - Wiley Online Library

Journal of Ecology 2009, 97, 97–108

doi: 10.1111/j.1365-2745.2008.01445.x

Understanding the dynamics of an undisturbed tropical rain forest from the spatial pattern of trees

Blackwell Publishing Ltd

Nicolas Picard1*, Avner Bar-Hen2, Frédéric Mortier3 and Joël Chadoeuf4 1

CIRAD, UR 37, BP 4035, Libreville, Gabon; 2Université Paris 13, SMBH-Lim & Bio, 74 rue Marcel Cachin, 93017 Bobigny Cedex, France; 3CIRAD, UR 39, Campus international de Baillarguet, TA 10/C, 34398 Montpellier Cedex 5, France; and 4INRA, Centre d’Avignon, Unité de Biométrie, Domaine St-Paul, site Agroparc, 84914 Avignon Cedex 9, France

Summary 1. Considering that the spatial pattern of trees is a footprint of the biological processes that drive their dynamics, increasing work has been undertaken to analyse spatial patterns and fit spatial point processes to them. When diameter is taken into account, the underlying point process is a marked point process. The question then is how to correctly model the dependence between the mark and location, and the different patterns at the different scales, to gain understanding of the underlying biological processes. 2. The data used comes from the Paracou rain forest in French Guiana (5°15′ N, 52°55′ W). The spatial pattern of trees in this forest exhibits regularity at small distances (c. 6 m) and clustering at larger distances (c. 30 m). The pattern is linked to diameter, with a shift from clustering to regularity as trees grow. 3. Two models of spatial pattern are used. The first one is pattern-driven in the sense that it breaks down the observed pattern into a mixture of regular, clustered and random contributions. The second one is process-based and uses a simple individual-based space-dependent model of forest dynamics as a simulation algorithm. It is obtained as the limit of this spatio-temporal model when time tends to infinity. 4. Both models are consistent with the observed pattern, with a better fit provided by the second model. Moreover this second model provides a biological interpretation of the observed pattern in terms of forest dynamics. However the first model’s implementation is simpler (both for simulation and for parameter estimation). 5. Synthesis. Modelling the spatial pattern of plants using spatial point processes gives insights into the biological processes that drive their dynamics. It allows the reconstruction of their dynamics given only a snapshot of plant locations. Very few solutions exist to model complex marked spatial patterns when point location and mark are dependent. We defined and compared two point processes that successfully modelled the spatial pattern of trees in a rain forest with interaction between tree location and tree size. Both models highlight competition (either symmetric or asymmetric) as a driving process towards regularity. The second model further reveals a selforganizing dynamic with a feedback effect of competition. Key-words: diameter-location interaction, French Guiana, marked point process, random field model, Ripley K-function, spatial pattern; virtual stand

Introduction The spatial pattern of plants has long been recognized as a footprint of the biological processes that drive their dynamics (Cale et al. 1989; Cole & Syms 1999; Belyea & Lancaster 2002; Cutler et al. 2008). A recurrent question in ecology then *Correspondence author. E-mail: [email protected]

is: Is it possible to infer the biological processes from the spatial pattern of the individuals? A convenient framework to analyse spatial patterns of plants is the theory of spatial point processes (Cressie 1993; Stoyan & Stoyan 1994), although early work was mainly based on quadrat counts and distancebased methods (Greig-Smith 1964; Pielou 1969). The most straightforward analysis of spatial patterns is based on summary statistics such as Ripley’s K function or

© 2008 The Authors. Journal compilation © 2008 British Ecological Society

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the pair correlation function (e.g. in relation to tropical rain forest, Forman & Hahn 1980; Forget et al. 1999; Dessard et al. 2004). A more thoughtful approach consists in modelling the spatial pattern of plants by a point process (e.g. in relation to tropical rain forest, Batista & Maguire 1998; Pélissier 1998; Neeff et al. 2005). In this approach, a biological interpretation is most often given to the simulation algorithm of the process. Apart from the ecological interest to understand the processes that drive population dynamics, modelling spatial patterns of trees using point processes also has a computational interest. The idea then is to produce realistic virtual forests for simulation-based studies (Stoyan & Penttinen 2000; Goreaud et al. 2004; Tscheschel & Stoyan 2006). A realization of the point process can be used as the initial state for an individual-based spatially-explicit model of forest dynamics, thus permitting either simulation-based studies of the properties of the model, or disaggregation from nonspatialized data (Goreaud et al. 2004). A point process that models a forest stand in a realistic way can also be used to optimize an inventory design (Stoyan & Penttinen 2000). Various point processes have been used to model the spatial pattern of trees in tropical rain forests. When addressing tree species separately, the pattern is most often clustered or random (Forman & Hahn 1980; Forget et al. 1999; Dessard et al. 2004) and Cox processes are then used (Batista & Maguire 1998). Addressing the whole community of tree species remains a challenging task as tropical rain forests have a high species richness and the interactions between species are complex (Loussier 2003; Illian et al. 2003, 2006). When disregarding the species, the most striking feature of natural rain forest is the regular pattern of large trees (Pélissier 1998; Neeff et al. 2005; Picard et al. 2008). Markov processes (mainly Strauss or soft-core processes) have been used to model this regular pattern (Neeff et al. 2005). Bivariate repulsive Markov processes have also been used to take into account the interaction between adult and juvenile trees (Pélissier 1998). Analysing the spatial pattern of trees jointly with their diameter brings us to marked point processes, where the mark is continuous (we would rather speak of multivariate point process if the mark was discrete, or of Boolean point process if the mark was a shape, see Wiegand et al. 2006). The marked approach almost always gives better insight into the underlying biological processes than the non-marked approach because the relative location of a plant with respect to its neighbours in dense natural ecosystems is almost always related to its size (Ford & Diggle 1981; Weiner et al. 2001; Stoll & Bergius 2005). The interpretation of summary statistics for marked point process in the forestry context has been clarified (Penttinen et al. 1992; Goreaud & Pélissier 2003; Parrott & Lange 2004). Marked point processes have also been proposed to model forest stands. These studies have been limited to temperate forests which have been intensely managed, with either simple regular patterns that could be modelled by marked Strauss or soft-core models (Fiksel 1984; Goulard et al. 1996; Mateu et al. 1998; Kokkila et al. 2002), or more complex patterns that were modelled by heuristic processes (Pielou 1960; Moeur 1997; Hanus et al. 1998).

To model a marked point pattern such as a set of tree coordinates with their associated diameter, the first step is to know whether the mark (that is diameter) is independent from point (that is tree) locations. Tests of the interaction between marks and locations have been developed (Schlather et al. 2004; Guan 2006; Renshaw et al. 2007). If mark is independent from location, then the spatial distribution of marks and the spatial distribution of points can be modelled separately using a random field model (Schlather 2002; Penttinen 2004). The random field model (also known as the model with external geostatistical marking, see Penttinen 2004) consists in superimposing a non-marked point process onto an independent random field. However, independence between tree diameter and tree location is rarely met in natural forests due to competition and other density-dependent processes. Other approaches are needed to model marked point patterns. A marked point process will be considered as an appropriate model if it reproduces the observed spatial pattern of plants. This does not mean that the model is ‘true’, in the sense that the processes that drive the outcome of the point process may still have no connection with the biological processes that drive the spatial pattern of plants. However, as long as the model is not invalidated by additional data, it may be considered as an in silico experimental tool to gain understanding of the processes that drive plant dynamics. In particular, it can be used to make predictions to be subsequently tested using additional data. This study aims at gaining understanding of the process that drive the dynamics of a natural tropical rain forest from a snapshot of the spatial pattern of its trees, using two marked point processes. These two processes are new in that they can face dependence between mark and location. The first point process is pattern-driven in the sense that it breaks down the observed pattern into a mixture of regular, clustered and random contributions. Fitting the model is then equivalent with finding the best weighting of regular, clustered and random contributions depending on the mark. The second point process is process-based in the sense that it integrates a priori knowledge on the biological processes that drive forest dynamics. The model then simply mimics these biological processes, and the observed pattern is supposed to be a realization of the limit of this spatio-temporal process when time tends to infinity. Each point process is used to model the joint distribution of diameter and spatial location of trees in an undisturbed rain forest in French Guiana, and to predict how the spatial pattern would vary if some characteristics of the forest stand were changed.

Methods STUDY SITE

The forest stand studied is within a tropical rain forest at Paracou (5°15′ N, 52°55′ W), near Kourou in French Guyana. Paracou is an experimental site dedicated to studying the effects of logging damage on stock recovery. It was created in 1984 in an area representative of

© 2008 The Authors. Journal compilation © 2008 British Ecological Society, Journal of Ecology, 97, 97–108

Dynamics and spatial pattern undisturbed forest. The site lies in a terra firme rain forest on the coastal plain with equatorial climate. A dry season occurs from August to mid-November. From March to April, a short drier period interrupts the rainy season (Gourlet-Fleury et al. 2004). The physiography of the site shows smooth slopes incised by minor streams. The experimental design of the site consists of three blocks of four 300 × 300 m permanent sample plots with a 25 m inner buffer zone. In each central 250 × 250 m square, all trees over 10 cm diameter at breast height (dbh) were identified and geo-referenced. Since 1984, girth at breast height, standing deaths, treefalls and newly recruited trees over 10 cm dbh have been monitored annually (Gourlet-Fleury et al. 2004). Data from plot 1 in 1984 was used for this study. The central hectare of this 6.25 ha plot is shown in Fig. 1.

SPATIAL PATTERN ANALYSIS

A spatial point pattern is considered as the outcome of a point process, that is a stochastic process that generates point locations (Cressie 1993). We hereafter suppose that the observed pattern follows from a homogeneous and isotropic point process, which means that the point distribution is invariant under any translation or rotation. Marked point processes additionally associate a mark (either quantitative such as diameter or height, or qualitative such as species) to each point. They jointly generate point locations together with their mark. Hereafter the mark is tree diameter and is thus strictly positive. Spatial pattern analysis relies on the estimation of first and second order characteristics of point processes. Within the first order characteristics, the intensity, denoted λ, represents the number of events per unit area. Within second-order characteristics, the pair correlation function, denoted g, describes the probability of finding two events a distance r from each other (see Appendix S1 in Supporting Information). The integral, K (r ) =

r

2πu g (u )du

eqn 1

0

99

defines Ripley’s K function. Intuitively, λK(r) is the expected number of extra events within a distance r from a randomly chosen event. Functions g and K describe the spatial distribution of points, without considering their mark. They have counterparts that include information on the marks (Stoyan & Penttinen 2000; Schlather 2001; Parrott & Lange 2004). A counterpart of the pair correlation function is the gmm function such that the average value of the product of the marks found in two infinitesimal circles with areas δ1 and δ2 at a distance r from each other (taking zero if the ball does not contain any point) equals (λm)2gmm(r)δ1δ2, where m is the expectation of the mark. The integral similar to eqn 1 where g is replaced by gmm defines Stoyan’s Kmm function (Stoyan & Stoyan 1994). The null model for point pattern analysis is the homogeneous Poisson process that distributes points at random. The pair correlation function for the homogeneous Poisson process equals one at every distance, and its K function thus equals πr2. When points are clustered at distance r, g(r) > 1 and K(r) > πr2, whereas g(r) < 1 or K(r) < πr2 indicates that the pattern is regular at distance r. Besag’s transform of the K function is L(r ) = K (r )/π − r. As detecting departure from randomness amounts to assessing whether L(r) is different from zero, it is actually more convenient to use L(r) than K(r). To test the null hypothesis, we used empirical 95% confidence bounds around L(r) computed by Monte Carlo technique using B = 100 simulations of the homogeneous Poisson process. The null hypothesis for marked point pattern analysis is the random labelling hypothesis, meaning that marks are assigned at random to points. Under the random labelling hypothesis, gmm(r) = g(r) and Kmm(r) = K(r). A transform of the Kmm function that is analogous to Besag’s transform for the K function is Lmm = K mm (r ) − K (r ). Detecting departure from random labelling then is equivalent with assessing whether Lmm(r) is different from zero. To test the null hypothesis of random labelling, we used empirical 95% confidence bounds around Lmm(r) computed by Monte Carlo technique using B = 100 simulations. Each simulation consists in permuting marks at random while points stay at the same locations. The random labelling hypothesis is stronger than the hypothesis of independence between marks and locations, as it further assumes that marks have no spatial structure on their own. Moreover, Kmm(r) < K(r) or Kmm(r) > K(r) has no unequivocal interpretation. For instance, Kmm(r) < K(r) may result either from the clustering of the smallest trees, or from the regularity of the largest trees (or from a combined action of both; Picard & Bar-Hen 2002). To understand the shape of Kmm, it is thus necessary to do further analysis. To this end, we analysed different size classes separately using the K function. Given a point pattern in a domain A, the intensity is simply estimated by dividing the number of points by the area of A. This estimate thus corresponds to tree density. To estimate the pair correlation function, we used a kernel estimator based on the Epanecnikov kernel, on the translation correction of edge effects, and on the bandwidth coefficient recommended by Stoyan & Stoyan (1994, eqn 15.15, p. 284). Ripley’s K function was estimated using Ripley’s estimator (Stoyan & Stoyan 1994, p. 282) whereas Stoyan’s Kmm function was estimated using Stoyan’s estimator (Stoyan & Stoyan 1994).

ASSESSING THE ADEQUACY OF A MODEL

Fig. 1. Central hectare of plot 1 of Paracou in 1984. Each circle gives the location of a tree and circle radius is proportional to tree diameter.

The adequacy of a model will be assessed by comparing its K and Kmm functions to the estimates of these functions obtained from the Paracou data. Let Kobs be the estimate of the K function using Paracou data, and let Kpred be the K function of the model. When no theoretical expression for Kpred is available, it will be approximated by simulation as


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Kpred ≈

1 B

∑

B i =1

(i) Ksim (r )

eqn 2

(i)

where Ksim is the estimate of the K function for the ith simulation of the model out of B simulations. The discrepancy between Kobs and Kpred is usually quantified by the sum of their squared difference (SSD) over the range [0, rmax] (Stoyan & Stoyan 1994): rmax

SSD =

( Kobs (r ) − Kpred (r )) dr 2

0

rmax

1 MSSD = B

∑ (K B

obs

(i) (r ) − Ksim (r )) dr 2

i =1

0

Provided that Kpred is computed using eqn 2, the MSSD breaks down into SSD plus the sum of the empirical variance (SEV) of Ksim over the range [0, rmax], MSSD = SSD + SEV where SEV =

0

This approach consists in splitting the forest stand into diameter classes, and then modelling each subset independently of the others using a random field model. Let k be the number of diameter classes, let Φi be a non-marked point process used to model the spatial pattern of trees in the ith diameter class, and let Zi be a random field model that describes the spatial distributions of diameters belonging to the ith class. The pattern-driven model is then defined as:

∪i=1∪x∈Φi [ x; Zi (x )] k

Recently, it has also been suggested to take into consideration the variability of the estimate of the K function generated by the model by considering the mean SSD over B simulations of the model (Diggle 1983; Ngo Bieng 2007):

rmax

PATTERN-DRIVEN MODEL

B ⎫ 2⎪ ⎪⎧ 1 (i) Ksim (r ) – Kpred (r ) ⎬ dr ⎨ ⎪⎩ B i=1 ⎪⎭

∑[

]

Using MSSD instead of SSD as a criterion of adequacy means that a good model is a model that not only fits on average the data (low SSD), but also generates low variability (low SEV). In other words, the best model according to MSSD is the one that provides the best trade-off between accuracy and precision, whereas the best model according to SSD is the one that provides the best accuracy. To test departure from a given model, the observed value Kobs (r) of the K function at Paracou was compared to the 2.5% and 97.5% quantiles of the estimator of the K function under this model. These quantiles were again obtained by simulation, using the B estimates (1) ( B) Ksim, ... , Ksim of the K function corresponding to B simulations of the model. All computations that we have just presented for the K function were done in a similar way using the Kmm function.

POINT PROCESS DEFINITION

Two marked point processes were defined to model the spatial pattern of trees at Paracou, namely the pattern-driven model and the process-based model.

Each non-marked point processes Φi is chosen among three possible models: a homogeneous Poisson process, a Matérn process, or a simple sequential inhibition (SSI) process. The first model generates random patterns, the second clustered patterns, and the third one regular patterns (Fig. 2). These three models were chosen because they are easy to simulate and because their summary statistics are known. By mixing them with appropriate weights, it is possible to produce complex patterns consisting of a mixture of randomness, clustering and regularity. The homogeneous Poisson process was already defined. The Matérn process depends on three parameters that are the parent intensity ω, the dispersion radius R and the fecundity γ (Stoyan & Stoyan 1994, p. 309). The names of the parameters refer to the following simulation algorithm and should not be confused with actual tree dispersal or fecundity: parent points are drawn according to an homogeneous Poisson process with intensity ω; each parent produces independently of each other a random number of daughter points according to a Poisson distribution with parameter γ; the daughter points are located uniformly in the disk of radius R centred on their parent. The Matérn point field is composed of the daughter points only. The intensity of the Matérn process with parameters (ω, R, γ) is ωγ. The SSI process depends on three parameters that are the intensity λ, the inhibition distance ρ and the survival parameter ν (Diggle & Cox 1983). Its simulation is straightforward, although some care has to be taken to ensure that λπρ2/| A | is small enough, where A is the simulation window and | A | its area. The number of events is drawn according to a Poisson distribution with parameter λ| A |. Events are then located sequentially: the location of the first event is drawn uniformly in A; given the locations of k events, a trial position for the (k + 1) th event is drawn uniformly in A; the trial k position is retained with probability g (r ), where ri is the i =1 SSI i distance between the trial position and the ith event, and gSSI is a survival function given by:

∏

v ⎧ gSSI (r ) = ⎨(r/ρ) ⎩ 1

(r < ρ). ( r ≥ ρ)

eqn 3

Fig. 2. One outcome of each of the base processes used to compose the pattern-driven model: (a) Matérn process with parameters ω = 0.01 m–2, R = 15 m, γ = 20; (b) Poisson process with intensity 0.02 m–2; (c) SSI process with parameters λ = 0.02 m–2, ρ = 4.8 m, ν = 100. © 2008 The Authors. Journal compilation © 2008 British Ecological Society, Journal of Ecology, 97, 97–108

Dynamics and spatial pattern Trial positions for the (k + 1)th event are drawn until the first acceptance. The interaction between two events in the SSI process thus depends on the overlap between the two disks with radius ρ/2 centred on each event. In the point process terminology, this corresponds to a model with core interactions. When ν = 0, the overlap between two disks (or cores) does not influence the outcome and the SSI is a homogeneous Poisson process. When 0 < ν < ∞, the overlap between two disks is possible (but less likely as ν increases): this corresponds to a soft-core model. When ν = ∞, the overlap between two disks is impossible: this corresponds to a hard-core model.

PROCESS-BASED MODEL

The process-based model is defined by its simulation algorithm, whereas its statistical properties remain unknown. The simulation algorithm mimics the biological processes of growth, mortality and recruitment that drive the forest dynamics. It thus corresponds to an individual-based spatially-explicit model of forest dynamics (Liu & Ashton 1995; Busing & Mailly 2004), with a stopping rule that ensures that a realization of the process corresponds to a stationary state of the model. Alternatively, the individual-based spatiallyexplicit model of forest dynamics can be seen as a space-time point process (Rathbun & Cressie 1994; Moeur 1997; Stoyan & Penttinen 2000; Druckenbrod et al. 2005). In the individual-based spatiallyexplicit model of forest dynamics that we consider here, a tree is defined that its diameter D and its spatial coordinates x. The forest stand is composed of the N individuals with a diameter greater than a minimum diameter d0. Time is discrete. The functions and processes used in this model were chosen as the most simple ones while maintaining the capacity of the model to produce realistic spatial patterns. They originate from the simplification of more complex models that were previously studied (Picard et al. 2001; Verzelen et al. 2006). The diameter growth depends on a distance-dependent asymmetric competition index L, that gives the number of trees greater than the subject tree within a distance ρ from it: L( D, x ) =

∑

N

i =1

1(|| x − xi ||) 1( D < Di )

eqn 4

where 1 is the indicator function, (D, x) are the characteristics of the subject tree and (Di, xi) are the characteristics of the ith tree in the forest stand. This competition index is classical and has been used in many models of forest dynamics (Biging & Dobbertin 1992; Verzelen et al. 2006; Caplat et al. 2008). The expectation of L should not be confused with λK (D)(ρ), where K (D) is Ripley’s K function for trees with a diameter greater than D. However, under the hypothesis of random labelling, the expectation of L equals λK(ρ)[1 − F(D)] where F is the distribution function of diameters (Verzelen et al. 2006). The diameter increment of a tree between two successive time steps is: ΔD = exp(−aL)

eqn 5

where a is a parameter that tunes the strength of competition. This equation implies that the time step is chosen so that the maximum diameter increment during one time step equal one unit of diameter. This growth equation is also standard (Caplat et al. 2008). The mortality rate is defined to ensure that the simulated diameter distribution will fit that observed at Paracou. At each time step, a tree has a probability to die equal to:

m( D ) = − ψ ′( D ) − ψ ( D )

f ′( D ) f (D)

101 eqn 6

where D is the diameter of the subject tree, f is the target distribution density of the diameters that is to be simulated, and the function ψ approximates the average relationship between diameter and diameter increment, that is the expectation of the individual relationship defined by eqns 4 and 5. The form of eqn 6 follows from the stationary solution (setting ∂f/∂t = 0) of the partial derivative equation ∂f ∂ =− ψf − mf ∂t ∂D

{ }

that approximates the individual-based model as well as ψ approximates the individual relationship between D and ΔD (Picard & Franc 2001). We used for ψ a polynomial of degree 2 with respect to the diameter: ψ(D) = α0 + α1D + α2D2 where the coefficients α0, α1 and α2 are estimated at each time step from the data set {(Di, ΔDi), i = 1 ... N} using a linear regression. Recruited trees, that are the trees whose diameter reaches the minimum diameter d0 between two successive time steps, have a diameter equal to d0. They are located according to a Matérn process with parameters (ω, R, γ). The flux of recruited trees, that equals ωγ | A | (where A is the simulation window), is set equal to the flux of dead trees. Thus, on average, the number N of individuals in the forest stand remains constant. The initial state of the simulation is drawn according to a random field point process. Trees are initially located according to a homogeneous Poisson process with intensity: λ0 = λ exp[μ(d1 − d0)]

eqn 7

where λ is the tree density at Paracou and d1 = 10 cm is the minimum diameter for inventory at Paracou. Their diameters are initially independently and identically distributed according to the empirical distribution of diameters. A realization of the process is defined as the state reached after a fixed burn-in period. We checked that the stationary state of the model was reached well before the 300th time step for a 6.25-ha plot, and thus fixed the burn-in to 300 time steps.

PARAMETER ESTIMATION

Let p be the number of parameters of the diameter distribution. Given the number k of diameter classes, the pattern-driven model k depends on p + k + n parameters, where ni = 2 if Φi is a i =1 i Matérn or a SSI process, and ni = 0 if Φi is a Poisson process. These parameters include the total intensity λ, k − 1 diameter breakpoints, and ni parameters for each process Φi. The intensity of Φi is not counted because it follows from λ, from the bounds of the ith diameter class, and from the diameter distribution. The process-based model depends on p + 6 parameters that are its intensity λ, the p parameters for the diameter distribution, the competition radius ρ, the competition parameter a, the minimum diameter for recruitment d0, the dispersion radius R and the fecundity γ of the Matérn process for recruited trees. Let θ be the set of all parameters of a model. The total intensity λ, that identifies with tree density, and the parameters of the diameter distribution are stand characteristics that were estimated directly from non-spatial tree measurements. The other parameters were estimated using a minimum contrast method based on the pair correlation function (Stoyan & Stoyan

∑


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1994, p. 304): the estimate of θ was the value of θ that minimized the SSD between gobs and gpred, rmax

3 = arg minθ

( g obs (r ) − gpred (r; θ))2 dr

eqn 8

0

where gobs is the estimate of the pair correlation function using Paracou data, and gpred is the pair correlation function of the model. As the number of parameters to estimate was quite high, the optimization problem 8 was solved using a combination of systematic scanning of some parameters and using the Nelder-Mead optimization algorithm (Nelder & Mead 1965) for the remaining parameters. As the Φi processes that compose the pattern-driven model are independent from each other, the pair correlation function for the pattern-driven model is: gpred (r; θ) =

∑

k

i =1

wi2 gi (r; θ) + 2

∑ ∑ k −1

k

i =1

j =i +1 i

w wj

eqn 9

where gi is the pair correlation function of Φi, and wi is the weight of Φi within the pattern-driven model. This weight equals: wi = F(di+1) − F(di), where F is the distribution function of diameters (that is the primitive that sums to one of the f function defined in eqn 6), and di is the lower bound of the ith diameter class (that is also the upper bound of the (i − 1)th diameter class). An intuitive proof of eqn 9 is given in Appendix S1. When Φi is a SSI process, its pair correlation function is approximately equal to gSSI defined by eqn 3. When Φi is a Matérn process, its pair correlation function is (Stoyan & Stoyan 1994): ⎧ ⎛ r ⎞ 2γ ⎪1 + h⎜ ⎟ (r < 2 R ) gMatérn (r ) = ⎨ λ( πR )2 ⎝ 2 R ⎠ ⎪ (r ≥ 2 R ) 1 ⎩ where h( z ) = arccos( z ) − z 1 − z 2 for z < 1 and λ is the intensity of the Matérn process. As no explicit expression of gpred was available for the process-based model, we followed Cressie’s (1993, p. 685) suggestion to replace it by an empirical estimate obtained by simulation: B

gpred (r; θ) ≈

1 i) g (sim (r; θ) B i=1

∑

(i)

where g sim is the estimate of gpred for the ith simulation of the model out of B simulations. The greater B is, the better the approximation is, but the longer computing time is. As the process-based model required long computing time, we used B = 100 replicates.

Results SPATIAL PATTERN OF TREES AT PARACOU

Plot 1 of Paracou in 1984 comprises 3840 trees with a diameter greater than 10 cm, which corresponds to a tree density of λ = 0.0614 m–2. Figure 3 shows the estimates of second-order characteristics for this plot. Figure 3a indicates three scales of spatial organization: at distances less than 6 m, the spatial pattern of trees is regular; between 15 and 60 m, it is clustered; beyond 60 m, it is random. Further analyses consisting of separate estimates of the K function for different diameter classes (not shown here to conserve space) show that:

Fig. 3. Estimates of second-order characteristics of the spatial pattern of trees at Paracou, in plot 1 in 1984. Black lines: estimates obtained from Paracou data. (a) Besag’s transform of Ripley’s K function; white line is the theoretical value of the function under the null hypothesis of an homogeneous Poisson process; grey area is the 95% confidence bounds under the null hypothesis, computed by Monte Carlo technique using B = 100 simulations of the Poisson process. (b) Transform of the Kmm function; white line is the theoretical value of the function under the null hypothesis of random labelling; grey area is the 95% confidence bounds under the null hypothesis, computed by Monte Carlo technique using B = 100 random permutations of the marks (point locations being invariant).

1. Clustering in the range 15–60 m is due to the smallest trees (with a diameter c. < 13 cm). 2. Regularity below 6 m is due to medium-sized and large trees (with a diameter between 13 and 45 cm). 3. The largest trees (diameter > 45 cm) are distributed at random. Figure 3b shows that random labelling is not verified for distances between 6 and 13 m. In this range of distances, Kmm is significantly below K. This decrease of Kmm is a consequence of the relationship between tree size and spatial pattern (i.e. smaller trees are clustered and larger trees are regularly spaced). The diameter distribution is approximately exponential with parameter μ = 0.0875 cm–1. The empirical variogram of diameter is flat (not shown here to conserve space) and can be


Dynamics and spatial pattern modelled as a pure nugget effect. Thus, diameter has no spatial structure on its own. As a consequence, the departure from random labelling that is shown by Fig. 3b also means that there is dependence between diameter and tree location.

PARAMETER VALUES

As diameter distribution was exponential with parameter μ, the target distribution in eqn 6 was: f(D) = μ exp[−μ(D − d1)] and the weights in eqn 9 were: wi = exp (μd1)[exp(−μdi) − exp(−μdi+1)] where di is the lower bound of the ith diameter class (with d1 = 10 cm and dk+1 = ∞). Moreover, as the diameter variogram at Paracou was flat, the covariance function of the random field Zi that describes the spatial distribution of diameters in the ith diameter class was set equal to a nugget covariance function. Hence the marks in the ith class were independently and identically drawn according to an exponential distribution with parameter μ truncated to this class. On the basis of the spatial pattern of trees at Paracou, we took k = 3 diameter classes for the pattern-driven model. Moreover, a Matérn process was used for the first diameter class, a SSI process was used for the second class, and a Poisson process was used for the third class. The patterndriven model thus had eight parameters whereas the processbased model had seven parameters. Table 1 lists the values of the parameters. The roles of the parameters γ, a and d0 of the process-based model could be intuitively understood as follows: γ controls the strength of clustering. The higher it is, the higher the maximum of the pair correlation function of the process is (this maximum is obtained when the distance tends to ρ+). The parameter a controls the strength of regularity. The higher it is, the smaller the minimum of the pair correlation function is (this minimum is obtained when the distance tends to zero). Finally, the diameter d0 controls the diameter range

Table 1. List of the parameters of the models fitted to plot 1 of Paracou in 1984

Symbol

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at which the pattern switches from clustering to regularity. Given γ and a, the smaller d0 is, the sooner regularity appears. The estimated value of d0 implies that there are on average N = 8437 trees with diameter greater than d0 in a simulated stand (see eqn 7). To get an idea of what represents the value of the parameter a in the process-based model, one can compute the value of the growth reducer exp(−aL) using the mean field approximation (which is equivalent to the random labelling hypothesis plus the hypothesis of random location, see Verzelen et al. 2006). The relationship between the growth reducer and the diameter of a tree then is a Gompertz function. Under these hypotheses, a tree with diameter 7.5 cm would have its growth equal to 50% of its potential growth. This increases to 95% for diameter 37.2 cm.

ADEQUACY OF THE MODELS TO PARACOU

Figure 4 compares the estimates of K and Kmm at Paracou to their predictions according to the pattern-driven model. The pattern-driven model is consistent with the Paracou pattern according to K. However the pattern-driven model underestimates Kmm around 35 m. Figure 5 compares the estimates of K and Kmm at Paracou to their predictions according to the process-based model. The process-based model is consistent with the Paracou pattern according both to K and Kmm. Table 2 gives the SSD, SEV and MSSD of K and Kmm for each model. The homogeneous Poisson process with random labelling, which is the model with no spatial structure at all, is also included as a reference. According to the K function, the best model is the process-based model when looking at the SSD, and it is the pattern-driven model when looking at the MSSD. This means that the K function predicted by the process-based model is closer on average to the observed K function than that predicted by the pattern-driven model, but it also has a higher variability than that of the pattern-driven model. As a comparison, the K function predicted by the Poisson process is much less variable but is also much farther on average from the observed K function. According to the

Definition

For both models (stand characteristics) λ Intensity (or tree density) μ Parameter of diameter distribution Pattern-driven model R Dispersion radius (Matérn) γ Fecundity (Matérn) ρ Inhibition distance (SSI) ν Survival parameter (SSI) d2 Diameter breakpoint between first and second class d3 Diameter breakpoint between second and third class Process-based model ρ Competition radius a Strength of competition Minimum diameter for recruitment d0 R Dispersion radius (Matérn) γ Fecundity (Matérn) © 2008 The Authors. Journal compilation © 2008 British Ecological Society, Journal of Ecology, 97, 97–108

Value

0.0614 0.0875

Unit

m–2 cm–1

30 17.57 6 0.2424 13 45

m m − cm cm

6 0.08 1 30 30

m − cm m −

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Fig. 4. Comparison between the second-order characteristics of the spatial pattern of trees at Paracou in plot 1 in 1984 and those predicted by the pattern-driven model: (a) Besag’s transform of Ripley’s K function; (b) transform of the Kmm function. Black line: estimate of the function from Paracou data. White line: prediction by the pattern-driven model. Grey area: 95% confidence bounds according to the pattern-driven model, computed by Monte Carlo simulation techniques using B = 100 replicates.

Table 2. Sum of squared differences (SSD), sum of the empirical variance (SEV) and mean sum of squared differences (MSSD) of K and Kmm for the pattern-driven model, for the process-based model, and for the homogeneous Poisson process, using rmax = 60 m SSD Model

Function

×10–4 (m5)

Pattern-driven Process-based Poisson Pattern-driven Process-based Poisson

K K K Kmm Kmm Kmm

0.48 0.32 10.69 5.14 0.79 8.34

SEV

MSSD

5.75 7.38 1.12 2.90 5.46 1.47

6.23 7.70 11.80 8.04 6.25 9.81

Fig. 5. Comparison between the second-order characteristics of the spatial pattern of trees at Paracou in plot 1 in 1984 and those predicted by the process-based model: (a) Besag’s transform of Ripley’s K function; (b) transform of the Kmm function. Black line: estimate of the function from Paracou data. White line: prediction by the process-based model. Grey area: 95% confidence bounds according to the process-based model, computed by Monte Carlo simulation techniques using B = 100 replicates.

Kmm function, the best model is the process-based model, whether we look at the SSD or the MSSD. The Kmm predicted by the pattern-driven model is again less variable than that predicted by the process-based model, but this time the gain in variability does not compensate for the worse fitting on average. These model comparisons do not take into account the number of parameters of the models, whereas the process-based model has one parameter less than the pattern-driven model.

MODELS PREDICTIONS

Assuming that the models are valid, predictions can be made regarding the changes of the spatial pattern of trees as the characteristics of the forest stand vary. Two characteristics of the forest stand are considered here: tree density λ, and mean


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Discussion

Fig. 6. Prediction of Besag’s transform of Ripley’s K function according to (a) the process-based model and (b) the pattern-driven model when stand characteristics (tree density or mean diameter) vary. Solid line: stand characteristics are those of Paracou; dashed line: tree density is half that of Paracou; dotted line: mean diameter is 1.5 times that of Paracou.

diameter (that is equal to 1/μ + 10 cm). Figure 6 shows the changes of Ripley’s K function for both models when tree density is divided by two while keeping the same mean diameter, or when μ is divided by two (which corresponds to an increase of the mean diameter from 21.4 to 32.9 cm) while keeping the same tree density. For both models, the decrease of tree density brings a strengthening of clustering in the range 15–60 m, with a stronger reaction for the process-based model than for the pattern-driven model. However the two models have opposite reactions to an increase of the mean diameter: for the patterndriven model, it brings a mitigation of clustering in the range 15–60 m, whereas for the process-based model it brings a strengthening of it. The regularity below 6 m for the patterndriven model is almost unaffected by the change of tree density or mean diameter. On the contrary, both the decrease of tree density and the increase of mean diameter bring a weaker regularity below 6 m for the process-based model.

Both the pattern-driven model, that relies on a decomposition of the observed pattern in a weighted superimposition of clustered, regular and random sub-patterns, and the processbased model, that mimics the biological processes of forest dynamics, are able to reproduce the pattern observed at Paracou. The process-based model provides a slightly better fit to the data. As further it has one parameter less than the pattern-driven model, Occam’s razor would suggest to use the process-based model rather than the pattern-driven model. The two models give different insights into the biological processes that drive forest dynamics. As the point processes (Matérn, SSI and Poisson) that compose the pattern-driven model are basically used to draw the pattern towards clustering, regularity or randomness, this model offers more a description of the spatial pattern at Paracou than an explanation of the processes that built it. Nevertheless, a biological interpretation can be given to each component point process. The SSI process, that is similar to soft-core processes in that it models the interaction between trees using an influence zone (here a disk with radius ρ/2) around each one, accounts for competition. This influence zone is one of the competition indices used in forestry to predict tree growth (Biging & Dobbertin 1992). More generally, there is an analogy between the competition indices used to predict tree growth and the interaction potentials used in Markov processes (Comas & Mateu 2007). An important point here is that the competition index used by the SSI process is symmetric, in the sense that each tree influences its neighbours irrespective of sizes (Caplat et al. 2008). Moreover the competition index acts on the deletion of an event (and its immediate relocation somewhere else), which corresponds to mortality. The corresponding biological interpretation is that mortality is density-dependent. The Matérn process belongs to the family of Neyman-Scott processes and thus inherits from them a dual interpretation. A Neyman-Scott process can be simulated by dispersing daughter points around parents points, thus mimicking the biological process of reproduction and dispersion of offspring around seed bearers. In addition, Neyman-Scott processes belong to the larger family of Cox processes (Møller & Waagepetersen 2004, p. 61) and can also be simulated, like any Cox process, by drawing a random field that gives the intensity of a heterogeneous Poisson process. The corresponding biological interpretation is that of an heterogeneous environment (different soil conditions, for instance) that conditions the density of the species. Thus, the same spatial pattern could equally be interpreted as the result of biotic (seed dispersion) or abiotic (specific response to the environment) processes. As the biological interpretation is attached to the simulation algorithm of the process, the fact that the same process can be simulated in different ways, thus giving different (and potentially contradictory) biological interpretations, is a definitive limit to blind ecological inference. Contrary to the pattern-driven model, the process-based model offers a more straightforward biological interpretation. Although the process-based model generates point patterns


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that evolve through time, some care has to be taken when interpreting this temporal evolution. Transients have no meaning: they are simply a burn-in period to reach the stationary state, as is usual in Metropolis-Hastings-like algorithms. As for the temporal distribution of the spatial pattern in the stationary state, it simply corresponds to the distribution generated by the point process. Nevertheless, contrary to the pattern-driven process that generates a one time snapshot of tree locations, a large tree that contributes to regularity in the process-based model was once a small tree that contributed to clustering. This means that, with the process-based model, there is consistency between the one time snapshot of the spatial distribution of trees and the longitudinal temporal distribution of a given tree. The spatial pattern of trees at Paracou shows a shift from clustering to regularity as tree grows. This shift has been observed many times, either for trees or for plants, and either in natural or in experimental conditions (Ford & Diggle 1981; Sterner et al. 1986; Weiner et al. 2001; Stoll & Bergius 2005). The process-based model gives an explanation of this shift, as the outcome of asymmetric competition between individuals: a tree that grows under a larger tree has a reduced growth with respect to a tree of the same size growing in open conditions. As a result, the former tree will stay longer in the same size class than the latter. Starting from a cohort of suppressed trees and a cohort of dominant trees of the same size, it means that a lesser proportion of the former will reach a given diameter than the latter, because mortality rate depends on diameter only. Hence trees growing near larger trees will tend to decrease in proportion in their size class as diameter increases, and their spatial pattern will become more and more regular. Thus a regular pattern can be the outcome of a density-dependent mortality, as hypothesized in the JanzenConnell model (Clark & Clark 1984; HilleRisLambers et al. 2002) and as implemented in the SSI process, as well as the outcome of a density-independent mortality coupled with a density-dependent growth, as implemented in the processbased model (He & Duncan 2000). The shift towards regularity of the spatial pattern of trees also corresponds to a shift towards lower values of the competition index. Then growth rates are higher than expected under the null hypothesis of mean-field (that is random labelling plus random location) (Turnbull et al. 2007). In other words, competition induces a spatial self-organization whose feedback effect is to limit competition. The spatial pattern observed at Paracou seems to be common for natural tropical forests, where large trees organize themselves in a regular way. This pattern has been observed elsewhere in Amazonia (Neeff et al. 2005), India (Pélissier 1998) or Central Africa (Picard et al. 2008). This pattern disappears when considering each species separately. This means that, even though species have different dynamics and ecological strategies, they collectively (but maybe not equally) contribute to a common spatial pattern. And, as the processbased model proves it, separating species is not required to explain this common pattern. The regular pattern of large trees is explained by both models as an outcome of competition.

The clustered pattern of small trees is less obvious to explain. Dispersion of offspring around seed bearer cannot be invoked in this context where species is not taken into account. A possible explanation is the heterogeneity of the environment, assuming that small trees have a stronger response to the heterogeneity of the environment than large trees. A subsequent study will test if the spatial pattern of soil characteristics coincide with that of saplings. Another possible explanation is that current clusters of small trees are the remnants of past windfalls that would have created large canopy gaps followed by an intense regeneration. The process-based model and the pattern-driven model can be used to predict changes in spatial pattern that would result from changes in stand characteristics (tree density or mean diameter). As the predicted changes are qualitatively different depending on the model used, this could be used to test the models using additional data. Further extension of the models can be proposed in connection with more realistic biological interpretations. For instance, the pattern-driven model presently relies on a symmetric competition index to model competition and does not define interactions between size classes. The former limitation could be removed by replacing the SSI process by a Markov process with an interaction potential that depends asymmetrically on marks (Renshaw & Särkkä 2001). The latter could be removed using a hierarchical process. Assuming for instance that, due to asymmetric competition, the spatial pattern of trees with diameter D depends on trees with diameters > D but not on trees with diameter ≤ D, one could define Φk, then Φk−1 conditionally to Φk, then Φk−2 conditionally to Φk and Φk−1, and so on (Hanus et al. 1998; Högmander & Särkkä 1999). Another approach to explore is the link between simulation algorithms and Markov point processes (Comas & Mateu 2007). At each time step (say, 1 year), the simulation algorithm of the process-based process erases some points (dead trees) and replaces them with new points (recruited trees). Continuing the analogy with the biological processes of forest dynamics, the number of dead (or recruited) trees during one year is nothing else but the annual cumulative sum of individual events (death in this case). From a statistical point of view, the process-based model can thus be seen as the balance at periodic time steps of a birth-and-death process. Now, birth-and-death processes are the basis of some algorithms to simulate Markov point processes (Møller & Waagepetersen 2004). A bridge may thus be built using birth-and-death processes between Markov point processes and simulation algorithm such as the process-based model.

Acknowledgements The authors are thankful to Rachid Senoussi for fruitful suggestions at the beginning of this work. Authors are also grateful to two anonymous referees for their helpful comments.

References Batista, J.L.F. & Maguire, D.A. (1998) Modeling the spatial structure of tropical forests. Forest Ecology and Management, 110, 293–314.


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Supporting Information Additional Supporting Information may be found in the online version of this article: Appendix S1. Pair correlation function for the pattern-driven model. Please note: Wiley-Blackwell are not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the article.

Received 6 February 2008; accepted 09 September 2008 Handling Editor: Frank Gilliam
