Improving the effectiveness of angular dispersion in plant neighbourhood models

Ecological Modelling 221 (2010) 1649–1654

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Improving the effectiveness of angular dispersion in plant neighbourhood models M.L.A. Richards a,∗ , M.J. Aitkenhead b , A.J.S. McDonald a a b

School of Biological Sciences, University of Aberdeen, Cruickshank Building, St Machar Drive, Aberdeen AB24 3UU, Scotland, UK The Macaulay Institute, Craigiebuckler, Aberdeen AB15 8QH, Scotland, UK

a r t i c l e

i n f o

Article history: Received 10 November 2009 Received in revised form 24 February 2010 Accepted 8 March 2010 Available online 3 May 2010 Keywords: Angular dispersion Competition index Neighbourhood models Spatial arrangement

a b s t r a c t Spatial arrangement can be an important factor affecting competition among plants. We evaluated three ways to improve the effectiveness of angular dispersion (AD) for describing spatial arrangement in plant neighbourhood models. First, we modified Zar’s (1974) AD formula by weighting each neighbour by its competitive influence. We calculated this using two different competition indices to derive an AD of competitive influence, rather than of equally weighted plant locations, around a subject plant. Secondly, we constrained the effect of AD on the neighbourhood model using an optimised parameter that defines the minimum value that AD can adopt. Thirdly, we included the direction in which competition is concentrated (the mean azimuth of the weighted AD) in the growth models. These developments were evaluated within a radial growth model of Scots pine and birch growing in semi-natural, spatially heterogeneous forest. Weighted AD resulted in significant improvements in predicted radial growth of target trees over the traditional measure of AD. The optimised parameter that defines the minimum value of AD consistently evolved values significantly higher than zero. This suggests that clumped and dispersed neighbourhoods do not differ in their negative effects on a subject tree as much as expected. The inclusion of directional components of the weighted AD did not improve the accuracy of the growth models. Weighting of the angular dispersion of neighbours improved the performance of local competition models. © 2010 Elsevier B.V. All rights reserved.

1. Introduction The spatial arrangement of plants can be an important factor affecting individual plant performance (Auld et al., 1983; Fowler, 1984; Silander and Pacala, 1985; Lindquist et al., 1994). Therefore, to improve the performance of plant growth models, spatial arrangement should be adequately assessed. The spatial arrangement of competitors can vary from uniformly dispersed to highly clumped. The cumulative competitive pressure exerted by neighbours is likely to be reduced when they have a clumped rather than dispersed spatial arrangement. The mechanistic basis for this is that plants tend, through their morphological plasticity, to grow in the direction of least competition (Ross and Harper, 1972). Also, interactions amongst the competitors themselves may affect subject plant performance: the influence of a competitor may be reduced when there is another competitor between it and the focal plant (Bergelson, 1993). Spatial arrangement in plant neighbourhood models is often described using angular dispersion (AD), introduced by Zar (1974), which is a measure of the aggregation of neighbours based on the

∗ Corresponding author. Tel.: +44 1224 273810; fax: +44 1224 272703. E-mail addresses: [email protected], [email protected] (M.L.A. Richards). 0304-3800/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2010.03.006

angles separating them. AD can be an important factor affecting plant performance (Mack and Harper, 1977; Silander and Pacala, 1985; Lindquist et al., 1994), but this is not always the case (Waller, 1981; Weiner, 1984; Vila et al., 1998; Wagner and Radosevich, 1998; Somanathan and Borges, 2000). AD is expressed as 1 − AC, where AC is calculated by the formula



 n

AC =

i=1

cos ai n

 n

2 +

i=1

sin ai

n

2 (1)

where ai is the azimuth of neighbour plant i measured from the target plant and n is the number of neighbours in the neighbourhood. AC is the angular concentration of the neighbourhood which can vary between 0 (a uniformly dispersed neighbourhood) and 1 (a highly concentrated or aggregated neighbourhood in which all neighbours are located in the same direction). A potentially important limitation of this formula for plant neighbourhood modelling is that each neighbour is given an equal weighting in the calculation. However, plant neighbourhoods are frequently composed of unevenly sized individuals of mixed species which are located at varying distances from the subject plant. In these cases the competitive influence of individual neighbours on the subject plant will vary widely. Therefore, it may be more relevant to weight each neighbour to reflect their individual influence on the subject tree and then to calculate the AD based

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on these weighted neighbour locations. Each neighbour can be weighted using a competition index which can then be integrated with the measure of angular dispersion (provided the competition index itself does not utilise angular information associated with position of the neighbours since this would lead to a circular model). Fig. 1 illustrates the problem using two different spatial arrangements of neighbours of various sizes and distances from the subject plant. The left diagram of Fig. 1A shows a target plant with two equal-sized neighbours, situated at different distances from the subject plant. The middle and right diagrams show the same neighbourhood represented by the traditional measure of AD and a weighted AD, respectively. Because AD is calculated from the angles separating neighbours, the neighbourhood can be represented as a series of points located on the rim of a disc. Each point has a weight, allowing the centre of gravity of the disc to be determined. Each disc balances at the point marked by an X in the diagrams, which is calculated as follows: the azimuth of point X is the mean azimuth of the neighbourhood and the distance of X from the centre of the disc is the value of AC from Eq. (1) (0 is the centre of the disc, 1 is on the rim of the disc). The centre of gravity of the neighbourhood calculated using the traditional AD is at the centre of the disc (indicating a uniformly dispersed neighbourhood) since each neighbour is given an equal weight regardless of its distance from the subject plant. In contrast, the weighted AD has a centre of gravity located near the rim of the disc (indicating a more concentrated AD) in the direction of the closest neighbour because this neighbour is given a greater weight as it has a greater competitive influence. Fig. 1B shows a neighbourhood of six plants, each of equal distance from the subject plant but of different sizes. The traditional AD results in a uniformly distributed neighbourhood whereas the weighted AD results in an aggregated distribution because the larger neighbours, which are all located on one side of the subject plant, have a greater competitive influence. Previous implementations of AD in neighbourhood models have allowed the value of AD to vary between 0 (complete aggregation of neighbours) and 1 (complete dispersion of neighbours). Typically, this value is then used to multiply a separate competition index to determine the overall competition index value for the subject plant (e.g. Silander and Pacala, 1985; Lindquist et al., 1994). This usage means that aggregation of neighbours (i.e. values of AD

Fig. 1. (A and B) Example spatial arrangements of plant neighbours (filled circles) around a subject plant (unfilled circles) and representations of their angular dispersion (AD) calculated using the traditional measure of AD and an AD in which each neighbour is weighted by its competitive influence on the subject plant. The size of the circles represents the size of the plant. X marks the “centre of gravity” of each angular dispersion. An X located in the centre of the circle represents a perfectly uniform AD. The closer the X is to the edge of the circle the more concentrated the AD. See text for more details.

tending towards zero) results in a very large reduction in the overall value of the competition index. This assumes that the AD of a neighbourhood has a very large impact on its competitive influence on the subject plant. The validity of this assumption can be addressed through the use of numerical optimisation techniques to determine an appropriate minimum value for the AD term. The minimum value would serve to constrain the overall effect which AD can have on the competition index value of the neighbourhood. This has apparently not been reported. AD can be used to determine the direction in which competition is most concentrated. Canham (1988) found that over a growing season, understory locations north of forest gaps with a radius of 5 m in the northern hemisphere receive up to eight times as much light as locations south of the gap, and that this directional effect increases with latitude. This suggests that knowledge of the direction in which competition is concentrated may improve growth model accuracy. The direction in which competition is most concentrated can be expressed as the mean azimuth of the angular dispersion, but has apparently never been used in plant growth modelling. In this study we assess three ways that may improve the effectiveness of AD measures in plant neighbourhood models. First, we introduce a weighted AD that describes the angular distribution of competition intensity around a subject plant. The weighted and traditional AD measures are integrated with numerically optimised competition indices which are then assessed for their contribution to a model of tree radial growth. Secondly, we introduce an optimised parameter which defines the minimum value that AD can adopt, in order to constrain the effect of AD on the overall competition index value. Thirdly, we investigate the importance of the direction in which competition is concentrated (the mean azimuth of the weighted AD) to the predictive accuracy of a tree radial growth model.

2. Methods 2.1. Data The data for the study came from semi-natural, spatially heterogeneous, even and uneven-aged forest stands in Glen Affric (57◦ 17 N, 05◦ 01 W), in the north-west Highlands of Scotland. The forest is dominated by Scots pine (Pinus sylvestris L.) and to a lesser extent silver birch (Betula pendula Roth) and downy birch (Betula pubescens Ehrh.). The two birch species are represented as a single entity (hereafter referred to as simply ‘birch’), because identification to species level can be problematic. The pine data used for this study are the same as those used by Richards et al. (2008). The DBH and annual radial growth of each subject tree were measured. Radial growth for the year 2003 was determined with a microscope from cores (1–3 were taken depending on the size of the tree) bored at 1.3 m above soil level. One hundred and eight Scots pine and 51 birch subject trees were selected randomly over a wide area and from a range of forest types (pine-dominated, birchdominated or mixed pine and birch). A national grid reference for each subject tree was determined using GPS and was used to obtain environmental data for each tree from GIS datasets. The Ordnance Survey 10m digital elevation model (DEM) was used to generate the GIS datasets. For each tree location, altitude, aspect, exposure (windiness) and soil moisture index were determined. Site exposure for each tree was calculated using the regression model of Quine and White (1994). Aspect and soil moisture index were calculated using the Terrain Analysis System (Lindsay, 2005). The soil moisture status for each tree was estimated using Beven and Kirkby’s (1979) ln(a/tan ˇ) topographic index of soil moisture.

M.L.A. Richards et al. / Ecological Modelling 221 (2010) 1649–1654

All neighbours with an elevation angle greater than 35◦ were measured. Elevation angle was taken as the vertical angle subtending a point 1 m above the base of the subject tree and the top-centre of the neighbour’s crown. The elevation angle, azimuth, distance, DBH and species of each neighbour were recorded. The vast majority of neighbours were pine and birch, although occasional individuals of other broadleaf species were present.

and

n sin i Ci i=1 Yj = n

where Ci , the competition value for each neighbour i, calculated using the formula of the competition index to which WAD is being applied. So, Ci for a conspecific neighbour in CI1-WAD is calculated using Heygi’s (1974) index as follows:



2.3. Measures of angular dispersion

n

Xj = and Yj =

cos i i=1 n

n i=1

sin i

n

(2)

(3)

where  i is the azimuth of neighbour i, measured from the location of the subject tree and n is the total number of neighbours. Weighted angular dispersion, WAD, used in the CI1-WAD and CI2WAD indices is calculated by modifying Zar’s (1974) formula. The rectangular coordinates of the mean competition azimuth are now calculated as Xj =

n cos i Ci i=1 n C i=1 i

Ci1 =

(4)

Di1

a 

Dj



1

(6)

Sib

1

Ci for a heterospecific neighbour in CI1-WAD is calculated as follows:



Ci2 =

Di2

a 

Dj



1

IC

Sib

(7)

2

Ci for a conspecific neighbour in CI2-WAD is calculated as follows:



Ci =

EAi1 − EAmin

c

(8)

90 − EAmin

Ci for a heterospecific neighbour in CI2-WAD is calculated as follows:



Ci2 =

EAi2 − EAmin 90 − EAmin

c

IC

(9)

The remaining calculations for both the traditional and weighted measures of angular dispersion follow Zar’s (1974) original method. All that is required is to substitute in the appropriate values of X and Y from Eqs. (2) to (5) above. The angular concentration (AC) is found by the formula: ACj =



X2 + Y 2

(10)

Angular (and weighted angular, WADj ) dispersion is then calculated as ADj = 1 − ACj

(11)

¯ and sine (sin ) ¯ of the angular The mean azimuth cosine (cos ) concentration and weighted angular concentration are calculated by the formulas: cos ¯ =

Angular dispersion AD used in CI1-AD and CI2-AD is calculated by Zar’s (1974) method. First, the rectangular coordinates X and Y of the mean neighbour azimuth are calculated

(5)

C i=1 i

2.2. Competition indices The traditional AD and new weighted AD measures were integrated with two different competition indices to form larger, composite indices (Table 1). Competition indices CI1, CI1-AD and CI1-WAD are based on the commonly used size-ratio distanceweighted index of Heygi (1974). Indices CI2, CI2-AD and CI2-WAD are based on Pukkala and Kolström’s (1987) elevation angle index. This index was chosen because the elevation angle captures information on both the size and distance of the neighbour in a single value and because this index has performed well in previous studies (Miina and Pukkala, 2000; Richards et al., 2008). In total, six optimisable parameters (IC, a, b, c, d, and EAmin ) were applied to the competition indices, though not all to each index (see Table 1). These parameters are described below. Both competition indices were modified to separate intra- and inter-specific competition. The inter-specific portion of each index is weighted by an optimised coefficient of inter-specific competition, IC. Parameters a, b, c and d were also added to allow the functional forms of the indices to be numerically optimised. The AD component (ADj and WADj ) of each index is normalised to a value between d and 1. If highly clumped neighbourhoods (a low value of AD) have a much lower competitive effect than equivalent, uniformly dispersed neighbourhoods, d should evolve a value close to zero and greatly reduce the overall competition index value. However, if AD has little effect on the subject tree, d should evolve a high value so the angular dispersion term will have little effect on the overall competition index value. We also introduced a parameter EAmin , that specifies the minimum elevation angle a neighbour must have to be included in the competition index and allowed this to be optimised. We also use EAmin to normalise the elevation angles of each competitor in Pukkala and Kolström’s (1987) index (CI2, CI2-AD and CI2-WAD in Table 1). All the optimisation parameters were optimised using the simulated annealing method described in Richards et al. (2008).

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Xj ACj

(12)

and sin ¯ =

Yj ACj

(13)

respectively, where Xj and Yj are rectangular coordinates from Eqs. (2) and (3), or (4) and (5), respectively, depending on whether angular dispersion or weighted angular dispersion is being used. ACj is the angular concentration from Eq. (10). The direction compo¯ are used to describe the nents of angular dispersion, cos ¯ and sin , directional distribution of competition around a subject tree. 2.4. Modelling procedure Here we provide a brief description of the modelling procedure. A more detailed description of the process is given in Richards et al. (2008). The models were determined by linking a simulating annealing program and an artificial neural network (ANN). The simulated annealing algorithm subjects the optimisable competition index parameters (which are given initial values) to evolutionary

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M.L.A. Richards et al. / Ecological Modelling 221 (2010) 1649–1654 Table 1 Definitions of competition indices. Index

Definitiona

CI1

Sum of ratios between the competitor and subject tree DBH (modified by a), multiplied by the inverse of the distance separating the competitor and subject

tree (modified by b): CI1j =

n1 

a  Di 1 Sb

1

Dj

+

n1

EAi

1

c

−EAmin

n2

EAi

+

90−EAmin

2

IC

i2 =1

n1 

a  Di 1 Sb

1

Dj

n2 

a  Di

+

1 Sb

2

Dj

i1

i1 =1

(d + ADj (1 − d))

IC

i2

i2 =1

As CI2 but multiplied by angular dispersion (constrained by optimisation parameter d):   CI2ADj =

n1

EAi

1

−EAmin

c

n2

EAi

+

90−EAmin

2

−EAmin

c

(d + ADj (1 − d))

IC

90−EAmin

i1 =1

i2 =1

As CI1 but multiplied by weighted angular  dispersion:



CI1WADj =



a

n1

Di

1

Dj i1 =1

CI2-WAD

c

As CI1 but multiplied by angular dispersion (constrained by optimisation parameter d):   CI1ADj =

CI1-WAD

−EAmin

90−EAmin

i1 =1

CI2-AD

IC

i2

i2 =1

Sum of normalized competitor elevation angles (modified

by optimisation parameter c): CI2j =

CI1-AD

1 Sb

2

Dj

i1

i1 =1

CI2

n2 

a  Di

1 Sb

+

i1

 n2 

a  Di 1 Sb

2

Dj

As CI2 but multiplied by weighted angular  dispersion: CI2WADj =

n1

EAi

1

−EAmin

90−EAmin

c

+

i1 =1

n2

EAi

2

IC

(d + WADj (1 − d))

i2

i2 =1

−EAmin

90−EAmin



c IC

(d + WADj (1 − d))

i2 =1

a Di = diameter at breast height of competitor tree i (cm); Dj = diameter at breast height of the subject tree j (cm); Si = distance between subject tree j and neighbouring tree i (m); EAi = elevation angle of competitor tree i (degrees); EAmin = optimisable parameter; minimum elevation angle used to select competitors (degrees); ADj = angular dispersion of competitors about subject tree j; WADj = weighted angular dispersion of competitors about subject tree j; 1, 2 = conspecific and heterospecific neighbours, respectively (relative to the subject tree); IC = optimisable coefficient of inter-specific competition; n = number of neighbours with an elevation angle > EAmin ; a, b, c = optimisable parameters; d = optimisable parameter that constrains the angular dispersion component between d and 1.

pressures in order to optimise each competition index for the prediction of annual radial growth. The competition index value for each subject tree is then calculated. An ANN program (written by the authors) accepts as input the competition index value and five other input variables (DBH, soil wetness, windiness, altitude and aspect) together with the corresponding radial growth output for each subject tree and trains an ANN (for 2000 training cycles). The network architecture used was that of a fully-connected feedforward neural network, with topology 8:16:16:1 (i.e. 8 input nodes, 16 nodes in each of two hidden layers, and a single output node). This architecture was used based on trial and error. The training algorithm used was backpropagation, which can be used to train a neural network rapidly and accurately. It was found that 2000 training cycles, with a learning rate of 0.1 for connection weight adjustment, produced a stable neural network that was not overfitted to the data. Following training, the ANN is then tested against independent data and the accuracy of the growth predictions is fed back to the simulated annealing algorithm in an iterative process. A total of 1000 annealing iterations was found to produce a fullyannealed system, i.e. no further improvement was made. Prior to each evolutionary process 80% of the subject trees were randomly selected for training the ANN. The remaining 20% were used to test the accuracy of the trained ANN. An ANN was employed because the competition-growth model involves many input variables that may exhibit subtle but significant nonlinear effects on growth which cannot be properly accommodated by conventional statistical techniques. For complex systems such as this ANNs have, in general, benn found to outperform conventional regression techniques (e.g. Lek et al., 1996; Razi and Athappilly, 2005; Yu et al., 2006). In addition, ANNs provide a flexible, self-adjusting approach that is particularly valuable

when the functional relationship between independent and dependent variables are unknown, as is the case with spatial competition indices and tree growth. The result of the coupled simulated annealing and neural network process, which creates a directed walk through the parameter space, is that the parameters will eventually achieve a global (or nearly global) fitness maximum (i.e. they will adopt values that give the optimum, or near optimum, growth prediction accuracy). Convergence of the simulated annealing algorithm to the global optimum is not guaranteed so the optimisation process was carried out 10 times for each competition index-species combination (i.e. for CI1 for pine, CI1 for birch, CI1-AD for pine, CI1-AD for birch, etc.). In addition to these models, two other models were created using two additional input variables to the ANN that describe the directional distribution (DD) of competition around the subject tree (cos ¯ and sin ¯ from Eqs. (12) and (13)). This was done to investigate the significance of the direction components in the growth model. These two models use the competition indices CI1-WAD and CI2-WAD (see Table 1) both as input variables to the ANN and to calculate the directional distribution of competition. The two models are referred to as CI1-WAD-DD and CI2-WAD-DD, respectively. To compare the accuracy of the growth model with and without the angular and weighted angular dispersion components and the directional distribution components, the root mean square error reduction (RMSER) relative to the corresponding model without those components was calculated using:



RMSER = 1 −

RMSE  2

RMSE1

× 100

(14)

where RMSE1 is the root mean square error of the model using CI1 or CI2 and RMSE2 is the root mean square error of the model using

M.L.A. Richards et al. / Ecological Modelling 221 (2010) 1649–1654 Table 2 Root mean square error (RMSE) and root mean square error reduction (RMSER) relative to CI1. Competition

Scots pine models

Birch models

Index

RMSE (mm)

RMSER (%)

RMSE (mm)

RMSER (%)

CI1 CI1-AD CI1-WAD CI1-WAD-DD

0.194 0.170 0.143 0.137

0 12.4 26.3 29.4

0.104 0.087 0.086 0.090

0 16.3 17.3 13.5

Table 3 Root mean square error (RMSE) and root mean square error reduction (RMSER) relative to CI2. Competition

Scots pine models

Birch models

Index

RMSE (mm)

RMSER (%)

RMSE (mm)

RMSER (%)

CI2 CI2-AD CI2-WAD CI2-WAD-DD

0.166 0.155 0.126 0.131

0 6.6 24.1 21.1

0.096 0.080 0.068 0.073

0 16.7 29.2 24.0

Table 4 Mean and standard error of the optimised values of parameter d (which defines the minimum value that the angular dispersion component of the competition indices can take); n = 10. Competition

Scots pine models

Birch models

Index

Mean

S.E.

Mean

S.E.

CI1-AD CI2-AD CI1-WAD CI2-WAD CI1-WAD-DD CI2-WAD-DD

0.56 0.63 0.57 0.55 0.58 0.61

0.03 0.04 0.03 0.03 0.04 0.03

0.38 0.41 0.34 0.31 0.40 0.32

0.02 0.02 0.02 0.02 0.02 0.02

either CI1-AD, CI1-WAD, CI1-WAD-DD or CI2-AD, CI2-WAD, CI2WAD-DD, respectively. 3. Results The RMSER for the CI1 and CI2 based models are shown in Tables 2 and 3, respectively. The inclusion of the traditional angular dispersion component or the weighted angular dispersion component always resulted in a reduction in RMSE. The weighted angular dispersion resulted in greater reductions in mean square error compared to the traditional angular dispersion in all cases, although the difference between CI1-AD and CI1-WAD for birch was only marginal (1%). The highest RMSER resulting from the inclusion of the traditional angular dispersion was 16.7% (CI2-AD for birch) compared to 29.2% for the weighted angular dispersion (CI2-WAD for birch). The inclusion of the directional distribution input variables resulted in reduced model accuracy in three out of the four models, with only a marginal improvement being found in CI1WAD-DD for pine. The mean optimised values and standard errors of parameter d are shown in Table 4. The mean optimised value of parameter d ranged from 0.55 to 0.63 for the pine models and 0.31–0.41 for the birch models. There are no obvious differences in the optimised values of d resulting from the type of base competition index used (i.e. CI1 or CI2). 4. Discussion The weighted angular dispersion (WAD) component consistently resulted in greater improvements in model accuracy than the commonly used measure of angular dispersion (AD) introduced by Zar (1974). This suggests that the AD of competitive influ-

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ence rather than the AD of unweighted neighbour locations can be more appropriate for modelling plant performance. Moreover, the competition indices that utilise WAD do not require any more data than the equivalent index using the traditional measure of AD. We tested WAD by combining it with two different competition indices but many other competition indices could be used in their place. There was no obvious difference in the performance of WAD resulting from the type of competition index it was combined with. However, we expect that the performance of WAD will depend on how effectively the competition index used to calculate the weighting summarises the competitive influence of each neighbour. We assessed the performance of the AD measures using numerically optimised competition indices because the optimisation process reduces the uncertainty in finding the best specification of a particular index (Miina and Pukkala, 2000; Richards et al., 2008). Previous implementations of AD in plant neighbourhood models allowed the AD value to vary between 0 and 1. This value is usually used to weight another competition index (as we have done here), making the assumption that highly clumped neighbourhoods (i.e. with an AD close to zero) will result in a large reduction in the competitive influence of the neighbourhood on the subject plant. In order to address this assumption we added an optimisable parameter (d) to the AD components which defines the minimum value the AD term can take. The simulated annealing algorithm used for the optimisation process evolved values of d significantly higher than zero (0.55–0.63 for pine and 0.31–0.41 for birch), suggesting that clumped neighbourhoods exert a moderate degree of competition on the subject plant. This finding shows that the usual practice of allowing AD to vary between 0 and 1 may underestimate the competitive effect of clumped neighbourhoods (i.e. those with an AD tending towards zero) on subject plants. Puettmann et al. (1993) showed that the AD method of Zar (1974) used in this study can overestimate dispersion when neighbours are located in two or more clumps. To overcome this limitation, Puettmann et al. (1993) developed an alternative measure of AD based on the variance of the differences between the azimuth of neighbouring plants. We chose Zar’s method as the basis of the WAD measure because it is the most widely used method and because we were interested in analysing the direction in which competition is most concentrated, which cannot be determined using the method of Puettmann et al. At the relatively high latitude of the study site (57◦ 17 N) we expected a strong directional variation in above canopy light availability (see Canham, 1988) and hence a strong directional influence on the competitive effect of individual neighbours. However, the inclusion of the directional distribution input variables (cos ¯ and ¯ to the growth models did not improve model accuracy. Two sin ) possible reasons for this include (1) the measure of the directional distribution of competition used in this study may be inadequate, or (2) the directional distribution of competition is unimportant in this forest (e.g. competition may be dominated by below-ground interactions rather than for light). Acknowledgements We thank Forestry Commission Scotland for providing access to the Glen Affric Forest Reserve. We are grateful to the anonymous reviewers for their helpful comments. This study was funded by the Natural Environment Research Council. References Auld, B.A., Kemp, D.R., Medd, R.W., 1983. The influence of spatial arrangement on grain yield of wheat. Aust. J. Agric. Res. 34, 99–108.

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