Richard Law (tel. +44 1904 328586; fax +44 1904 328505; e-mail RL1@york.ac.uk).
1The effects of neighbours on the growth of individual plants are fundamental to dynamics in plant populations and can be described by means of mathematical functions, so-called competition kernels, in formal spatiotemporal models. Little is known about the form and components such functions should have.
2We evaluate some properties of kernel functions using data on the growth of Arabidopsis thaliana plants in replicated, even-aged stands of many individuals. Because of the essential non-independence of plant growth in stands, we employed a Bayesian hierarchical modelling approach to estimate values and uncertainties of kernel parameters in location-dependent models of interacting plants.
3During the experiment plant size and a simple measure of neighbourhood crowding became strongly correlated, plants tending to be small where local crowding was intense, indicating that local competition was an important process in the growth of the plants.
4Competitive interactions between plants of different sizes were strongly asymmetric, the larger individual acquiring a disproportionately greater share of resources. Competition increased with plant size and attenuated rapidly at distances of a few centimetres, but the exact shape of the attenuation function was less important.
5Kernel functions with the same kind of structural features were similar in their predictive ability. However, a simple zone-of-influence model, based on overlap of pairs of individuals, with competition favouring the larger individual, was arguably the most parsimonious.
6Neighbourhood competition in stands of even-aged plants may be successfully captured with relatively simple kernel functions. The results should inform and enhance the formal theory of spatiotemporal plant population and community dynamics. Bayesian hierarchical modelling is a powerful tool with which to analyse complex, spatially dependent data, and has potential as a widely applicable statistical approach for plant ecology.
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Plant interactions are predominantly local in space. When plants grow together, they compete with their immediate neighbours for limited resources above or below ground, or for both. Their fates are delicately coupled: how much each plant grows depends on its own size, the sizes of its neighbours and the distances to these neighbours. As the plants change in size, their continuing growth is affected by the size increments they and their neighbours have previously made. Such mutual dependence operates whether the plants are monospecific stands of crops, or wild plants growing in natural communities.
For several reasons, ecologists need to know more about this coupled growth. First, local spatial interactions are potentially important determinants of yield in agriculture and forestry (e.g. Baumann et al. 2000; Weiner, Griepentrog & Kristensen 2001). Secondly, plant growth is an important process in plant population and community dynamics, and yet barely dealt with in the formal spatiotemporal theory on this subject (e.g. Law & Dieckmann 2000; Bolker et al. 2003). At the heart of research on spatiotemporal dynamics are functions, which we call competition kernels, which describe how individual plants interact with their neighbours. With knowledge of these functions, ecologists would be better placed to answer some basic quantitative questions about competition: (i) How are the negative effects of competition partitioned among plants of different sizes? (ii) How do interaction strengths attenuate with distance? (iii) How is competition altered by growth of the plants? (iv) How do several neighbours combine their effects on the growth of a plant?
In this paper, we therefore examine the spatial development of experimental stands of Arabidopsis thaliana (L.) Heynh., and evaluate some properties that kernel functions might be expected to have, by comparing the functions with the observed growth of plants within stands. A. thaliana was chosen as a convenient experimental tool, and one for which some information on growth of interacting plants is already available (Purves & Law 2002; Stoll & Bergius 2005). This species has the advantages that rosette size provides a non-destructive measure of plant weight and that hundreds of plants can be grown together in relatively small amounts of space and time.
Doing work of this kind raises a basic statistical problem. As the growth of each plant is linked to the growth of other plants, the standard assumption of stochastic independence of sample units does not apply (Mitchell-Olds 1987). We therefore described these dependences explicitly by coupling the growth of individual plants through a dynamic growth model with neighbourhood interactions. We then estimated parameters of the growth model, given our experimental data, within a Bayesian framework. Classical statistics deal with likelihood of the data given a set of parameters within a model. This likelihood is typically maximized with regard to the parameters (maximum likelihood estimation). In contrast, Bayesian theory assumes the parameters to follow certain prior probability distributions based on background knowledge. This allows the calculation of posterior probability distributions, which describe the probability of parameters given the data. In Bayesian hierarchical modelling, this procedure is repeated on several levels where the prior distributions of the parameters are assumed to be driven by hyperparameters, which may again follow prior distributions. The hierarchical structure of the model reflects structures found in nature and makes this a naturally admissible modelling approach. Here, spatial structure and dynamics of neighbourhood interaction were explicitly included in the model, such that they did not cause a problem in the estimation procedure.
Although Bayesian data analysis is not yet widely used in ecology (Ellison 2004), recent applications of Bayesian techniques to spatial problems in plant ecology, including neighbourhood-dependent survival of trees in a tropical rain forest (Hubbell et al. 2001), tree fecundity (Clark et al. 2004) and species diversity (Gelfand et al. 2005), suggest these techniques have a lot of potential. We show in this paper that the techniques can be applied to quite intricate problems of the dynamics of neighbourhood-dependent plant growth, and generate insights into the quantitative dependences that would be hard to obtain by other means.
Materials and methods
The study was based on spatially referenced plants, grown together over time, the size of each plant being measured at short intervals to provide information on its growth trajectory. Knowing the location of the plants, the growth trajectory of each plant could then be investigated in the context of the size and location of its neighbours.
Two genotypes of A. thaliana with contrasting morphologies were selected in a preliminary test from a set obtained from the Nottingham Arabidopsis Stock Center (NASC). Both showed similarly slower rates of development than wild type but genotype 1 (NASC N334) had a more compact rosette than genotype 2 (NASC N443), which developed long petioles and loose rosettes from the earliest stages. The rosettes of genotype 1 eventually had diameters greater than those of genotype 2.
We used the maximal distance dmax through the rosette as a convenient non-destructive measure of plant size to follow individual plant growth over time. One may think of this measure in terms of a line drawn through the centre of the rosette and rotated until the distance across the rosette reaches its maximum. To scale this to above-ground dry weight, we carried out a preliminary experiment, growing 64 isolated plants of each of the two genotypes in a glasshouse. The plants were destructively harvested over the growth period, dried and weighed. There was a close relationship between the natural logarithm of above-ground plant weight (y) and the maximal distance dmax.
For genotype 1:
y = 2.42 loge(dmax) + 0.58 (r2 = 0.98)( eqn 1 )
For genotype 2:
y = 2.33 loge(dmax) + 0.68 (r2 = 0.98)( eqn 2 )
We used these functions to transform raw measures of rosette size to the logarithm of above-ground plant weight in the main experiment, weight being envisaged as multiples of a small quantity (1 µg), a scaling which ensured y > 0.
The main experiment followed the growth of plants living together in stands of many interacting individuals. Four replicate stands of each genotype in monoculture were established in polypropylene boxes of 60 × 40 × 12 cm3 filled with Levington F2s peat compost treated with Imidacloprid (Intercept 5GR, The Scotts Company, Marysville, OH 43041, USA). Seeds were sown at an average density of 4000 m−2 in computer-generated spatial point patterns with points located independently at random, using perforated transparent sheets. After emergence, the realized locations of the individuals were recorded. The plants were grown under controlled growth conditions at 23.5 °C during 16 hours of 215 µmol m−2 s−1 artificial light and at 21.7 °C during 8 hours of darkness. The plants were kept well watered and the average air humidity was 70%. The stands were frequently rotated to prevent environmental gradients in the growth rooms from affecting the results.
The growth of individuals was recorded in a central area of 29 × 29 cm2. Analyses were carried out on the plants in a central part, 21 × 21 cm2, the plants in the 4-cm boundary around this being used only as neighbours of central ones. The maximal distance through the rosette of each individual was measured at fixed census times, twice a week for the first 3 weeks after germination and approximately weekly for the next 3 weeks until the plants started to flower. During the experimental period, only 8 out of 2135 plants died. Another 10 plants remained much smaller than could be explained by competition alone but were present until the end. Both dead and small plants were treated as outliers and excluded from the analysis.
mathematical model of neighbour-dependent plant growth
where size is measured as the natural logarithm of above-ground dry weight, scaled to be positive, and the rate of change is equivalent to the relative growth rate, as routinely used in plant growth analysis (Hunt 1982). The first two terms on the right-hand side describe neighbour-independent growth in terms of a Gompertz model, where θ1 can be thought of as an intrinsic growth rate due to the uptake of resources, and θ2yi as a metabolic loss proportional to the size of plant i. The model above just describes growth of a single stand; the full indexing by genotype and replicate is brought in for the statistical analysis below.
Using the data from the main experiment and the approach outlined below, we checked that growth was of a Gompertz type using a Richards growth model (Seber & Wild 1989), which has a flexible shape parameter δ encompassing several growth models as special cases. The parameter δ was not significantly different from 1, suggesting the data adequately described by a Gompertz growth function, in keeping with previous work on A. thaliana (Purves & Law 2002).
The last term on the right-hand sides of equations 3 couples the growth of neighbouring individuals by means of a kernel function F(yi, yj, dij), which in general depends at least on the size yi of the target plant, the size yj of the neighbour and the distance dij between them. The kernel function is summed over all neighbours and scaled by a parameter θ3 to give the overall effect of the neighbourhood on growth. Little is known about the form such kernel functions should take, and we therefore considered a number of properties they might be expected to have (Table 1). The functions were evaluated in a more or less hierarchical order, starting with model 0, which lacked explicit dependence on neighbours, and then introducing effects of neighbour distance, size and competitive asymmetry. Attenuation of competition with distance was described by means of Gaussian (kernels A1–A6) or hyperbolic functions (kernels B1–B3) with a parameter θ4 describing how quickly the attenuation occurred. In addition, we used a simple zone-of-influence (ZOI) model (Opie 1968; Weiner, Stoll, Muller-Landau & Jasentuliyana 2001) based on area of overlap of two circles with radii proportional to the size of the target and neighbour plant (kernels C1–C3). Calculations on interactions between plants were based on pairs of individuals rather than higher-order overlaps (Freckleton & Watkinson 2001; Weiner, Stoll, Muller-Landau & Jasentuliyana 2001) because the latter computations would have been too intensive for the Bayesian statistical analysis needed. Asymmetry of competition was dealt with by means of a tanh function. The tanh function allows the total loss of growth of two plants to be divided up along a continuum from equal sharing by both (θ5 = 0) to all the loss being concentrated on the smaller plant (θ5 → ∞).
Table 1. Kernel functions to describe neighbourhood interactions in stands of Arabidopsis thaliana with overall uncertainty as estimated by Markov chain Monte Carlo. The summed root squared error (sRSE) was used to quantify residual variation and is shown with 95%-credibility intervals
Notation in the statistical analysis below is simplified by collecting all parameters of the growth model into a vector θ = (θ1, … , θk), the first three elements being as in equations 3, and subsequent elements being parameters of competition kernels (Table 2). The statistical analysis deals only with the plant sizes at fixed census times in the experiment, and we obtained the modelled sizes υit,θ at these census times t for a given θ as follows. First, a numerical Euler integration of equations 3 corresponding to a given θ was carried out over the whole period of the experiment. Plant sizes at the first census were used as the initial values for the integration so that differences in initial sizes, which can readily come about through small variations in germination time and early growth, were taken into account. The integration was done on the plants in the central area of each stand, the boundary plants being used only as neighbours; lacking information on neighbours of boundary plants, the sizes of boundary plants between observations were obtained by linear interpolation. This integration generated a set of growth curves in continuous time for the n coupled plants. Secondly, we extracted from the growth curves the plant sizes at the discrete census times; for census t, this yields the modelled sizes of n plants υ1t,θ, … , υnt,θ for a given θ. These modelled sizes at fixed census times were used in the statistical analysis that follows.
Table 2. Information on model parameters and their prior distributions in the statistical model. N represents a normal distribution with mean and variance as indicated in parentheses, whereas Γ represents a gamma distribution
Testing properties of kernel functions entailed fitting growth models with given kernels and model parameters θ (Equations 3, Table 1) to the data. Because the spatial dependence of plant sizes invalidated any assumption of independence of the sample units, we used a Bayesian approach. Bayesian statistics are based on Bayes’ theorem, which originally described the probability of an event A given another event B, but can be generalized to probability distributions. As noted, in a Bayesian setting we assume the parameters θ to be random variables with a given prior distribution P(θ). Bayes’ theorem now links the posterior distribution P(θ | y) of the parameters, given the data, to the prior distribution in the following way:
( eqn 4 )
where P(y | θ) is the likelihood of y given θ. The prior probability distribution P(θ) reflects information about the parameters available before the experiment has been conducted and P(y) is the marginal probability density across the parameter space. As P(y) is simply a normalizing constant independent of θ, the posterior is proportional to the likelihood times the prior. For more details on Bayesian statistics and their application see Gilks et al. (1996) and Gelman et al. (2004).
With the data organized in replicated stands of two genotypes, parameters were estimated at the stand level and at the genotype level. The hierarchical statistical model is schematically represented by the directed acyclic graph of Fig. 1.
Likelihood function and prior probabilities for the parameter values were calculated as described below. We assumed that the observed size ygqit of an individual i of genotype g in replicate stand q at time t was normally distributed around the modelled value with some unknown precision τ, i.e. that . Here θgq = (θ1, … , θk)gq is a vector of k parameter values for replicate stand q and genotype g (Fig. 1). Because of the normal distribution of y around υ, the likelihood P(y | θ, τ) of the set y of all observed values, given all the parameters θ, τ (θ now including all stands and both genotypes) equals the normal probability density function
( eqn 5 )
the summation being over every target plant at every census in all replicate stands of both genotypes, and N being the total number of observations. This formulation does not deal with the possibility of some correlation remaining among the residuals at different census times within plants, so we did a simple check of returning a single sum of the squared residuals from each plant, making N the number of plants. We found that the effect on estimates was small, understandably because the number of censuses relative to the number of plants was small.
For each replicate stand of each genotype, the parameter values θgq were uncorrelated and assumed to be normally distributed around some mean value with precisions (Fig. 1), adding to the posterior the likelihood P(θ | µθ, τθ) similar to equation 5.
In order to complete the specification of a full probability model we require prior distributions on all top parameters in the hierarchical model, i.e. on , and τ. Priors for were chosen to be normal, with means a guided where possible by previous information on growth of A. thaliana (Table 2) and with large variances b. As we had no information on variances, priors of the precision parameters were chosen to be wide gamma distributions with parameters c, d for and e, f for τ (Fig. 1). Applying Bayes’ theorem, the joint posterior probability is the product of the likelihood of y, τ and of the parameter means with their prior probabilities.
Most joint posterior probabilities are intractable analytically and are typically evaluated using Markov chain Monte Carlo (MCMC) techniques. MCMC is the random sampling from a probability distribution (Monte Carlo) based on constructing a Markov chain where the next step only depends on the current stage of the chain (Gilks et al. 1996). The state of the chain after many steps is then used as a sample from the desired distribution. We used a random-walk Metropolis-Hastings algorithm (Chib & Greenberg 1995). In each iteration, new parameter values were proposed from a normal distribution around the old values, accepting each new value with a probability calculated as the ratio between the posterior probabilities of the new and the old value (Gilks et al. 1996). Three chains of 10 000 iterations with over-dispersed starting values were run. In most cases, visual inspection suggested convergence after 2000 iterations and usually the first 3000 iterations were discarded as ‘burn-in’. Mean parameter values of the combined chains with summary statistics and convergence tests were calculated in R using library BOA (Bayesian output analysis; Smith 2004). To describe the fit of the model to the data we recorded the root squared error summed over all stands and genotypes (sRSE) as a measure of residual variation (Table 1).
spatial organization of plant size
A preliminary test for the build-up of a relationship between plant size and the overall effect of neighbours (neighbourhood weight) was made by examining the correlation between these variables. This required a competition kernel function to provide a measure of neighbourhood weight. For the sake of illustration, we used a function in which the effect of a neighbour decayed as a Gaussian function of distance, multiplied by the size of the neighbour (Table 1, kernel A2). The value of this kernel was computed for each plant and the product moment correlation coefficient between plant size and kernel value was obtained. Non-independence was tested statistically using the distribution of correlation coefficients obtained from repeated independent randomizations of the plant sizes; this entailed holding the locations of the plants as fixed in space so that the spatial point pattern was unaffected, and then randomising the plant sizes across these fixed locations.
spatial organization of plant size
At an early stage of growth, the correlation coefficients of plant size and neighbourhood weight were close to zero and showed no sign of spatial structure in plant size (Fig. 2a,b). However, as time progressed, the observed correlation coefficients became negative (Fig. 2c,d). This leaves little doubt about the significance of local neighbourhood competition: by the end of the experiment plants tended to be small when there was a large weight of neighbours and vice versa.
evaluation of kernel functions
As might be expected, the primary determinant of plant size was time after germination as expressed by the intrinsic (Gompertz) growth of the plants (Table 3). The intrinsic growth parameter θ1 had a value of about 1 d−1, being more than 10 times that of θ2 in all kernels. Both parameters tended to be greater in genotype 2 than in genotype 1, in keeping with the tendency of genotype 2 to increase in radius faster early on. In contrast, the size to which plants would tend based simply on the ratio θ1/θ2 (i.e. when growth rate would be zero in the absence of the kernel in equations 3), was higher for the more compact genotype 1 in all kernels.
Table 3. Estimated mean genotype values of the parameters in the kernel functions shown in Table 1 for stands of two genotypes of Arabidopsis thaliana (NASC N334 and N443). Medians with their 95%-credibility intervals are shown; units are given in Table 2
Estimated parameters: median (2.5, 97.5 percentile)
θ1: 0.895 (0.838, 0.952)
θ1: 0.929 (0.855, 1.002)
θ2: 0.0784 (0.0638, 0.0917)
θ2: 0.0857 (0.0713, 0.0994)
θ1: 0.922 (0.866, 0.979)
θ1: 0.962 (0.896, 1.033)
θ2: 0.0781 (0.0644, 0.0923)
θ2: 0.0864 (0.0718, 0.1014)
θ3: 1.70 × 10−2 (1.24 × 10−2, 2.17 × 10−2)
θ3: 1.75 × 10−2 (1.26 × 10−3, 2.27 × 10−2)
θ4: 1.071 (0.606, 1.557)
θ4: 0.939 (0.484, 1.409)
θ1: 0.903 (0.850, 0.960)
θ1: 0.941 (0.869, 1.012)
θ2: 0.0748 (0.0609, 0.0880)
θ2: 0.0815 (0.0675, 0.0952)
θ3: 2.47 × 10−3 (1.52 × 10−4, 6.54 × 10−3)
θ3: 1.99 × 10−3 (1.99 × 10−4, 5.76 × 10−3)
θ4: 1.244 (0.755, 1.564)
θ4: 1.634 (1.172, 2.113)
θ1: 0.901 (0.843, 0.957)
θ1: 0.939 (0.867, 1.008)
θ2: 0.0747 (0.0601, 0.0883)
θ2: 0.0828 (0.0683, 0.0967)
θ3: 2.11 × 10−3 (1.21 × 10−4, 6.00 × 10−3)
θ3: 1.91 × 10−3 (9.04 × 10−5, 5.72 × 10−3)
θ4: 1.154 (0.268, 2.488)
θ4: 1.373 (0.483, 2.646)
θ1: 0.939 (0.884, 0.996)
θ1: 1.005 (0.929, 1.075)
θ2: 0.0795 (0.0658, 0.0935)
θ2: 0.0902 (0.0758, 0.1045)
θ3: 1.80 × 10−3 (9.37 × 10−5, 5.41 × 10−3)
θ3: 1.67 × 10−3 (8.29 × 10−5, 5.36 × 10−3)
θ4: 1.626 (1.365, 1.882)
θ4: 2.039 (1.840, 2.194)
θ5: 2.771 (1.947, 4.732)
θ5: 2.062 (1.562, 2.528)
θ1: 0.935 (0.877, 0.990)
θ1: 0.999 (0.932, 1.065)
θ2: 0.0789 (0.0655, 0.0925)
θ2: 0.0892 (0.0754, 0.1034)
θ3: 4.43 × 10−3 (4.14 × 10−3, 4.81 × 10−3)
θ3: 3.10 × 10−3 (2.76 × 10−3, 3.48 × 10−3)
θ4: 1.583 (1.030, 2.106)
θ4: 2.133 (1.889, 2.370)
θ5: 3.674 (0.995, 7.067)
θ5: 1.844 (0.945, 3.058)
θ1: 0.936 (0.876, 0.996)
θ1: 0.997 (0.921, 1.070)
θ2: 0.0791 (0.0651, 0.0931)
θ2: 0.0889 (0.0744, 0.1031)
θ3: 4.94 × 10−3 (2.27 × 10−4, 1.62 × 10−2)
θ3: 4.95 × 10−3 (2.32 × 10−4, 1.59 × 10−2)
θ4: 1.291 (0.432, 2.626)
θ4: 1.032 (1.914, 2.558)
θ5: 4.317 (0.672, 20.17)
θ5: 1.617 (0.585, 3.402)
θ6: 2.306 (0.775, 3.790)
θ6: 1.085 (0.459, 2.926)
θ1: 0.902 (0.847, 9.587)
θ1: 0.945 (0.876, 1.017)
θ2: 0.0746 (0.0606, 0.0882)
θ2: 0.0822 (0.0690, 0.0961)
θ3: 3.63 × 10−3 (3.43 × 10−4, 8.22 × 10−3)
θ3: 3.23 × 10−3 (2.95 × 10−4, 7.50 × 10−3)
θ4: 1.078 (0.324, 2.040)
θ4: 0.772 (0.447, 1.227)
θ1: 0.899 (0.854, 0.947)
θ1: 0.942 (0.885, 1.002)
θ2: 0.0745 (0.0613, 0.0882)
θ2: 0.0827 (0.0691, 0.0963)
θ3: 2.48 × 10−3 (1.61 × 10−4, 6.54 × 10−3)
θ3: 3.18 × 10−3 (2.49 × 10−4, 7.48 × 10−3)
θ4: 0.939 (0.328, 1.586)
θ4: 0.586 (0.286, 0.924)
θ1: 0.938 (0.880, 0.994)
θ1: 1.003 (0.937, 1.069)
θ2: 0.0794 (0.0657, 0.0928)
θ2: 0.0899 (0.0759, 0.1042)
θ3: 8.90 × 10−3 (2.65 × 10−4, 1.64 × 10−2)
θ3: 5.03 × 10−3 (2.60 × 10−4, 1.64 × 10−2)
θ4: 1.734 (0.961, 2.616)
θ4: 2.009 (1.398, 2.757)
θ5: 4.695 (1.221, 10.23)
θ5: 2.073 (0.973, 3.383)
θ1: 0.800 (0.749, 0.848)
θ1: 0.819 (0.729, 0.914)
θ2: 0.0631 (0.0488, 0.0766)
θ2: 0.0690 (0.0546, 0.0840)
θ3: 2.29 × 10−3 (1.40 × 10−4, 6.24 × 10−3)
θ3: 1.99 × 10−3 (1.08 × 10−4, 5.92 × 10−3)
θ1: 0.804 (0.757, 0.850)
θ1: 0.841 (0.759, 0.920)
θ2: 0.0636 (0.0494, 0.0742)
θ2: 0.0720 (0.0572, 0.0861)
θ3: 2.31 × 10−3 (1.33 × 10−4, 6.11 × 10−3)
θ3: 2.03 × 10−3 (1.12 × 10−4, 5.92 × 10−3)
θ5: 1.408 (1.149, 2.293)
θ5: 1.966 (1.562, 2.538)
θ1: 0.816 (0.763, 0.873)
θ1: 0.865 (0.759, 0.951)
θ2: 0.0657 (0.0515, 0.0798)
θ2: 0.0751 (0.0582, 0.0911)
θ3: 1.37 × 10−2 (1.58 × 10−3, 2.97 × 10−2)
θ3: 3.56 × 10−2 (8.61 × 10−3, 5.88 × 10−2)
θ7: 0.798 (0.398, 1.142)
θ7: 0.773 (0.607, 1.153)
When neighbourhood competition was made explicit by introducing the kernel functions (all models except 0), sRSE was reduced by up to about 5% (Table 1). Although differences were small, sRSE of different kernel functions had clearly distinct credibility intervals, enabling some comparison of different kernels with different properties. Parameter θ3, which scaled the overall effect of the kernel functions, was small. Nonetheless, the effect of the kernel could still be substantial because the summation over neighbours could be large in the presence of many neighbours, and because a linear effect of neighbours on log weight corresponds to an exponential effect on untransformed weight.
Of the terms in the kernels, asymmetry usually had the largest effect on the residual variation, the parameter θ5 for both genotypes being clearly different from zero in all kernels in which it was included, with an estimated value of approximately two. This is indicative of strong asymmetry in competition: if one plant in a pair was about three times the weight of the other, competition would be partitioned such that about 99% of its effects would apply to the smaller plant and about 1% to the larger plant.
Distance terms in the kernels also contributed to the reduction of sRSE. The distance parameter θ4 was estimated to be between 0.8 and 2.1 cm, implying a rapid attenuation of competition with distance, such that most interactions occurred with closest neighbours. For instance, with θ4 = 1.5 cm, the strength of competition between two plants 4 cm apart would fall to about 0.001 of its value at displacement zero in the Gaussian kernel (4 cm was the width of the borders used in the experiment and was therefore the upper limit allowed in the analysis). The effect of distance did not peak too abruptly at the origin. This is suggested by the fact that, when the power term in the exponential function was made a parameter of the statistical model (kernel A6), estimated values tended to be concentrated around a value 2 (Gaussian), rather than 1 (exponential). Of the two genotypes, the more diffuse genotype 2 had a larger area of competitive interaction in the majority of kernels, as indicated by the higher distance parameter θ4 (Table 3).
The reduction of sRSE attributable to distance came about irrespective of whether a Gaussian or hyperbolic function of distance was used (kernels A vs. B). ZOI kernels (C1–C3) were somewhat different in that plant size and distance between plants is built into their structure through the area of overlap. It was notable that ZOI kernels were among the most effective in reducing sRSE. A parameter scaling rosette diameter to the diameter of the ZOI in kernel C3 was not different from unity and did not improve the model's fit.
The sRSE could be slightly reduced by introducing neighbour size (A2, A3 compared with A1). Terms in the kernels factoring in a direct effect of both target and neighbour size had relatively little effect on sRSE (A3 compared with A2, B2 compared with B1). Evidently what mattered more was the size of the target relative to the neighbour, incorporated through the asymmetry term in kernels A4, A5, A6, B3, C2.
On the grounds of parsimony, the ZOI kernel function (C2), based directly on the radial overlap of pairs of plants and asymmetry in partitioning the effect of overlap, was arguably the best compromise. This function gave the greatest reduction of sRSE, while introducing only two extra parameters (overall weighting and asymmetry).
The experimental results demonstrate the importance of competition with neighbouring plants on the growth of plants in crowded stands. Within about 25 days, spatial structure in the size of plants had developed, with the result that plant size and a measure of neighbourhood crowding were negatively correlated, i.e. large plants tended to be found more where neighbourhood competition was relatively weak (Fig. 2). Such a relationship between plant performance and neighbourhood weight is not surprising, although success in demonstrating the phenomenon has been rather variable in the past. Early work by Kira et al. (1953) found the weight of regularly spaced plants to be correlated with the mean weight of the plant's six nearest neighbours, and several subsequent studies also found a relationship between plant growth and neighbourhood properties (Mithen et al. 1984; Weiner 1984; Silander & Pacala 1985; Stoll et al. 1994; Hubbell et al. 2001; Stoll & Bergius 2005). However, such relationships have sometimes proved harder to demonstrate in the field (Martin & Ek 1984; Coomes et al. 2002), especially when plants are regularly spaced (Lorimer 1983; Wimberly & Bare 1996; but see also Larocque 2002).
The analysis presented here brings new quantitative insight to neighbourhood-dependent plant growth, showing that simple models could successfully encapsulate some important features of the competitive dynamics of replicated stands of the two genotypes. The major determinant of plant size was clearly the intrinsic growth, well described by the Gompertz equation. Although relatively little additional variation was attributable to the coupled growth of plants through competition kernel functions (Table 2), there was a strong build-up of spatial structure over time (Fig. 2) and certain properties of local competition consistently emerged from the analysis. First was the importance of asymmetry of competition. The credibility intervals for asymmetry parameters were always well away from zero, with values indicative of strong asymmetric competition, as one might expect in a system where competition was primarily for light (Schwinning & Weiner 1998). The potential importance of asymmetry in local competition is widely recognized in the literature (Schwinning & Weiner 1998; Freckleton & Watkinson 2001; Bauer et al. 2004; Stoll & Bergius 2005), but quantitative estimates of its value from stands of interacting plants are scarce. Second was the attenuation of competition with distance. This took place over rather short distances of 1–2 cm according to our estimates of the distance parameters. The exact shape of the attenuation function (Gaussian, hyperbolic or exponential) was found to be of less importance, probably because all these functions were flexible enough to capture the signal present in the data. This is in line with Purves & Law (2002), who found the exact choice of the kernel function not to be critical. Hence, structural components of the kernels (e.g. asymmetry, size, distance) were more important than the exact mathematical form. This corroborates a number of studies, which, because of the lack of comparative studies, used summed neighbour sizes weighted by distance (Stoll et al. 1994) or squared distance (Weiner 1984), a weighted mean distance to neighbours (Silander & Pacala 1985; Lindquist et al. 1994) or a hyperbolic function of neighbour weight and distance (Coomes et al. 2002).
Although all models with the same structural components had comparable predictive ability, a case could be made for a simple ZOI model (kernel C2) on the grounds of parsimony. This function is based on overlap of pairs of circles, with radii as measured on the rosettes, and with competition favouring the larger individual. It is parsimonious because it introduces into the growth model just two extra parameters, θ3 for the overall scaling of neighbourhood and θ5 describing the strength of asymmetry. To illustrate the function in action, we have integrated equations 3 with the estimated parameter values of genotype 1, and have picked two plants with contrasting neighbourhoods from the first replicate stand, to show how they grow (Fig. 3). The first plant was relatively large from the start of the experiment, had relatively few overlapping neighbours, and was larger than all but two of them on day 10. Its growth curve was changed relatively little by incorporating the kernel function to allow for other plants in its vicinity. The second plant was rather smaller at the start, and had 11 overlapping neighbours larger in size on day 10. It is well known that plant growth is much influenced by events early in life (Ross & Harper 1972; Fowler 1984; Nagashima 1999) and, as one might expect, the growth of this plant was more curtailed by its neighbours as time went on. This effect of neighbours was successfully encapsulated by the ZOI kernel function. By the last census, the kernel was responsible for a twofold reduction in (untransformed) dry weight in the first plant and about a fivefold reduction in the second.
The differences between the two genotypes used were marked and consistent through most models. Coupled growth models, especially simple ZOI models as presented here, are therefore a promising tool in studying competitive interactions in multispecies populations, at least in the context of rosette plants. Plants with morphologies other than rosettes may well require other approaches.
We believe that a Bayesian statistical analysis of the kind applied here has potential to resolve a number of problems that, because of intricate spatial dependences of individuals, have looked intractable in the past. Spatial dependence is a recurring feature of ecological field data; in fact, the ecological signal of interest in the present study was precisely to do with the nature of this spatial dependence. Our approach was to put the spatial dependences explicitly into a dynamic growth model, to construct a statistical model embracing the non-independence, and to work directly with it. The combined model may be very complex. For instance, in the present study, the hierarchical model was based on replicated stands of two genotypes and encapsulated uncertainty within and between stands. The growth model for each replicate stand involved solutions of a system of over 200 ordinary differential equations, with initial conditions given by observed plant sizes at the start.
A caveat about the analysis is that we reached the limits of computational power for describing interactions among individuals just by taking them in pairs. In reality, multiple overlaps among plants were no doubt common over much of the growth period (Freckleton & Watkinson 2001; Weiner, Stoll, Muller-Landau & Jasentuliyana 2001). Such overlaps are visible, for example, in the circles shown in Fig. 3, and the evident success of measuring neighbourhood weight as a sum of effects on pairs of individuals is surprising at first sight. However, a circle or some other simple index can be no more than a rough approximation for the intricate interactions among neighbouring plants as they shade one another and compete for nutrients (Busing & Mailly 2004). The extent to which lowest-order interactions predominate could usefully be tested in future work using approaches such as the field of neighbourhood developed by Berger & Hildenbrandt (2000).
Neighbour-dependent plant growth has some interesting broader ecological implications. In particular, it emphasizes the inherent tendency of plants to distribute their biomass over space, growing relatively little under crowded conditions and much more when space is plentiful. Individual plants are also able to avoid shade and distribute their biomass into space under less influence of neighbours (Stoll & Bergius 2005). As the signals of spatial structure in first-order statistics (size hierarchies) appear to be relatively weak (Weiner, Stoll, Muller-Landau & Jasentuliyana 2001), ecologists will need to turn to second-order summary statistics of spatial point processes, such as mark correlation functions (Stoyan & Penttinen 2000), to see these signals in plant biomass. In the case of A. thaliana here, plant biomass became somewhat uniformly distributed over space during the experiment as a consequence of asymmetric competition (results not shown). Such behaviour has been noted in growth of A. thaliana (Stoll & Bergius 2005) and in stands of forest trees (Stoyan & Penttinen 2000). An interesting consequence of this plasticity in plant growth is that there should be some uncoupling of area-based primary productivity, an ecosystem process, from the birth and death processes of populations. The division between ecosystem ecology and population ecology is sometimes thought to be an artificial one (Jones & Lawton 1995), but arguably some separation between them comes about as a natural consequence of neighbourhood-dependent plant growth.
Bayesian hierarchical modelling is a powerful tool to evaluate complex spatially dependent data and has yielded some quantitative answers to the questions posed in the Introduction. Clearly competition is an important process in the growth of stands of A. thaliana. (i) Competition is strongly asymmetric. (ii) Competition is local and attenuates to zero over a scale of centimetres. (iii) Competition increases with plant size and the effect of size can be accounted for by a simple ZOI function. (iv) Summing the effects of pairwise interactions leads to a measure of neighbourhood weight that can explain some variation in growth. However, a comparison with models that allow higher-order dependences will be needed to establish how good an approximation this simple summation is. Neighbourhood interactions are an important process in plant population dynamics (Murrell & Law 2003), and the knowledge gained from this study should inform and enhance the formal theory of spatiotemporal plant population and community dynamics in the future.
The authors thank Christian Asseburg, Drew Purves and Jon Pitchford for comments on the manuscript, Peter Lee and Dale Taneyhill for advice on MCMC, and Colin Abbott and Chris Lancaster for technical assistance. Michael Raghib suggested the use of a tanh function to describe competitive asymmetry. This research was made possible by grants from the Swiss National Science Foundation and the Walter-Hochstrasser Foundation.