## Introduction

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?

Yet surprisingly little is known about competition kernels. When kernel functions are used in modelling growth of interacting plants they tend to be based more on assumptions than on observations (e.g. Yokozawa & Hara 1999). When models have been fitted to spatial data, it is usually the final plant size that is of interest rather than continuous plant growth through time (e.g. Mitchell-Olds 1987; Pacala & Silander 1990; Lindquist *et al*. 1994; Stoll & Bergius 2005). In a few cases the growth of individual plants has been followed over time (Nagashima *et al*. 1995; Nagashima 1999), but without dealing directly with the dynamic nature of neighbourhood competition as plants grow. Exceptions are some work on crop plants (Benjamin 1999) and extensive work on forest trees (e.g. Soares & Tomé 1999). Neighbourhood effects on tree growth were often observed to be small (Lorimer 1983; Martin & Ek 1984; Wimberly & Bare 1996), at least partly because of the functions traditionally used (Larocque 2002).

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.