## Introduction

Herbivory is one of the dominant biological mechanisms organizing natural habitats (Crawley 1983). These habitats are characterized by environmental heterogeneity that affects both the distribution, and interactions between species (Begon, Harper & Townsend 1996). For consumers, such as herbivores, environmental heterogeneity in food resource quality, for example, may structure populations in space (McLaughlin & Roughgarden 1992; McCauley, Wilson & de Roos 1996; Hambäck, Schneider & Oksanen 1998). In seeking to understand spatial structuring of populations, though, it is not enough just to consider environmental heterogeneity alone. The population dynamics of the species themselves have important consequences for spatial structuring (Rhodes, Chester & Smith 1996; Tilman & Kareiva 1997; Bascompte & Solé 1998). In natural systems it may often be difficult to disentangle the spatial-structuring consequences of population dynamics and environmental heterogeneity. The comparatively uniform nature of many extensive farmland habitats provides an excellent opportunity to disentangle such effects.

Terrestrial gastropod molluscs are generalist herbivores with a broad range of acceptable foods (Grime, Stewart & Dearman 1968; Dirzo 1980). They can influence the species composition of plant communities (Cottam 1985; Glen, Cuerden & Butler 1991a; Hanley, Fenner & Edwards 1995a,b, 1996a,b), the frequency of polymorphisms (Crawford-Sidebotham 1972; Cates 1975) as well as causing considerable crop damage (Port & Port 1986; Glen, Milsom & Wiltshire 1989). These complex effects of mollusc herbivory are due to interactions between the mollusc and plant species, with individuals aggregating under clumps of cocksfoot grass (*Dactylis glomerata* L.) for example (South 1965), and environmental heterogeneities, such as soil temperature and moisture. Even in apparently uniform plant habitats, however, over-dispersed distributions of molluscs may form (Airey 1984). Despite the relatively homogeneous nature of many farmland habitats, in comparison to more natural situations, mollusc herbivore spatial patterns and damage to plants are dynamic (Barnes & Weil 1944, 1945; South 1965; Glen *et al*. 1989; Glen, Wiltshire & Milson 1992), and possibly due to the population dynamics of these species.

Spatial patterns of individuals have been treated as having two aspects: parametric intensity and form (Pielou 1977; Thrush 1991). Parametric intensity is evaluated through approaches that assign the sample count data to frequency distributions; a good and often-used example is the sample variance to mean ratio. Assessments of form, however, attempt to include the locational information in a sample, using auto-correlation or variation with distance techniques (Sokal & Oden 1978a,b; Cliff & Ord 1981; Legendre 1993), in order to describe gradients or patches in the observed pattern. Typically, correlograms, based upon parametric statistics such as Moran's *I* (Moran 1950) or Mantel *r*, are plotted with distance (Legendre & Fortin 1989). From such correlograms inferences about the significance, dimension, shape and anisotropy of gradients or patches in the spatial pattern may be made (Legendre & Fortin 1989). Furthermore, these flexible techniques, together with trend surface methods, using low order polynomials, can be used to extend the description of form across spatial scales (scaling-up), even though the dimension of the sampler (grain), the intersample distance (lag) and overall area encompassed (extent) of the sampling are fixed (Thrush *et al*. 1997).

Such approaches have proved extremely successful at describing spatial form in terrestrial and marine molluscs. Sokal & Oden (1978a) showed contrasting spatial forms of allozyme diversity in the snail *Helix aspersa* Müller in two adjacent city blocks in Bryan, Texas, which were consistent with possible differences in the numbers of founding individuals in each population. In marine bivalves, Irlandi (1994) and Irlandi, Ambrose & Orlando (1995) found that continuous beds of seagrass afforded higher probabilities of survival than small, isolated patches. Legendre *et al*. (1997) used a combination of intensity, auto-correlative and surface trend analyses to identify important pattern generating processes for two sandflat bivalves and the scales on which the processes operated. Simulation and mathematical modelling of these results provided encouragement that these small and medium scale dynamics could be synthesized into large scale predictions for the management of molluscs and other sandflat organisms (McArdle, Hewitt & Thrush 1997; Schneider *et al*. 1997; Thrush *et al*. 1997).

Parametric intensity and the methods commonly used to evaluate form do not utilize all the spatial information in a sample and often fail to describe the spatial arrangement of the counts in a sample. Spatial arrangement is an aspect of form and describes the distribution of the counts in a sample. The simplest way to consider arrangement is to use the checkerboard example of Sokal & Oden (1978a,b). For this example, the cells of the board represent individual sampling points, each containing a count of individuals. If all individuals were found to be aggregated into one cell then, by definition and intuition, the arrangement would be spatially aggregated. Conversely, if the counts in each of the cells were the same or, to relax the criteria somewhat, should all the rows and columns of the board sum to a similar number, then the arrangement across the board would be defined as being spatially regular. Arrangements that are intermediate between spatial regularity and aggregation would, therefore, be spatially random.

Evaluating these spatial arrangements is important because processes that lead to spatial aggregation are likely to be biologically very different from those that lead to randomness or regularity. Parametric intensity fails to differentiate between such arrangements because for a given sample of *n* counts taken across a checkerboard, there are approximately *n*! unique ways of arranging the sample counts to the board. It is theoretically possible, for example, for a sample to have a variance that greatly exceeds the mean but which exhibits a spatially regular arrangement. Similarly, correlograms and other distance methods may also fail to describe arrangement. For example, Legendre & Fortin (1989) considered a surface made up of similar, binomial bumps arranged regularly with a characteristic distance between each bump. Legendre & Fortin (1989) produced correlograms of this manufactured surface from which the regular pattern of bumps could have been predicted. However, if the bumps were rearranged to give a spatially random arrangement, with the same mean distance between bumps, a similar correlogram would have resulted.

The recently developed, non-parametric Spatial Analysis by Distribution IndicEs (SADIE) techniques were specifically created to analyse the spatial arrangements of biological count data, by utilizing the location information in the sample (Perry 1995a,b, 1998). The aim of the present study was to explicitly compare parametric intensity and SADIE approaches to evaluate the spatial dynamics of two terrestrial gastropod mollusc herbivores, the slugs *Deroceras reticulatum* (Müller) and *Arion intermedius* Normand, with contrasting biology. *D. reticulatum* and *A. intermedius* both grow to hermaphrodite adults and lay eggs in batches in the soil. Populations of *D. reticulatum* undergo more than one, but fewer than two generations per year (Hunter & Symonds 1971). Individuals of *D. reticulatum* are behaviourally sexual, depending on their physiological status (South 1982), and reproduce whenever meteorological conditions are favourable, with two peaks of egg laying, generally in spring and autumn (Carrick 1938; Hunter & Symonds 1971). *D. reticulatum* may over-winter as egg, juvenile and adult stages. The juvenile and adult stages are very surface active. Adult individuals may move many meters per night (Duval 1970; Bailey 1989), although the net distance of migration, per night, is normally considerably less than this (South 1965; Pinder 1969; Hogan 1985; Glen, Wiltshire & Butler 1991b). On the contrary, *A. intermedius* is self-fertile (Davies 1977; Foltz *et al*. 1982) and univoltine (Jennings & Barkham 1975; South 1989), with eggs being laid in the autumn to hatch from autumn to early spring. Few adult individuals survive the winter. Sampling with refuge traps suggests that *A. intermedius* exhibits relatively low surface activity, by comparison with *D. reticulatum* (Glen & Wiltshire 1986).

To compare parametric intensity and SADIE approaches, however, there is the need to define a simple but strict nomenclature to prevent confusion. In this paper, the terms over- and under-dispersion are used solely to describe the parametric, frequency distributions of sample counts; indicating that the sample variance exceeds the sample mean and the variance is less than the mean, respectively. Where the sample variance of the frequency distribution approximates the mean, the distribution is described as dispersed. The terms aggregated, regular and random are commonly used as synonyms for over-, under- and dispersed frequency distributions, respectively. These terms would imply an explicitly spatial arrangement to the measures of intensity that is inappropriate. Here the terms aggregated, regular and random are purely used for the spatial arrangement of the sample counts.

In the present study, the frequency and spatial dynamics of *D. reticulatum* and *A. intermedius* were followed in a field of winter wheat from March 1997 to March 1998. *A priori*, the field was assumed to be homogeneous, providing a predictable and uniform food resource that would allow us to investigate aspects of the spatial structuring and frequency consequences of the population dynamics of these species with low environmental heterogeneity. The approach was to sample both slug species, using a geometric series of grids at four spatial lags, from 0·25 m to 16 m, a common spatial grain of 0·25 m × 0·25 m, and a soil sampling technique that gave direct and repeatable estimates of the absolute local density of slugs in the upper 10 cm of soil (Glen *et al*. 1992; Symondson *et al*. 1996). We anticipated that coherent spatial arrangements should become apparent at some of these lags. Such spatial arrangements may be described and compared between *D. reticulatum* and *A. intermedius*, to test the similarity in spatial arrangement with spatial lag, and across time. In turn, the careful comparison of arrangement and association between the two species, at each spatial lag, will test for the species specificity of spatial arrangement.