## Introduction

Carabid beetles are important polyphagous predators in many ecosystems and have been the subjects of significant ecological research, particularly as beneficial agents in agriculture (den Boer 1977; Thiele 1977; Luff 1987; Lövei & Sunderland 1996). Carabids are considered to be generalist and opportunist predators, which orientate principally by microhabitat cues, feeding on almost any suitable prey they encounter and which they are able to subdue. However, there is growing evidence that carabid beetles orientate towards areas where specific prey species are abundant (e.g. Bryan & Wratten 1984; Wallin & Ekbom 1994). The carabid *Pterostichus melanarius* is a common polyphagous predator on arable land (Sunderland 1975; Wallin & Ekbom 1988; Symondson *et al*. 1996; Thomas, Parkinson & Marshall 1998). Symondson *et al*. (1996) demonstrated that slugs (Mollusca: Gastropoda) were important prey for *P. melanarius*, by showing that in areas of high slug biomass more beetles were caught in pitfall traps and that these beetles had ingested more slug material than in areas with lower slug biomass. These findings suggested a stronger relationship between slugs and *P. melanarius* than simply that of opportunism. However, the differences in beetle numbers and slug biomass were related to different cultivation methods, with highest numbers and biomass recorded in the absence of cultivation (Symondson *et al*. 1996). Thus, it is possible that the association between *P. melanarius * and slugs could have resulted from both responding independently to the effects of cultivation, and with beetles preying to a greater extent on slugs in less disturbed areas because there simply happened to be more slugs there.

The location of individuals, with respect to each other, is extremely important for the dynamics of ecological interactions (Hassell, Comins & May 1991; Silvertown *et al*. 1992; Powell *et al*. 1995; Grenfell & Bolker 1998). The study described in this paper was designed to investigate the spatial association and trophic relationship between *P. melanarius * and slugs [predominantly *Arion intermedius* Normand and *Deroceras reticulatum* (Müller)] in a uniform arable field during June and July, when adult *P. melanarius* were present at high density (Thomas *et al*. 1998). We ask whether the locations of the slug and beetle individuals are consistent with slug predation by *P. melanarius* being opportunistic or whether there is evidence of beetles responding in a more directed manner to the presence of slugs.

Our approach was to map the distribution of slugs and *P. melanarius* (both the total population of beetles and those that had ingested slug protein) in a field of winter wheat at a number of spatial scales up to a dimension of 80 × 64 m. We believed that, at some of these scales, coherent spatial structuring in the form of slug and *P. melanarius* patches would become apparent. In this case, opportunistic predation by *P. melanarius* would be observable as apparent independence between the carabid and slug spatial distribution patterns (*sensu*Perry 1998). A significant spatial association between the *P. melanarius* and slug distributions, however, could indicate that either *P. melanarius * and the slugs were responding to a common spatial variate or factor, or that the carabids were directly responding to slug density. Thus, the formal tests for our stated question are that: (i) there is a statistically quantifiable spatial association between the distributions of *P. melanarius * and slugs; (ii) the *P. melanarius* have ingested slug protein; and (iii) the spatial distributions of *P. melanarius* and slugs are not associated with other spatial variates or factors.

There are, however, considerable statistical problems in using an explicitly spatial approach to test the null hypothesis, resulting mainly from the practical difficulties of sampling for slugs and carabids. Consider an attempt to produce a map of *P. melanarius* density using a grid of sampling points of pitfall traps. As pitfall trapping relies on the carabids approaching and falling into traps, the numbers trapped do not reflect the absolute local densities of *P. melanarius*, but rather what has been termed the ‘activity-density’ of the carabids for some distance, or spatial scale, about each trap (Ericson 1977). The scale of the activity-density is a function of the *P. melanarius* migration rate and how easy it is for *P. melanarius* to move through the environment (Greenslade 1964; Honek 1988). Across a field of winter wheat, and during a single sampling, these factors might be assumed to be constant. However, even in an apparently homogeneous field, where the scales of beetle activity-density overlap more than one trap, individual beetles could fall into any trap within their range of movement. In this case, all the traps within the scale of activity-density are sampling the same activity-density and would be spatially autocorrelated.

This type of spatial autocorrelation is essentially a problem of the scale of sampling. At a small spatial scale, the traps interfere with each other because the same individual is capable of appearing in any one of a number of traps. At larger spatial scales, the pitfall traps are independent because their scales of activity-density do not overlap, and thus the *P. melanarius* captured in one trap do not interact with those in other traps. In this case, and all other things being equal, the activity-density of the sample reflects the absolute local density of the carabids, and the sampling is reproducible. At these large spatial scales, a graphical plot of activity-density across the grid of traps produces a map that may be used for analysis of spatial pattern and association and parametric analyses. It is important to note, however, that at these large scales the catches of *P. melanarius* in the pitfall traps need not be independent. If a *P. melanarius* patch structure is evident across such scales, then trap catches will necessarily be autocorrelated. It should be stressed, however, that the sampling is independent and the pitfall traps are independently sampling from a common spatial distribution, the patch.

It is not possible, a priori, to define the scale at which pitfall trap catches become independent, particularly because this is dependent upon carabid activity, which, in turn, changes with environmental conditions (Greenslade 1964; Honek 1988; Wallin & Ekbom 1994; Honek 1997). Thus, the lowest spatial scale at which maps may be produced has to be determined using sample data. In this report we have sampled for *P. melanarius* using a series of nested sampling grids, with a geometrically increasing intersample distance, to establish the scale at which the pitfall traps become independent. We define this as the lowest spatial scale, of those sampled, at which the trap catches are spatially random with respect to one another.

We sampled slug populations using a soil sampling technique that has few of the problems of pitfall trapping and gives estimates of the absolute local density of slugs in the upper 10 cm of soil (Glen, Wiltshire & Milsom 1992; Symondson *et al*. 1996). Thus, at all spatial scales, plots of the distribution of slugs may be used to describe spatial pattern and association. For parametric analyses, however, we must again invoke the criterion that the sampling is spatially independent. As with sampling for *P. melanarius*, we employed a geometric series of nested grids to establish the scales of independence for the slugs and produced maps of absolute local slug density.