Parametric intensity and the spatial arrangement of the terrestrial mollusc herbivores Deroceras reticulatum and Arion intermedius


Dr David A. Bohan, IACR-Long Ashton Research Station, Department of Agricultural Sciences, University of Bristol, Long Ashton, Bristol, BS41 9AF, UK. Tel: (44) 1275 549329, Fax: (44) 1275 394007, E-mail:


1. Parametric intensity and spatial arrangement analyses were used to investigate the spatial pattern of the slugs Arion intermedius and Deroceras reticulatum.

2. The spatial lag of sampling (distance between sampling points) was shown to be unimportant in the intensity analyses. Rather, the 0·25 m grain scale was imposed on the whole sampling. The observed slope of the variance to mean relationships was common to both species, possibly determined by egg laying in batches at 0·25 m. However, the variance of the sample, for a given mean, was lower in summer. This corresponded with a reduction in the proportion of zero counts, which could be due to slug movement, possibly increased by predator activity, acting at the 0·25 m scale.

3. By contrast with the intensity analyses, the lag scale was important for spatial arrangement. At 0·25 m, in March 1997, the A. intermedius and D. reticulatum juveniles were aggregated, presumably about where egg batches were laid. At higher scales, the arrangements of D. reticulatum became spatially random, and A. intermedius resolved to a patch arrangement at the 16 m scale.

4. Over time, the D. reticulatum spatial arrangements remained random and independent of the previous sampling date. From March to July 1997, the A. intermedius patch persisted. A crash in abundance of both species, between July and October 1997, appeared to destroy the patch, but subsequent association suggested that the patch persisted until March 1998. The arrangements of the species were independent of one another on all sampling dates.

5. These species-specific spatial arrangements were independent of all measured environmental factors and consistent with differences in the local reproduction, survival and migration of A. intermedius and D. reticulatum.

6. This comparative study indicates that the terms aggregated, random and regular should have separate definitions for parametric intensity and spatial arrangement. Furthermore, spatial scale has different meaning in intensity and arrangement analyses. Spatial arrangements are not described by parametric intensity. Spatial arrangements change with spatial scale. Temporal changes in intensity need not manifest as changes in spatial arrangement.


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.

Materials and methods

Experimental field

The study was conducted in the middle of a flat, triangular 1·7 ha field, approximately 3 km west of IACR-Long Ashton Research Station, Bristol, UK. The field (Field 54, Island Orchard) has a well-drained loamy soil and is bordered on all sides by managed, mixed hedgerows. Prior to the investigation, the field had a crop of winter oilseed rape, which was harvested in the summer of 1996. In 1996–97 and 1997–98 the crop was winter wheat.

Sampling design

Slug populations were sampled on 17 March, 23 June, 21 July, 21 October and 28 November 1997, and 17 March 1998. The sampling used a geometric series of grids based upon a two-dimensional array of 5 × 5 sampling points with four spatial lags (0·25 m, 1 m, 4 m and 16 m); this being the distance between the centres of adjacent samples (Fig. 1a). At the 16 m lag, a 5 × 6 array was used, giving a total experimental extent of 64 m × 80 m within the field.

Figure 1.

Schematic diagrams of the position and location of the soil samplings. (a) The sampling points for the soil sampling, with the 0·25 m lagged grid nested within the 1 m grid, and so on, up to the 16 m grid. the position of the smaller grids within the 16 m grid was determined by random numbers taken from the Uniform distribution, with the configuration shown being used in June 1997. (b) Schematic for the location of the sampling origin points in the 16 m lagged grids. the point 1 is the origin point for the soil sampling grid (a). in order to avoid oversampling, the sampling grid was moved in sequence from point 1, in March 1997, 1·5 m south to point 2, in June, and so on to point 6, in March 1998.

The origin of the 16 m grid was fixed within the field. Because of the destructive nature of soil sampling for slugs, however, the 16 m sampling grid was moved by 1·5 m, about an origin point, on each sampling date (Fig. 1b). This movement avoided over-sampling and was not considered to be a significant effect in the analysis. The grids with smaller lags were nested within the larger grids. Again, due to the destructive nature of slug sampling, the position of these grids changed with the sampling date. Thus, the precise positions of the 0·25 m grid within the 1 m grid, the 1 m grid within the 4 m grid, and the 4 m grid within the 16 m grid, were determined by random numbers from the Uniform Distribution. It should be noted that samples were taken at only the 16 m lagged grid in November 1997.

Sampling technique

Slugs are predominantly found in the upper 7·5 cm of soil, except in very dry or cold conditions (Hunter 1966). Slugs in the surface 10 cm of soil were sampled using a soil sampling and flooding technique (Glen et al. 1992) modified from South (1964) and Hunter (1968). At each sampling point, a 25 cm × 25 cm × 10 cm (length × breadth × height) steel template was pushed into the ground and undercut with a spade. Each soil sample was transferred to a plastic tub (40 cm diameter, 20 cm high), which had three holes close to the base covered with 1 mm mesh. The tubs were covered with a wooden lid and transferred to a soil-flooding unit, where they were stood in troughs filled with a shallow (2 cm) depth of water. A drip feed from a glasshouse irrigation system was introduced to each trough and the drip rate was adjusted so that the soil samples would be flooded gradually over a period of 8–10 days. The flooding forced the slugs to the soil surface, where they were collected daily and individually identified, weighed and marked with the sample location. The 25 cm × 25 cm dimension of the soil sampler gave the grain for the sampling. Note that across the 0·25 m lagged grid, soil sampling resulted in the complete removal of the upper 10 cm of soil over an area of 1·25 m × 1·25 m.

Moisture, temperature and plant assessments

Measurements of soil moisture and the crop were made on each date when soil samples were taken, at each sampling point across the 16 m lagged sampling grid. The percentage moisture of the soil was assessed by extracting a trowel-full of soil from a depth between 7 cm and 10 cm. Each wet sample was weighed before being dried at 110°C, for 2 days. The dry sample was then re-weighed. A Soil Moisture Index was then calculated as the rounded grams of water in every kilogram of soil. The spatial variability in wheat growth was assessed from the number of stems within a 25 cm × 25 cm quadrat and the maximum height of the plant within the quadrat, at the 16 m lagged grid, on 17 March 1997. On 23 June and 21 July, the dry weight of 10 randomly sampled wheat ears, at each 16 m sampling point, was measured. The soil temperature, at 7 cm depth, was measured every 3 hours (8 readings per day) at 16 points across the 16 m lagged sampling grid, using solid-state, automatic data loggers (Tiny Talk, Gemini Data Loggers (UK) Ltd, Chichester, UK). For analysis, day-degrees were calculated as the cumulative, arithmetic mean of the 8 daily temperature readings at each point, for the 7 days prior to slug sampling.

Spatial and statistical analysis

We were interested in the spatial position of each datum in the sampling grid (arrangement) and the spatial association of the data sets, for example across time or between the slug species. For this the recently developed, non-parametric suite of Spatial Analysis by Distribution IndicEs (SADIE) algorithms were used (Perry 1995a,b, 1998).

To evaluate spatial arrangement, SADIE_C uses a transportation algorithm taken from the linear programming literature (Kennington & Helgason 1980) to calculate the shortest total distance to regularity, for an observed sample, by moving the sampled individuals to neighbouring or more distant cells until a spatially regular arrangement is achieved. Each move of an individual between cells increases the total distance that the algorithm has moved the sample. Of all the possible summed combinations of moves that the algorithm could make, only a subset will yield the shortest distance to regularity. It is this observed shortest total distance to regularity, which by definition requires the least amount of moves, that is used as the test statistic.

To test this statistic, SADIE_C then conducts a specified number of randomizations of the sample. For each randomization, the algorithm randomly assigns each sample count to a new cell, thereby creating a new spatial arrangement. The shortest total distance to regularity is then permuted for the randomization, and across all randomizations a frequency distribution of permuted distances to regularity is created. It is against this frequency distribution that the observed shortest total distance to regularity is tested. An observed distance that lies in the largest 2·5% of permuted distances to regularity is not significantly different from being spatially aggregated; in effect, a significantly large number of moves away from regularity. Similarly, observed distances that lie within the shortest 2·5% of permuted distances are not significantly different from being spatially regular. All other distances are effectively spatially random. SADIE_C completes these comparisons automatically, assigning each observed sample a corresponding probability (Pa), which may be interpreted in the standard manner. The algorithm also achieves a sample- and density-independent index of arrangement (Ia) that may be used for descriptive and comparison purposes. Values of Ia in excess of unity denote spatial aggregation, approximating unity indicate randomness, and less than unity indicate regularity (Perry 1995a). A modification of this method (SADIE_A) allows the comparison of two data sets sampled across the same grid to test for association (Perry 1998). Randomizations of a scaled data set, combining the two observed data sets, give a sample- and density-independent index of association (Iass) and a probability of association (Tass). Values of Iass greater than unity indicate spatial association, and similarity between the observed data sets, whilst values less than unity indicate dissociation. It should be noted that both SADIE tests, of spatial distribution and association, are two-tailed. Pa and Tass should be interpreted accordingly. For all SADIE analyses reported here, a minimum of 400 randomization simulations were conducted, using a standard seed for the SADIE random number generator.

The utility of SADIE is not confined to a regular grid. Samples of the positions of individuals or samples taken across any spatially explicit sampling structure may be analysed. SADIE deserves a fuller description than can be given here. Those interested in the technicalities of the transportation algorithm and the calculation of Ia, Iass and Tass are directed to the linear programming and SADIE literature.

Standard parametric techniques were also used to describe the sample data. The sample data were modelled using the GLM procedures in Genstat 5 for Windows, Version 4·1 and in GLIM 3·77. The relationship between the sample variance and mean, for both D. reticulatum and A. intermedius at a given lag, was investigated by ancova (Crawley 1993). The log-transformed sample mean was fitted to the sample variance using the Gamma frequency distribution and the logarithmic link function, with slug species, sampling date and sampling lag as co-factors. The relationship between the proportion of zero counts and the mean was similarly modelled by ancova, using the Binomial frequency distributions, the Logit-link function and the total number of sample points in each lag as the binomial denominator. The November 1997 and March 1998 data were pooled to avoid an artefact of fitting sampling lag to the November 1997 data, for which only the 16 m lag data were collected. The standardized residuals of each model were checked for large deviance, linearity and leverage (McCullagh & Nelder 1989; Crawley 1993).


Parametric intensity of slug abundance with time and lag

A total of 1375 D. reticulatum and 670 A. intermedius were sampled in Field 54, on the six sampling dates. Across the grid with a 16 m lag, the abundance of D. reticulatum and A. intermedius changed with time (Fig. 2). From March to July 1997, D. reticulatum numbers increased, whilst the numbers of A. intermedius declined steadily over the same period. The abundance of both species declined substantially between July and October 1997 then remained low. The final abundance of A. intermedius in March 1998 was about one-twentieth of that observed in March 1997, whilst D. reticulatum declined to about one-quarter of the original abundance.

Figure 2.

Plots of the mean and standard errors in the abundance of A. intermedius (●) and D. reticulatum (▴), across the 16 m sampling lag, for each sampling date.

The ancova-analysed sample data were found to conform to Taylor's Power Law (Taylor 1961; Taylor, Woiwod & Perry 1978; Anderson et al. 1982), with the log-transformed sample mean accounting for some 75·8% of the error in the log-linked sample variance (F1,36 = 137·26, P < 0·001). By deletion, the change in deviance for the co-factor of slug species showed that D. reticulatum and A. intermedius did not have significantly different variance to mean relationships (F1,35 = 0·22, P > 0·05). No interaction was observed between the co-factor of species and the sample mean (F1,31 = 0·60, P > 0·05). Interestingly, the co-factor of sampling lag was not significant (F3,32 = 0·68, P > 0·05); it appeared that for a given sample mean, no change in the sample variance was noted with changes in the spatial lag of sampling. The interaction of lag with the co-factor of species was not significant (F3,25 = 1·45, P > 0·05), neither was the interaction of lag with the sample mean (F3,28 = 2·31, P > 0·05). Sampling date was found to be a highly significant effect (F4,36 = 8·2481, P < 0·001), showing that the variance to mean relationships changed with the date of sampling.

Scrutiny of the minimal ancova output showed that the sampling dates fell into two distinct groups with similar variance to mean relationships; one group of the summer dates of June and July, with relatively higher means and lower variances (Fig. 3), and one group of all the other sampling dates. A new co-factor was consequently created, termed sampling season, to reclassify the sampling dates as either belonging to the summer samplings or all others. When the co-factor of sampling season was fitted to the data, the change in deviance between the co-factors of sampling date and sampling season was not significant (F3,36 = 0·40, P > 0·05), and sampling season was highly significant (F1,36 = 31·76, P < 0·001). Fitting the minimal model of significant variates and co-factors (Fig. 3) showed that the exponent b, of the power law, was close to 1·5 for both sampling seasons. Rather, it was the underlying variance of the power law, parameter a, that changed with sampling season.

Figure 3.

The relationship between the logarithms of the sample variance (s2) and sample mean (m), for A. intermedius and D. reticulatum, collected across the four spatial sampling lags in two sampling seasons. the lines represent the ancova-fitted Taylor's Power Law (log s2= a + b log m, where a and b are constants) to the A. intermedius (○) and D. reticulatum (▵) in summer (a = 0·110, SE = 0·193; b = 1·498, SE = 0·075), and the A. intermedius (●) and D. reticulatum (▴) on the other sampling dates (a = 1·286, SE = 0·107; b = 1·498, SE = 0·075).

Marked changes in the parameter a, may be associated with changes in the proportions of zero sample points in the sampling data set (J. Perry, personal communication). Such changes were investigated by fitting the sample mean, for a particular lag and date, to the number of zero sample points in the sampling. By deletion, the sample mean was found to account for 82% of the variability in the proportion of zero sample points in the sample (F1,38 = 166·42, P < 0·001) and, as expected, the proportion of zeros fell with increasing sample mean. The species co-factor was found not to be significant (F1,38 = 0·07, P > 0·05), and no significant interaction between species and the sample mean was found (F1,36 = 1·40, P > 0·05). As expected, the season of sampling was again found to be a significant effect (F1,38 = 17·23, P < 0·001), but showed no interaction with the mean (F1,37 = 0·23, P > 0·05) or slug species (F1,35 = 1·63, P > 0·05). The three-way interaction between the sample mean, slug species and sampling season also proved non-significant (F1,34 = 0·17, P > 0·05).

The minimal model (Fig. 4) showed that with increasing sample mean the proportion of zero sampling points in the sample fell. The proportion of zeros, for a given mean, was similar for both species. Most importantly, though, the proportions of zero sampling points in a sample were significantly lower in June and July, for a given mean, than in the samples taken on all other sampling dates.

Figure 4.

the relationship between the proportion of the number of zeros within a sample (p) and sample mean (m), at each of the four spatial lags, for A. intermedius and D. reticulatum in two sampling seasons. the lines represent the ancova fitted Logits (log [p/(n – p)]= a + bm, where n is the number of sampling points in a sample, and a and b are constants) to the A. intermedius (○) and D. reticulatum (▵) in summer (a = 0·615, SE = 0·213; b = − 0·855, SE = 0·082), and the a. intermedius (●) and D. reticulatum (▴) on the other sampling dates (a = 1·486, SE = 0·1525; b = − 0·855, SE = 0·082).

Spatial arrangement of slug abundance with time and lag

For D. reticulatum, SADIE_C analysis showed that spatial randomness was the rule, with spatially random distributions of individuals at all spatial lags on all sampling dates except at the 0·25 m lag in March 1997 (Table 1). The significant spatial aggregation observed in March 1997, at the 0·25 m lag, probably resulted from aggregations of juvenile individuals that had recently hatched from egg batches (Fig. 5). Indeed, the individuals within these aggregations all weighed below 25 mg (x¯= 4·83 mg, s2 = 9·54).

Table 1.  SADIE_C statistics for the spatial arrangements of the slugs A. intermedius and D. reticulatum with the spatial lag of sampling
Sampling date Arion intermedius
Sampling lag
0·25 m
1 m4 mDeroceras reticulatum
Sampling lag
16 m
0·25 m1 m4 m16 m
March 1997Ia1·2680·8791·3771·7581·4221·1201·0751·176
March 1998Ia0·6371·0560·9800·9400·3590·9700·8621·021
Figure 5.

Colour density maps of the spatial arrangements of abundance for A. intermedius and D. reticulatum, in March 1997, with sampling lag: (a) the arrangement of D. reticulatum at 0·25 m; (b) D. reticulatum at 1 m; (c) D. reticulatum at 4 m; (d) D. reticulatum at 16 m; (e) the arrangement of A. intermedius at 0·25 m; (f) A. intermedius at 1 m; (g) A. intermedius at 4 m; (h) A. intermedius at 16 m. the maps were produced by the linear interpolation of nearest neighbour sample densities.

For A. intermedius, a similar tendency to spatial aggregation was observed at the 0·25 m lag in March 1997 (Fig. 5, Table 1). Again, the individuals within the aggregations were small (x¯= 7·05 mg, s2= 32·33) and presumably represented the residual aggregations of the egg masses laid in autumn 1996. In contrast with D. reticulatum, however, the arrangement of A. intermedius changed with the sampling lag and date. In March 1997, the tendency to spatial aggregation at the 0·25 m lag resolved to spatial randomness at 1 m, and spatial aggregation at the 4 m and 16 m lags (Table 1). On the 16 m lagged grid, highly significant aggregation was observed from March to July 1997, which then disappeared from October 1997 to March 1998; although there was some indication of aggregation in November 1997 (Fig. 6). The arrangement of A. intermedius individuals became spatially random, at the 0·25 m lag, from June 1997 onwards. The distributions of A. intermedius at the 1 m and 4 m lags were also spatially random from June 1997 onwards, except for significant spatial aggregation at the 1 m lag, in June, and at 4 m, in July. Interestingly, at the spatial scales below 16 m, the observed aggregation of A. intermedius traversed the lags of 0·25 m, 1 m and 4 m as the sampling date moved from March 1997, through June to July. It is possible that this spatial and temporal dynamism in the arrangement pattern could have resulted from dispersal of the March 1997 cohort of individuals, hatched from particular egg masses. However, because neither the extent of these grids nor the sampling locations were coincident, due to the problems of destructive sampling, a test of this hypothesis was not possible.

Figure 6.

Colour density maps of the spatial arrangements of abundance for A. intermedius and D. reticulatum, at the 16 m sampling lag, with sampling date: (a) the arrangement of D. reticulatum in March 1997; (b) D. reticulatum in June; (c) D. reticulatum in July; (d) D. reticulatum in October; (e) D. reticulatum in November; (f) D. reticulatum in March 1998; (g) the arrangement of A. intermedius in March 1997; (h) A. intermedius in June; (i) A. intermedius in July; (j) A. intermedius in October; (k) A. intermedius in November; (l) A. intermedius in March 1998. the maps were produced by the linear interpolation of nearest neighbour sample densities.

At the 16 m lag, SADIE_A tests of the similarity of the spatial arrangements over time were possible. The spatial aggregation of A. intermedius, observed in June, was found not to be significantly different from the arrangement in March 1997 (Table 2). In turn, the aggregated arrangements observed in June and July were also not significantly different. Thus, a coherent aggregation, or patch population structure, of A. intermedius persisted from March to July 1997 (Fig. 6g–i). After July, direct comparisons of the arrangements of A. intermedius between July and October and between October and November were not possible because too few individuals were sampled in October for SADIE_A simulation. Comparison of the July and November samples showed, though, that the two arrangements were not significantly different. From November 1997 to March 1998, independence of the two arrangements was noted but, remarkably, a comparison of the March 1997 to March 1998 sample data showed that the two spatial arrangements were similar (Fig. 6g,l). Matching tests for the similarities of the D. reticulatum spatial arrangements, across the 16 m lagged grid, showed that all the arrangements, from whichever sampling date, were independent (Fig. 6a–f). Thus, the spatially random arrangements of D. reticulatum, at 16 m, changed significantly, and were independent, on each sampling date (Table 2).

Table 2.  SADIE_A statistics for the significance of association between the spatial arrangements on particular sampling dates, for the slugs A. intermedius and D. reticulatum across the 16 m lagged grid
Sampling datesArion intermedius
Deroceras reticulatum
March 1997 vs. June1·4770·0091·0020·487
June vs. July1·5870·0011·0210·402
July vs. OctoberNANA0·9900·706
October vs. NovemberNANA1·1060·145
November vs. March 19981·0380·2330·9850·550
July vs. November1·1130·0061·0060·386
March 1997 vs. March 19981·0950·0510·9680·759

A comparison of the arrangement pattern of A. intermedius and D. reticulatum at the 16 m lag, showed no significant difference between the species in March 1997 (Table 3). For all subsequent months, the arrangement patterns of A. intermedius and D. reticulatum were spatially independent, suggesting that A. intermedius and D. reticulatum might not, under these conditions, directly interact.

Table 3.  SADIE_A statistics for the spatial association of A. intermedius and D. reticulatum, sampled on the 16 m-lagged grid on particular sampling dates
SpeciesDeroceras reticulatum
ArionMarchIass= 1·150     
intermedius1997Tass= 0·013     
June Iass= 1·022
Tass= 0·388
July  Iass= 1·047
Tass= 0·403
October   NA
November    Iass= 1·030
Tass= 0·329
1998     Iass= 0·952
Tass= 0·740

Spatial arrangement of slugs in relation to environmental factors

We tested whether the observed spatial patterns of the two slug species were associated with plant and/or environmental variates. The index of soil moisture, at 10 cm depth, was found to be temporally dynamic. In March and November 1997, spatially random distributions of the soil moisture index were observed across the 16 m spatial lag (March 1997 Ia = 1·046, Pa = 0·365, x¯= 269·63 g kg−1, s2 = 138·93; November Ia = 1·117, Pa = 0·219, x¯= 238·30 g kg−1, s2 = 2926·01). For the remaining sample dates, either significant or strong indications of spatial aggregation were found for the moisture of the soil (June Ia = 1·316, Pa= 0·065, x¯= 191·90 g kg−1, s2 = 319·40; July Ia = 1·262, Pa = 0·080, x¯= 146·97 g kg−1, s2= 825·90; October Ia = 1·470, Pa = 0·009, x¯= 251·70 g kg−1, s2= 1497·11; March 1998 Ia = 1·882, Pa < 0·001, x¯= 230·90 g kg−1, s2= 240·09). For those dates where enough slugs were present to permit SADIE_A analyses to be completed, the arrangements of A. intermedius and D. reticulatum were found to be independent of soil moisture at 7–10 cm depth (Table 4). The day-degrees of soil temperature were found to be highly under-dispersed, with the day-degree variance, across the 16 m sampling lag, being much lower than the observed mean (Table 5). This pattern of day-degrees implies that, apart from fluctuations in the weather over the preceding 7 days, the soil temperature was similar at all sampling points across the 16 m-lagged grid; a pattern that cannot explain the spatial arrangements of A. intermedius and D. reticulatum.

Table 4.   SADIE_A statistics for the spatial arrangements of the Soil Moisture Index and slug populations across the 16 m-lagged grid
Sampling dateArion intermedius
Deroceras reticulatum
March 19970·9560·6610·9630·651
March 1998NANA1·2330·075
Table 5.  The variance and mean of the cumulative day degrees of temperature for the 7 days preceding the date of slug sampling
Sampling dateDay-degrees of temperature, °C
Variance, s2
March 199767·8191·034
March 199869·1755·693

The March 1997 arrangement of plant height was found to be spatially random (Ia = 0·873, Pa = 0·723, x¯= 37·57 cm, s2= 6·32). The spatial arrangement of plant stem density indicated a strong tendency to aggregation (Ia = 1·381, Pa = 0·031, x¯= 81·00, s2= 266·55). In turn, the weight of the wheat plant ears, in June and July 1997, were arranged in a spatially random manner (June Ia = 1·079, Pa = 0·258, x¯= 9078·33 mg, s2= 1407062·64; July Ia = 0·934, Pa = 0·583, x¯= 17290·67 mg, s2= 9988627·13). The arrangement patterns of A. intermedius and D. reticulatum were found not to be associated with these plant variates. The arrangement of A. intermedius was independent of ear weight in June (Iass= 0·985, Tass= 0·808) and July (Iass= 0·993, Tass= 0·473), and plant height (Iass= 1·010, Tass= 0·404) and stem density (Iass= 0·986, Tass= 0·617) in March 1997. Similarly, the arrangement of D. reticulatum was independent of ear weight in June (Iass= 0·990, Tass= 0·665) and July (Iass= 0·993, Tass= 0·706), and plant height (Iass= 0·858, Tass= 0·868) and stem density (Iass= 0·983, Tass= 0·720) in March 1997.


As might be expected for a species that reproduces whenever conditions are favourable, the abundance of D. reticulatum was found to increase from March to June 1997. Over the same period, A. intermedius numbers declined, as mortality in the field reduced the numbers of slugs from a maximum abundance of eggs, laid the previous autumn during the sole annual bout of A. intermedius reproduction. By October 1997, the abundance of both species had collapsed to low levels where they remained through to March 1998. The reasons for these marked declines are unknown but could include the effects of post-harvest cultivation in September 1997 and predation by carabid beetles, or the declines may be part of a general decline from high slug abundances previously noted in arable crops (Glen et al. 1991b; Glen et al. 1992).

For this paper, the importance of these dynamics was that they provided a broad and species-specific range of sample densities across which parametric intensity and spatial arrangement could be compared.

Parametric intensity

The variance to mean relationships, for these marked changes in slug abundance, were found to conform to Taylor's Power Law. The slope of the Taylor plots, parameter b, was common to both A. intermedius and D. reticulatum, at approximately 1·5, and was similar for all sampling seasons and at all spatial lags. The underlying variance of the Taylor plots, parameter a, was similarly common to both species and all lags, but changed with the sampling season. Interestingly, this change in the underlying variance of the plots seemed to be associated with a drop in the proportion of zero counts in the sample data, in the summer months. The slope of 1·5 would indicate that the variance of the slug count data was in excess of the mean. This is true of most biological data where there are typically many low counts and a few high counts within a sample (Anderson et al. 1982; Taylor 1961; Taylor & Taylor 1977). In essence, this means that where individuals of A. intermedius and D. reticulatum did occur, they tended to occur in high numbers. It should be stressed, however, that these high counts do not necessarily denote spatially aggregated arrangements.

These parametric results are intriguing. To explain them we simply postulate that A. intermedius and D. reticulatum share a common variance to mean relationship, because of similar life histories. Once this postulate is adopted, it then becomes important to explain why the slope of the relationship does not change with the spatial lag of sampling, what the common life history parameters may be, and why the underlying variance changes with the sampling season.

It is generally expected that with increasing spatial lag (or extent), increases in sample variance would be observed (Wiens 1989). However, Wiens (1989) was discussing heterogeneous landscapes where, as the extent (or lag) of the sampling increased, spatial pattern would be included that was not present at smaller lags or extent. Here, however, the slope of the Taylor plots did not change with increases in the lag and this may be attributed to the relatively homogeneous habitat encompassed by the extent of the sampling. Wiens (1989) also noted that sample variance is influenced by grain. In our study, though, the 0·25 m sampler grain was implicitly built into, and imposed upon, the whole sampling approach. For the parametric intensity analyses then, the slug counts were really only a sample of the 0·25 m sampler grain, taken across a relatively uniform field.

Spatial analysis of data from March 1997 (Table 1 and Fig. 5a,e), would indicate strong aggregations of juvenile A. intermedius and D. reticulatum at this 0·25 m sampler grain. We have interpreted these aggregations as individuals recently hatched from egg batches, and suggest that the grain of 0·25 m was appropriate for sampling slug egg batches. The slope of the Taylor plots would therefore chart the underlying relationship between the mean size and variance, not of egg batches but of eggs laid in batches, for A. intermedius and D. reticulatum in March 1997. These were over-dispersed, and for a given mean batch size could possibly be described by the Negative Binomial distribution, as has been shown for D. reticulatum (Warley 1970). Presumably, the habit of laying eggs in batches, and sometimes in clumps of batches (South 1965), establishes the slope of the variance to mean relationship for A. intermedius and D. reticulatum, which then persists across all sampling seasons.

The marked reduction in the underlying variance of the Taylor plots, as described by parameter a, in the summer season, is more difficult to explain. Any explanatory mechanism would have to operate at the 0·25 m scale and account for a reduction in the number of zero sample data. It is possible that the movement of growing individuals away from the sites of egg laying could, by itself, account for the reduction in zero counts in the summer months. However, another potential mechanism could be predation by adults of the carabid beetle Pterostichus melanarius Illiger, which is a known predator of slugs (Symondson et al. 1996). This carabid is mainly active from June through to September, and its abundance in pitfall trap catches has been shown to be associated with a density-dependent reduction in the rate of growth of the slug population (Bohan et al. 2000). Using a subset of the slug population data presented here, the effect of the carabids appeared to be to decrease slug populations in locales of high density, at a 0·25-m grain, whilst in areas of lower slug density there were fewer carabids and slugs numbers increased. The predation effect of P. melanarius thus worked across the 0·25 m scale, possibly leading to a reduction of the over-dispersion in the count data. The reduction in zero counts could be accounted for if predation modifies the distribution of the surviving slugs. If those slugs that encounter P. melanarius, and survive, move in the order of 0·25 m or more away from the predator, then predation could account for the reduction both in over-dispersion and zero counts in the sample data.

This discussion of the parametric analysis would suggest a surprisingly simple story. The distributions of A. intermedius and D. reticulatum were similar, presumably because of similarities in the egg-laying habit of the two species, at the 0·25 m scale. Changes in the parametric distribution of A. intermedius and D. reticulatum, with season, applied to both species and there was no increase in count information from increasing the between sample point distance. However, such simplicity in the parametric intensity analysis is difficult to reconcile with the species-specific spatial arrangement.

Spatial arrangement and association

From the common habit of laying eggs in batches, both A. intermedius and D. reticulatum exhibited distinct aggregations of juvenile individuals, at the 0·25 m lagged grid, in March 1997, presumably about the sites where eggs were laid the previous autumn. At the sampling lags of 1 m and above, the aggregations of D. reticulatum juveniles at 0·25 m were found to resolve into spatial randomness. For A. intermedius, however, the strong tendency to spatial aggregation at 0·25 m was observed to become significant spatial aggregation at 4 m, traversing through spatial randomness at the 1 m scale, to form a distinct spatially aggregated arrangement, or patch, at the 16 m lag. Thus, from a common spatial arrangement at the 0·25 m lag, distinct spatial arrangements were observed for the two species at higher lags.

Across the 16 m lag, in March 1997, the spatial arrangements of A. intermedius and D. reticulatum were associated with high numbers of A. intermedius and D. reticulatum in the western-most portion of the field. These initial similarities in arrangement for the two species did not persist over time. Comparison within each species showed that the spatial arrangements of D. reticulatum were spatially independent over time, whilst from March to July 1997 the arrangements of A. intermedius were significantly associated, and the A. intermedius patch persisted. A significant association between the March 1997 and 1998 arrangement might also suggest that the A. intermedius patch persisted for up to 1 year. Thus, from a common spatial arrangement in March 1997, at 16 m, distinct and species-specific spatial arrangements were observed for the two species over time. These species-specific patterns could either have been due to heterogeneities in the field environment and/or the different population biology of the two species working through an active tension between rates of reproduction, mortality and movement.

Environmental heterogeneity

Although the field site was assumed to be relatively uniform, and the intensity analysis of slug abundance would support this assumption, the field environment was not truly homogeneous. Soil moisture and the plant variates were all found to be under-and over-dispersed, and dispersed on different sampling dates. The spatial arrangement of these variates also changed with the date of sampling, with spatial aggregation and randomness being observed. There was, however, no clear pattern to the environmental data, except for soil temperature, which showed parametric under-dispersion, and on no date was there any statistical association between the spatial arrangements of the environmental variates and either A. intermedius or D. reticulatum.

If the environmental variates had limited the spatial distributions of A. intermedius and D. reticulatum, then association between the slug species might have been expected. However, after March 1997, the 16 m scale distributions of A. intermedius and D. reticulatum were spatially independent. It could be argued, not that the environmental variates had no effect on the spatial distribution of the slugs, but that the environmental variates did not limit the spatial arrangement of A. intermedius and D. reticulatum during the period of study and the data characterize to a close degree the spatial and temporal dynamics of A. intermedius and D. reticulatum in a homogeneous environment. This statement does not account for the association of the two slug species in March 1997 across the 16 m grid. We believe that the similarity of the A. intermedius and D. reticulatum arrangements was a historical consequence of survival within Field 54 prior to the study. From March 1997 onwards, however, the observed spatial dynamics of A. intermedius and D. reticulatum did not appear to be due to environmental heterogeneity alone. Rather, it would seem most likely that it was differences in the population biology of the two species that produced the species-specific arrangements.

Population biology

The habit in both A. intermedius and D. reticulatum of laying eggs in batches is likely to produce the common spatially aggregated arrangements observed for the two species across the 0·25 m lag. Persistence of a single patch of A. intermedius at the 16 m lag from March to July 1997, and possibly until March 1998 may be because individuals of A. intermedius have low dispersal potential (Glen & Wiltshire 1986) in comparison with the scale of the patch. By contrast, D. reticulatum is thought to be more surface-active than A. intermedius (Glen & Wiltshire 1986), it can move several metres per night (Bailey 1989) and reproduces whenever conditions are suitable (Hunter & Symonds 1971). Together, and in concert with local survival and recruitment, these population processes could lead to patterns of distribution that do not coalesce to a patch, with increasing spatial scale, and that are spatially independent with sampling date, as observed. It should be borne in mind, however, that the location of the 16 m sampling grid was moved by 1·5 m on each successive sampling date. Whilst this relocation was assumed to be unimportant in the analysis, it is possible that the spatially random arrangements observed at lower spatial scales were superimposed on the spatial arrangement observed at 16 m, generating the independence observed for D. reticulatum on successive sampling dates.

Spatial scaling

The finding that A. intermedius was aggregated into egg batches across the 0·25 m grid and randomly arranged at 1 m would suggest that the general arrangement of A. intermedius was as a randomly broadcast set of batches spaced approximately 1 m apart. Across the 16 m grid, however, a strong patch structuring was present. This marked change in arrangement was certainly not predictable from the sampling at the 0·25 m and 1 m lags. Rather, the arrangements across each lag were distinct and different.

Wiens (1989) classified spatial pattern in terms of domains of scale that had coherent form, separated by often discontinuous transitions. The different arrangements of D. reticulatum and A. intermedius, at each lag, belong to arrangement domains similar to those described by Wiens (1989). This means that arrangements studied at one spatial scale are distinct from arrangements at other spatial scales. In specific cases, identifiable behavioural or organizing mechanisms may allow predictions of arrangements to be scaled up. However, unlike the description of form by the use of variation with distance methods, we know of no general methods for scaling up spatial arrangements.

This has implications for the terminology used for describing spatial arrangements. For general descriptions of form, the anthropocentric scales of grain, lag and extent are extremely useful because they may be related (scaled-up) to the biological scales that are the subject of study. For evaluating spatial arrangements, though, these scales of sampling are inadequate. Only where the grain, lag and extent prove to be appropriate to the domains of coherent arrangement will meaningful arrangements of individuals and populations be highlighted. It is crucial, therefore, for the individual researcher to make informed decisions on where these domains exist.

Spatial dynamics

This analysis of the distributions of D. reticulatum and A. intermedius highlights four quite general findings that concern spatial pattern. First, scale should be defined very carefully. The sampling scales of grain, lag and extent should be treated as being different from the scales of the individual or population.

Secondly, it is clear that parametric intensity does not describe the spatial arrangement of D. reticulatum and A. intermedius. Both D. reticulatum and A. intermedius show a common slope to the Taylor plots, which could indicate that the frequency distribution of slug counts was similar for both species. Here, however, the implied aggregation of the over-dispersed frequency distributions neither translates to explicitly spatial aggregations nor to common spatial arrangements of the two slug species in the field. A correspondence does appear to exist between parametric and spatial scales, though, at the 0·25 m lag. At this dimension, the grain of sampling and the spatial lag coincided, and resided within the spatial domain of egg batch laying. Thus, the parametric over-dispersion of slug eggs laid in batches was found to reflect the spatial aggregation of the egg batches.

This leads to the third general finding that different spatial arrangements were apparent at different spatial scales. In March 1997, distinct spatial aggregations of both D. reticulatum and A. intermedius were found on the 0·25 m scale. With increasing spatial scale, though, other arrangements of slugs were noted. Spatial randomness was observed for D. reticulatum at all scales, whilst for A. intermedius spatially random arrangements at low spatial scales resolved to significant spatial aggregation at the 16 m scale, from March to July 1997.

The final general finding, which is closely related to finding 2, was that significant changes in parametric intensity with time did not necessarily manifest as changes in spatial arrangement. Both A. intermedius and D. reticulatum showed markedly lower sample variances, for a given mean, in the summer months of June and July 1997 than for all other sample months. These changes in the relationship between sample mean and sample variance were not readily apparent in the spatial arrangements of A. intermedius and D. reticulatum, however.


This comparative study shows clearly that parametric intensity and spatial arrangement describe quite different but complementary aspects of a sample data-set. The intensity analyses show how sample variance changes with the mean, on scales (grain, lag and extent) defined by the sampling method, but do not describe the spatial arrangement of the data. For that, explicitly spatial approaches, such as SADIE, are needed. SADIE highlighted that A. intermedius and D. reticulatum exhibited species-specific spatial arrangements, which were consistent with differences in the biology of the two slugs and were not determined by environmental heterogeneity; spatial dynamism that could not have been expected from considering the parametric intensity analyses alone. Together, these spatial-arrangement and parametric-intensity analyses have provided a considerable advance in our understanding of spatial and temporal arrangements and demography of slug herbivores, and this combination of methods provides a powerful tool of wide applicability in ecology.


This work was funded by the Biotechnology and Biological Sciences Research Council and the Ministry of Agriculture, Fisheries and Food of the United Kingdom. In addition the authors would like to thank Prof. Joe Perry, Dr Phil Brain and Dr George Thomas for their comments and encouragement during this research, together with Dr Gordon Port and an anonymous referee who suggested improvements to the manuscript.

Received 17 January 2000;revisionreceived 5 July 2000