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Dimensions of spatial heterogeneity
Although the aggregation of individuals has been a recurring theme in ecology for many decades (Raunkiaer 1934; Cole 1946, Taylor 1984, Perry et al. 2002), until recently the lack of adequate spatial analytical methods has limited the examination of spatially explicit phenomena (Liebhold et al. 1993; Perry et al. 2002). The wide array of possible approaches to the measurement and interpretation of aggregation that are now available were recently reassessed in light of current analytical developments (Coomes et al. 1999, Dale et al. 2002, Perry & Dixon 2002). However, these reviews do not consider the quantification of aggregation per se but rather any spatial pattern described in ecology. We use the term ‘spatial heterogeneity’, sensu Wiens (2000), to broadly encompass the array of spatial patterns (aggregation being only one of these) that can be described (Table 1, Fig. 1). Spatial heterogeneity can formally be defined as « discontinuities in space » (Wiens 2000), or pattern in spatial data (Liebhold & Gurevitch 2002), and may be quantified for either abundance (count), or occurrence (presence-absence) data. Although, spatially-referenced occurrence data can be transformed (with a loss of fine scale spatial information) to abundance per unit area (Perry & Dixon 2002; Perry et al. 2002), the use of untransformed occurrence data is common in ecology (Coomes et al. 1999; Plotkin et al. 2002 Wiegand & Moloney 2004). Therefore, defining terminology for spatial heterogeneity in both abundance and occurrence data will further contribute to unambiguous definitions in spatial ecology. Nonetheless, potential differences between the forms of spatial heterogeneity describe are likely to be greater for abundance than occurrence data, because spatial references are accompanied by recorded variable. Consequently, we focus on the differences between different degrees of spatial explicitness using abundance data (see Coomes et al. 1999; Plotkin et al. 2002; Wiegand & Moloney 2004 for detailed coverage of measures used to describe occurrence data).
The term ‘aggregation’ has commonly been used to denote the grouping of elements or a contagious condition of spatial heterogeneity. However, this term does not distinguish between the dimension (thus the level of spatial explicitness) used to quantify this form of spatial heterogeneity (i.e. the method used) (Wiens 2000). Because the form of spatial heterogeneity that is identified (e.g. overdispersed versus underdispersed, or regular versus aggregated, Table 1) may differ depending on the measure used to identify it (Perry 1998), the term ‘aggregation’ has become potentially misleading. Therefore, to allow unambiguous reporting of results, a need for formalised terminology to describe different forms of spatial heterogeneity has arisen. Here we use ‘aggregation’ as a loose, generic term for any grouping of elements (which is one form of spatial heterogeneity), and use the terminology outlined in Table 1 for reference to specific dimensions, measures and forms of spatial heterogeneity. The terms provided are largely a synthesis of those in the literature, and their distinction here is intended to avoid confusion between the data type, category and form of spatial heterogeneity described by different measures of spatial heterogeneity (Table 1). In the following paragraphs I outline the three categories that vary in the degree to which they are spatially explicit, each generally represented by a single commonly used measure of spatial heterogeneity in abundance and occurrence data. The three dimensions are spatially non-explicit, spatially semi-explicit and spatially explicit heterogeneity (Table 1). In addition, for abundance data we list a method for each dimension of spatial heterogeneity used to correlate the spatial heterogeneity in two data sets.
Statistical heterogeneity and nearest neighbour distance (quantified for abundance and occurrence data respectively) can be considered measures of spatially non-explicit heterogeneity, because records are taken across a series of sampling points in a site, which are not spatially referenced (e.g. Williams et al. 2001) (Table 1, Fig. 1). In fact statistical heterogeneity can be said to be totally independent of spatial pattern. Nonetheless, not using spatial information (spatially non-explicit heterogeneity) can be seen as the preceding step of using spatial information (spatially semi- and explicit heterogeneity, Table 1) in a classification of spatial data use. Spatial structure (abundance data) (Legendre & Legendre 1998) and spatial distribution (occurrence data) (Perry 1995a), are measures of spatially semi-explicit heterogeneity, because although spatial dependencies or patterns can be accounted for (Perry et al. 2002), the heterogeneity described is not explicitly related to any particular location within the study site (i.e. the exact pattern of specific locations or areas within a site are unknown) (Wiens 2000). Spatial non-randomness and point-spatial clustering are measures of spatially explicit heterogeneity (Table 1, Fig. 1). These measures are spatially explicit because spatial heterogeneity may be related to particular sample points or areas within the study arena (Wiens 2000). For example, with spatial non-randomness the position of areas of comparatively high counts can be described, while point-spatial clustering describes the position and size (number of individuals) of groups of individuals (Plotkin et al. 2002).
The traditional, spatially non-explicit approach to the measurement of heterogeneity, statistical heterogeneity (Table 1), is merely the relationship between the variance and the mean of the frequency distribution of counts and can be quantified by the Poisson index of dispersion (Perry 1998) (Table 1, Fig. 1). Animal abundance, usually with many zero and few large counts, is also considered overdispersed when fit by the negative binomial distribution (NBD) (Bliss & Fisher 1953; Williams et al. 2001). In such cases the index of aggregation of the NBD, k, is less than unity (Bliss & Fisher 1953). However, the ability of k to describe ecologically relevant spatial pattern has long been contested (Taylor et al. 1979). For example, the inverse of k behaves inconsistently over ranges of over dispersion even when the NBD fits the data, indicating the inadequacy of k to describe statistical heterogeneity (Taylor et al. 1979). Also, the NBD is not suitable for quantifying aggregation in patches of variable size (Sevenster 1996), or predicting abundance from occupancy in some cases (Warren et al. 2003).
Furthermore, Perry & Hewitt (1991) and Perry (1995b) consider overdispersion to be of limited interest when investigating the spatial heterogeneity of individuals, because overdispersion in biological data is virtually universal (Taylor et al. 1978). The relationship between two variables described by statistical heterogeneity can only be determined by correlation (Table 1). Although the two data sets may share spatial references, spatial references are not included in the quantification of the relationship. Any correlation detected will therefore be spatially non-explicit.
Cocoon sampling
Within each plot every tree was carefully searched for cocoons. Cocoons were inspected to determine the fate of the pupa inside the cocoon, i.e. i) parasitised, ii) alive, iii) dead as a result of unknown causes, or iv) successfully emerged. This was indicated respectively by the i) presence or ii) absence of small emergence hole(s), iii) light weight of the cocoon or iv) a single large anterior emergence hole (pers. obs.). Parasitoid species responsible for parasitism may be identified from the shape and size of emergence holes left in the cocoon wall of a parasitised pupa (Veldtman et. al 2004). The number of pupae and parasitised pupae per tree were counted.
The position of each tree within a plot was measured at the main trunk of the tree with a hand held Global Positioning System (GPS). For trees in close proximity to each other the direction and distance between the two trees were noted and assigned to one of three categories (half, quarter and a tenth of the third (last) decimals of a minute) based on hand drawn maps which specifically documented this fine scale distribution of trees. These spatial co-ordinates were used in all spatial analyses. For the investigation of the spatial pattern of parasitism, only sampling points (trees) with at least one pupa were included in analyses, as parasitism events can logically not be observed if there are no pupae. All counts of pupae or parasitised pupae were thus made per tree. At each site, pupae parasitised by different species of parasitoid were either analysed individually, or collectively (‘all species’) as a measure of total parasitoid mortality (see also Heads & Lawton 1983; Williams et al. 2001). Additionally we also considered the proportion of parasitised pupae (parasitism rate from here on), which was transformed into integers by multiplying by ten and rounding off.
GENERAL INTRODUCTION
References
CHAPTER 1: Predicting population dynamics and natural enemy responses from herbivore life history and defensive traits
Introduction
Methods
Results
Discussion
References
CHAPTER 2: The parasitoids of southern African wild silkmoths (Lepidoptera)
References
CHAPTER 3: Aggregation and spatial heterogeneity: from one dimension to the next
Introduction
Methods
Results
Discussion
References
Appendix
CHAPTER 4: Spatially explicit host-parasitoid relationships: density dependence revisited
Introduction
Methods
Results
Discussion
References
CHAPTER 5: Variability in cocoon size in southern African wild silk moths: implications for sustainable harvesting
Introduction
Methods
Results
Discussion
References
CHAPTER 6: Fine-scale pupal abundance and distribution patterns of Gonometa postica and G. rufobrunnea (Lepidoptera: Lasiocampidae)
Introduction
Methods
Results
Discussion
References
GENERAL CONCLUSION
References
Appendices
