Identifying aggregation and association in fully mapped spatial data.
Traditional studies of competition have used a range of experimental procedures to determine the impact of one species on the growth or reproduction of another (reviews by Law and Watkinson 1989, Goldberg and Barton 1992). These experiments are usually able to address questions of the form: What is the outcome of competition in well-mixed populations? However, in order to assess the strength of interactions between species in natural communities we need to know how often individuals of the different species interact. That is, we need to know how spatial structure mediates the strength of interactions between species, and, in turn, we need to know how interspecific competition affects the spatial structure (Harper 1977, Silvertown et al. 1992). Several recent studies have advanced our understanding of how spatial pattern and competitive interactions are interrelated (Mahdi and Law 1987, Mahdi et al. 1989, Pacala and Silander 1990, Silvertown et al. 1992, Herben et al. 1993, Law et al. 1993, Cain et al. 1995, Pacala et al. 1996, Rees et al. 1996).
Of course, the spatial pattern of a community is not only determined by competitive interactions. A combination of processes that may include the initial distribution of colonists, the history of disturbance, patterns of clonal growth and seed dispersal, and the underlying heterogeneity of the abiotic environment could influence the observed spatial pattern (Law et al. 1993). It is not possible to determine unambiguously whether an observed arrangement of plants has resulted from a particular process (Pielou 1961, 1974), but pattern analysis may lead to inferences about which processes are important (Watt 1947, Greig-Smith 1952, Kershaw 1959, Kenkel 1988, Rebertus et al. 1989, Duncan 1991, Leemans 1991).
Aspects of pattern that interest ecologists include the degree to which a given species forms intraspecific clumps, the extent of associations between species, and the persistence of pattern between sites and over time. Methods can be broadly divided into those based on counts (Fisher et al. 1922, Greig-Smith 1952, Hill 1973, Dale and MacIsaac 1989, Dale and Blundon 1990) and those based on precise locations of plants within quadrats (Ripley 1981, Diggle 1983). In this paper we seek to develop the latter approach.
Plants of a given species may be aggregated, random, or regularly arranged in space. Some of the most powerful tests of aggregation are based on drawing circles of fixed radius and calculating either (1) the probability of having at least one conspecific plant within the circle (Diggle 1983) or (2) the mean number of conspecific neighbors within the circle (Ripley 1981). The simplest way to test the significance of these aggregation statistics is to compare their values against the distribution of values generated by Monte Carlo simulation. An appropriate simulation process is the two-dimensional Poisson process, which produces patterns exhibiting complete spatial randomness (CSR sensu Diggle 1983).
Similar tests may be used to analyze bivariate patterns. Two species, i and j, are defined as being "negatively associated" when we observe fewer neighbors of species j around plants of species i than would be expected around randomly chosen points ([K.sub.12]-function, Ripley 1981). "Positive association" is similarly defined, but involves an excess rather than a deficit of heterospecific neighbors. The degree of association is scale-dependent, so, for example, two species may be negatively associated on a fine scale, but exhibit no association on a larger scale. Again Monte Carlo simulation is the simplest way to test significance, but more care is needed in choosing a randomization process. The difficulty is that the variance of the null estimator is dependent upon the degree of aggregation of the two species (Lotwick and Silverman 1982). Ideally the simulated patterns should preserve the characteristics of the intraspecific spatial patterns, while randomizing the positions of species j with respect to plants of species i. Several approaches to simulation have been proposed, but a consensus on which is most appropriate has yet to be reached (Lotwick and Silverman 1982, Watkin and Wilson 1992, Palmer and van der Maarel 1995).
An alternative method of exploring relationships between two patterns is based on identifying nearest neighbors. Species are said to be "segregated" when the ratio of conspecific to heterospecific nearest neighbors is greater than expected and "attracted" when the ratio is less than expected (Pielou 1961, Cuzick and Edwards 1990, Dixon 1994). Segregation of species has implications for neighborhood competition, because it results in a dearth of interspecific interactions, and may thereby slow rates of competitive exclusion (Pacala and Silander 1990). When edge effects are negligible, the expected ratio is simply the ratio of the abundance of the two species, and a random shuffling of plant identities provides Monte Carlo simulations with which to estimate variance (Dixon 1994). Note that segregation depends upon the degree of negative association and upon the extent of intraspecific aggregation.
In this paper, we describe a method of locating the positions of plant clusters in fully mapped coordinate data. The method uses a test of single-species aggregation developed by Diggle (1983) in conjunction with a form of cluster analysis (Howard 1990). We use the method to provide a description of aggregated patterns based on the size distribution of their constituent clumps. We also describe how knowledge of the distribution of clump centers can aid the analysis of association between species. A Monte Carlo simulation is described that is based on randomizing the positions of clump centers. The grouping of plants into clumps removes fine-scale spatial autocorrelation, so tests of segregation between clump centers provide information about underlying associations.
The methods are illustrated by four examples: (1) a series of single-species patterns generated by a simple clumping process, (2) a series of two-species patterns that exhibit different associations between species on the fine and course scales, (3) a complex pattern generated by a spatially explicit population model, and (4) some field data from a community of sand dune annuals. We discuss the merits and limitations of the method in comparison with the techniques currently available.
Outline of approach
Consider a pattern that consists of individuals grouped into several clumps. The proposed method ascertains the positions of the clumps and then describes the pattern in terms of the number of individuals contained in each clump and the radius of each clump. It works by repeatedly replacing pairs of close-together individuals by single points (clump centers) at their mean position. This process leads to a progressively less aggregated pattern, because close-together pairs are being systematically removed. We continue amalgamating until we produce a pattern that is indistinguishable from two-dimensional Poisson, and information about clump sizes is then taken from this pattern.
Tests for single-species aggregation using distances to nearest neighbors
At each step of the amalgamation process, we need to decide whether the remaining pattern of clumps is indistinguishable from random. Let P(R [less than or equal to] r) be the probability that the distance from a plant to its nearest neighbor, R, is less than or equal to a specified distance, r. One approach is to compare P(R [less than or equal to] r) with the probability expected when individuals have no effect on the conditional probability that a neighbor will be present (defined as complete spatial randomness by Diggle 1983). When a species has an aggregated distribution, P(R [less than or equal to] r) exceeds the expected probability at low r [ILLUSTRATION FOR FIGURE 1 OMITTED].
When points are generated by a two-dimensional Poisson process on an infinite plane ([Rho] individuals/[mm.sup.2]), the expected number of individuals within a circle of radius r is [Rho][Pi][r.sup.2]. The probability that there are no individuals within a circle of radius r is simply exp(-[Rho][Pi][r.sup.2]), and therefore the probability that an individual has a nearest neighbor less than a distance r away is
[G.sub.[P.sup.[infinity]]](r, [Rho]) = P(R [less than or equal to] r) = 1 - exp (- [Rho][Pi][r.sup.2]). (1)
This is a cumulative density function (cdf) with the property that [G.sub.[P.sup.[infinity]]](0, [Rho]) = 0 and [G.sub.[P.sup.[infinity]]] ([infinity], [Rho]) = 1. Differentiation of this expression gives the probability density function of the distance from a plant to its nearest neighbor.
In practice, we have to work within quadrats rather than on infinite planes. Some individuals are nearer to an edge than to their nearest neighbor, making edge corrections necessary (Ripley 1981, Cressie 1991, Haase 1995). Edge corrections were estimated by simulating 200 two-dimensional Poisson processes in a quadrat (500 mm x 500 mm). For each simulated pattern we calculated P(R [less than or equal to] r) for radii within the range 0-100 mm, including all plants in the calculations, even those close to the edge. The mean cdf, [G.sub.PQ](r, [Rho]), was then calculated as the mean of 200 simulations. A plant near the edge of a quadrat is likely to have fewer than [Rho][Pi][r.sup.2] neighbors, and consequently [G.sub.PQ](r, [Rho]) [less than or equal to] [G.sub.[P.sup.[infinity]]]](r, [Rho]) is expected to hold for all values of r [ILLUSTRATION FOR FIGURE 1 OMITTED]. The edge correction, [Xi], was calculated as
[Xi](r, [Rho]) = [G.sub.PQ](r, [Rho]) - [G.sub.[P.sub.[infinity]]](r, [Rho]). (2)
Edge corrections were calculated for population sizes, N, of 20, 40, 60, . . ., 1000 individuals, and nonlinear regression was used in a trial and error fashion to find a function that accurately described the relationship between r (1-100 mm), N (20-1000), and [Xi] [ILLUSTRATION FOR FIGURE 2 OMITTED].
The degree of clumping was tested by calculating the distance from each plant (including those near the edges of the quadrat) to its nearest neighbor and using this distribution of distances to generate a cdf, G(r, N). The observed cdf was then compared with that expected under complete spatial randomness, [G.sub.PQ](r, N). We chose to compare G(r, N) and [G.sub.PQ](r, N) by calculating the maximum difference between them:
dw = max[G(r, N) - [G.sub.PQ](r, N)] (3)
where max denotes the maximum values across all radii [ILLUSTRATION FOR FIGURE 1 OMITTED]. This is the Kolmogorov-Smirnov test statistic given by Diggle (1979).
We chose to include all plants in our estimation of G, and then to make edge corrections to the theoretical curve. A simpler approach is to consider only those plants that are closer to their nearest neighbor than to an edge, obviating the need to correct the theoretical curve (Diggle 1983, Cressie 1991). We chose a more complex method because the cluster analysis considers all plants within a quadrat, and it was important to be consistent when repeatedly switching between Diggle's test and the clustering algorithms within the clump recognition process.
Significance tests of aggregation using Monte Carlo simulation
The next step is to test whether the dw value of a given pattern is different from the null distribution of dw values predicted by Monte Carlo simulation. One thousand patterns were generated using a two-dimensional Poisson process, and the G(r, N) of each pattern was compared with [G.sub.PQ](r, [Rho]) to obtain the null distribution of dw. The upper tail of the null distribution was used to calculate critical values (e.g. the 95th percentile gave the critical value [dw.sub.crit] for P = 0.05).
An approach that is more efficient than running simulations for each pattern is to run simulations for a range of population sizes and then find a function that relates [dw.sub.crit] to N. We ran 1000 simulations for patterns containing 20, 40, 60, . . ., 1000 individuals, and used a nonlinear regression to fit the function shown in Fig. 3. The method described so far is no different from that given by Upton and Fingleton (1985), except that we have provided functions for edge corrections and threshold values of dw, greatly reducing the time needed to make each test. This time saving is important, because the clump recognition process relies on the repeated use of Diggle's test.
A method of finding clump centers
The process that reduces a pattern to its constituent clump centers uses a method of cluster analysis developed by Sokal and Michener (1958), known as the Weighted Pair Group Centroid Method (WPGMC). Initially we have a map containing the positions of plants, each of which is defined as a cluster composed of a single individual. At each step, a pair of clusters is chosen by calculating the distance between each cluster and its nearest neighbor, weighting this distance by the number of individuals contained in both clusters, and then selecting the smallest weighted distance. The selected pair is then amalgamated to form a single cluster that is positioned at the "center of gravity" (or centroid) of all the individuals from which it is composed.
As an example of the clump recognition process, we use a pattern that contains 280 individuals that are aggregated (dw = 0.252, [dw.sub.crit] = 0.037). Each cycle of the process reduces the values of dw and slightly increases the values of [dw.sub.crit], until the two curves cross after 210 amalgamations [ILLUSTRATION FOR FIGURE 4 OMITTED]. The termination of clumping is dependent upon the probability at which one decides to accept that a pattern is indistinguishable from CSR (henceforth referred to as [P.sub.stop]).
Measures of aggregation
The clump recognition process produces the following measures of aggregation:
1) [Mathematical Expression Omitted] is the mean number of individuals per clump.
2) [Mathematical Expression Omitted] is the average distance of plants from their respective clump centers. It is calculated by taking the mean distance
within each clump, and then taking the mean across clump centers, weighted by the number of plants each clump contains. Clumps containing single plants are defined as having zero radius.
3) A is the amalgamation index, defined as the number of reductions needed to produce a random pattern as a proportion of the number of individuals in the original pattern. A is related to [Mathematical Expression Omitted] by
[Mathematical Expression Omitted]. (4)
In order to gain a better understanding of the statistics derived from the clump recognition process, it is useful to compare them with tests based on counting the number of neighbors within a specified distance of target plants (Ripley 1981, Diggle 1983). If there are [N.sub.i] plants of species i within a quadrat of area Q, and if the plants are randomly located, then the mean number of plants within a distance r of a randomly chosen plant will be E([N.sub.ii]) = [Pi][r.sup.2]([N.sub.i] - 1)/Q. The degree of aggregation of a pattern may be expressed by the following index:
AI(r) = [summation of] [N.sub.ii](r) where 1 to [N.sub.i]/[[Pi][r.sup.2][N.sub.i](1 - [N.sub.i])/Q] (5)
where [N.sub.ii] is the observed number of conspecific neighbors within a distance r of each plant of species i. When AI [greater than] 1, clumping is indicated, while AI [less than] 1 indicates regularity. The significance of any deviation from the expected value of [N.sub.ii] is tested by Monte Carlo simulation, using a two-dimensional Poisson process to generate random patterns.
Measurement of interspecific association
Two species, i and j, are defined as being negatively associated when we observe fewer neighbors of species j around plants of species i than would be expected around randomly chosen points ([K.sub.12]-function, Ripley 1981). Let there be [N.sub.i] plants of species i, and [N.sub.j] of species j within a quadrat of area Q. If the two patterns are independent of one another, the number of plants of species j within a distance r of a randomly chosen plant of species i is expected to be E([N.sub.ij]) = [Pi][r.sup.2][N.sub.j]/Q. It can be shown that E([N.sub.ij]) does not depend on whether the intraspecific patterns of the two species are random, clumped or regular; it depends only on the degree of dependence of one pattern on the other.
The degree of association is expressed by using the following index:
[SI.sub.ij](r) = [summation of] [N.sub.ij](r)/([Pi][r.sup.2][N.sub.i][N.sub.j]/Q) where 1 to [N.sub.i] (6)
where [N.sub.ij] is the number of neighbors of species j within a distance r of a plant of species i. When SI [greater than] 1, positive association is indicated, while SI [less than] 1 indicates negative association.
The ideal process for Monte Carlo simulation should preserve the characteristics of intraspecific patterning while randomizing the pattern of one species with respect to the other. The positions of the clump centers that we have identified are, by definition, indistinguishable from random (within a specified probability), and patterns generated by randomizing the positions of the clump centers should retain many aspects of the intraspecific pattern. On these grounds, the randomization of clump centers was used for Monte Carlo simulation. The computer randomizes the positions of the clump centers of species j and then reassembles a plant map using information that it has retained regarding the position of each plant relative to its clump center. The sum of the individual [N.sub.ij] values is calculated for each simulated pattern, and the median of 400 patterns is used as the expected value. Confidence intervals are generated by considering the values of the 2.5 and 97.5 percentiles. Edge effects do not affect the significance tests because edge plants were included in the analysis of both observed and simulated patterns.
Measurement of interspecific segregation
Two species are segregated if the probability of having a conspecific nearest neighbor is greater than expected by chance. Dixon (1994) recommends using the following relationship as a measure of the degree of segregation between two species:
[S.sub.ii] = ln[[n.sub.ii]/[n.sub.ij]/([N.sub.i] - 1)/[N.sub.j]] (7)
where [N.sub.i] and [N.sub.j] are the number of plants of species i and j respectively, [n.sub.ii] is the number of plants of species i with conspecific nearest neighbors, and [n.sub.ij] is the number of plants of species i with heterospecific nearest neighbors (lowercase letters distinguish these measure from [N.sub.ii] and [N.sub.ij] used in Eqs. 5 and 6). The numerator is the odds of conspecific neighbors, while the denominator is the odds of conspecific neighbors that is expected by chance. It follows that values of [S.sub.ii] [greater than] 0 indicate segregation, and that values of [S.sub.ii] [less than] 0 indicate attraction. It is straightforward to test the significance of segregation, because Dixon (1994) has derived a statistic (henceforth referred to as Dixon's C-statistic) that has an asymptotic [[Chi].sup.2] (df = 2), making Monte Carlo simulation unnecessary.
Single-species patterns generated by a Poisson cluster process
The first example uses simulated patterns to demonstrate the efficacy of the clump recognition process to locate the positions of clump centers. The clumped patterns were generated using a two-dimensional Poisson process to position a given number of clump centers within a square of 500 mm length [ILLUSTRATION FOR FIGURE 5 OMITTED]. We then arranged a fixed number of plants at random within a radius X of each clump center. Patterns in Fig. 5a-d were generated with 30 clumps, each containing 10 plants, using X = 5, 15, 25, and 35 mm, respectively. Fig. 5e-h were generated with 60 clumps, each containing 5 individuals, again using X = 5, 15, 25, and 35 mm.
When patterns contained clumps that were distinct from one another, the process accurately estimated the number of clumps and their positions, as well as [Mathematical Expression Omitted] (Table 1). The results were robust to the choice of stopping probability (within the range 0.15-0.75). The process was less able to determine the number of underlying clump centers when clumps were diffuse and intermingled. From Table 1 it can be seen that the method is most effective when the mean distance between clump centers is twice the radius of the clump (it can be shown that for a 500 mm square quadrat, the mean distance between clump centers is 250/[-square root of no. clumps]). For intermingled patterns, the clump recognition process predicted too many clump centers, and their number was dependent upon the choice of stopping probability. We tried other types of cluster analysis described in Sneath and Sokal (1973), but none was more successful than the WPGMA method used. The "complete linkage" method, in which the distances between two clusters is determined by the greatest distance between any two plants within them, produced very similar results to WPGMA, while the unweighted PGMA method performed less well. The problem arises because some plants from neighboring clumps are juxtaposed by chance, and the clump recognition process distinguishes them as clump centers. It is well known that pattern analyses are unable to unequivocally identify the process (Pielou 1974). Our method provides an accurate description of intermingled clumps, but is unable to identify the cluster centers from the pattern-generating process.
The efficacy of randomizing clump centers to produce Monte Carlo simulations
The simulated patterns used to test interspecific association should preserve the intraspecific characteristics of each species. We have proposed that randomizing the positions of clump centers may simulate such patterns, and we used the eight patterns shown in Fig. 5a-h to test its efficacy. For each pattern, the clump centers were identified using [P.sub.stop] = 0.60, and 100 patterns were simulated by randomizing the positions of clump centers. Information retained about plant positions relative to clump centers was then used to construct new plant maps. The aggregation index for each pattern were remarkably similar to the median aggregation index of 100 simulations (repeated for r = 10, 30, and 50 mm), demonstrating that randomization of the positions of clump centers effectively preserved intraspecific patterning [ILLUSTRATION FOR FIGURE 6 OMITTED]. Only at small radii do the simulations appear biased; there is a slight tendency to over-aggregate at r = 10 mm [ILLUSTRATION FOR FIGURE 6 OMITTED].
Computer generated patterns involving two species
The next thing to demonstrate is how clump centers may be used in bivariate pattern analysis. The single-species patterns shown in Figs. 5e, g, both of which contain 60 clumps, were changed into two-species patterns by assigning a species identity to each clump. This was done in one of three ways: (1) Fifteen randomly selected clumps were assigned to species i, and the nearest clump to each of these was also assigned to species i, producing a negative association between the two species (within circles of r [approximately]50 mm). (2) Fifteen randomly selected clumps were assigned to species i, and the nearest clump to each of these was assigned to species j. The process was repeated, producing positive associations between clumps of the two species. (3) Thirty randomly chosen clumps were assigned to species i, and the remainder were assigned to species j, producing no association between the clumps [ILLUSTRATION FOR FIGURE 7 OMITTED].
First the patterns were analyzed using association indices (Eq. 6). We generated 95% confidence intervals by Monte Carlo simulations that randomized the position of clump centers of species, while leaving the other species unchanged. These clump centers were derived by running the clump recognition process on the maps of each species using a critical probability of [P.sub.stop] = 0.60. (Note that we did not attempt to distinguish clump centers containing mixtures of species.) The values of [SI.sub.ij] at r = 30 mm were consistent with associations that we had imposed. The association indices (AI), with 95% confidence intervals in brackets, were 0.17 (0.11-0.36) and 0.49 (0.32-0.89) for the negatively associated patterns [ILLUSTRATION FOR FIGURE 7A, D OMITTED]; 1.30 (1.03-1.76) and 1.31 (1.04-1.68) for the positively associated patterns [ILLUSTRATION FOR FIGURE 7B, E OMITTED], and were 0.86 (0.61-1.38) and 1.01 (0.73-1.47) for the nonassociated patterns [ILLUSTRATION FOR FIGURE 7C, F OMITTED]. The confidence intervals generated by randomizing the positions of clump centers were considerably larger than those generated by randomizing the positions of individual plants. In fact the latter approach would have [TABULAR DATA FOR TABLE 1 OMITTED] incorrectly predicted that the species in pattern 7c were negatively associated.
The intraspecific aggregation resulted in strong segregation between species. Every pattern had a positive index of segregation [S.sub.ii] and a highly significant Dixon's C-statistic [ILLUSTRATION FOR FIGURE 8 OMITTED]. Tests of segregation between clump centers provide an alternative way of looking at associations, because the segregation resulting from intraspecific aggregation has been removed. Consistent with our expectations, clump centers of negatively associated patterns had [S.sub.ii] [greater than] 0 [ILLUSTRATION FOR FIGURE 7A, D OMITTED], while those of positively associated patterns had [S.sub.ii] [less than] 0 [ILLUSTRATION FOR FIGURE 7B, E OMITTED]; these patterns generated significant Dixon's C-statistics. Patterns exhibiting no association had [S.sub.ii] [approximately equal to] 0, and nonsignificant C-statistics [ILLUSTRATION FOR FIGURE 7C, F OMITTED]. The results were robust to our choice of stopping probability when the analyses were repeated for [P.sub.stop] = 0.15, 0.30, 0.45, and 0.60 [ILLUSTRATION FOR FIGURE 8 OMITTED]. The only discrepancies were nonsignificant C-statistic for patterns in Fig. 7a, d when the stopping probability was set at 0.60, resulting from the clump recognition process amalagamating groups of clump centers into single clusters.
Spatially explicit population models
In the previous examples we used patterns generated by simple processes that produced clumps of similar radius and number. In the next example we generate a pattern by modeling processes that may be important for the long-term coexistence of species in natural communities. A simulation is used in which there is a trade off between a species' competitive ability and its potential seed production. The seed production (F) of a plant of species i is related to crowding by
F = [F.sub.i]/1 + [summation over i] [[Alpha].sub.ij][N.sub.ij] (8)
where [F.sub.i] is species i's maximum seed production, [[Alpha].sub.ij] is a competition coefficient describing the interaction between species i and species j, and [N.sub.ij] is the number of plants of species j within a radius [r.sub.j] of the target individual. We simulated the dynamics of four species whose competitive abilities and interaction radii ([r.sub.j]) were ranked 1 [greater than] 2 [greater than] 3 [greater than] 4, and whose maximum fecundities were ranked 4 [greater than] 3 [greater than] 2 [greater than] 1. For each generation of the simulation, Eq. 8 was used to calculate the seed production of an individual, and these seeds were dispersed from the plant using a radially symmetric negative exponential function that was identical for all species. The process was run for 4000 generations, and the final pattern was then analyzed [ILLUSTRATION FOR FIGURE 9 OMITTED].
Since we gave each species the same dispersal parameter, we might expect them to have similar clumping characteristics if dispersal alone determined pattern. In fact, the values of [Mathematical Expression Omitted] and [Mathematical Expression Omitted] differ greatly between species. Species 1 and 2 (good competitors producing few seeds) formed many small clumps, giving rise to low values of [Mathematical Expression Omitted] and [Mathematical Expression Omitted] (Table 2). At the other extreme, species 3 and 4 (weak competitor producing many seeds) formed large interconnected clumps, giving rise to high values of [Mathematical Expression Omitted] and [Mathematical Expression Omitted]. As would be expected from such aggregated patterns, each species is strongly segregated from all others (P [less than] 0.001 for all patterns, Table 3).
The strong competitive impacts of species 1 and 2 affected the patterns by creating "hallows" of unoccupied space around their clusters [ILLUSTRATION FOR FIGURE 9 OMITTED]. All pairwise indices of association were less than one at r = 30 mm, indicating negative association between species at that scale, but only the associations between species 1 and 2 and between 1 and 4 were significant. Note that the significance test was adjusted to accommodate multiple comparisons by dividing the critical probability (P = 0.025) by the number of comparisons made. Consistent with these results, tests of segregation on clump centers also indicated negative associations, although none of the C-statistics were significant (Table 3).
Field data from a community of dune annuals
Our final example is an analysis of field data recorded on coastal dunes in Norfolk, UK, during May 1995. The data were collected to describe competitive interactions using neighborhood analyses (D. A. Coomes and M. Rees, unpublished data). We recorded the precise location of annual plants within six square quadrats of 500 mm length, one of which is shown in Fig. 10. The species present were Cerastium semidecandrum, Phleum arenarium, Saxifraga tridactylites, and Valerianella locusta. The following analyses were conducted using [P.sub.stop] = 0.60, and the sensitivity of the results to [P.sub.stop] is left to the Discussion.
Almost all of the single-species patterns showed significant aggregation (21 of 24 patterns), and values of [Mathematical Expression Omitted] differed consistently between species across the six quadrats, with means [+ or -] 1 SE for Phleum, Cerastium, Saxifraga, and Valerianella of 1.08 [+ or -] 0.04, 1.43 [+ or -] 0.04, 2.42 [+ or -] 0.33, and 3.52 [+ or -] 0.92, respectively. Intraspecific aggregation caused segregation among the species, with most pairwise tests giving [S.sub.ii] values significantly greater than the null expectation (Table 4). Note that the test of significance was based on whether [S.sub.ii] values were consistently greater than zero across the six quadrats, and not upon Dixon's C-statistic. Segregation was strongest between the two most aggregated species, Saxifraga and Valerianella, while there was no segregation between the weakly aggregated species, Phleum and Cerastium (Table 4).
Despite the segregation caused by fine-scale aggregation, tests of association suggest that species were positively associated, albeit weakly. All pairwise tests of segregation between clump centers gave negative [S.sub.ii] values, but none of the implied positive associations was significant (Table 4). Similarly, all association indices were [greater than]1 (measured at r = 10, 30, and 50 mm), and the Cerastium-Saxifraga and Cerastium-Valerianella associations were significant positive at P = 0.05 (Table 4). The most plausible explanation for the positive associations is that patches within each quadrat are especially condusive to the growth and survival of annual plants in general. The lack of negative associations at the fine scale suggests that there is little specialization of species onto different substrate types within the quadrat, and that competition has little impact on spatial structure. The fine-scale aggregation may well have resulted from local seed dispersal, and the resultant segregation between species could be important in reducing the impact of interspecific competition.
Upton and Fingleton (1985) indicate that a range of methods should be used to analyze patterns, because each reveals a different facet of the structure. What information does the clump recognition process provide that is not already available from other approaches?
Comparison of local density to mean-field density (i.e. the aggregation index, AI) provides a meaningful measure to plant ecologists interested in density-dependent competition. It is for this reason that aggregation and association indices are used in this paper, rather than the transformed measures provided by Ripley's K-function (Ripley 1981, Haase 1995). While aggregation indices are useful, they do not detect all [TABULAR DATA FOR TABLE 2 OMITTED] [TABULAR DATA FOR TABLE 3 OMITTED] clumped patterns. For example, Baddeley and Silverman (1984) generated a pattern by placing a number of plants "[N.sub.s]" into cells "S" of a grid using the distribution P([N.sub.s] = 0) = 1/10, P([N.sub.s] = 1) = 8/9, and P([N.sub.s] = 10) = 1/90; most cells contained one plant, a few remained empty, and very few contained a tight cluster of 10 plants. Despite the aggregated nature of this distribution, Baddeley and Silverman showed it had a K-function concordant with complete spatial randomness. When we analyzed several realizations of this process using the clump recognition process, we found highly significant aggregation, and successfully reduced each of the 10-plant clumps to single-clump centers. The other clear benefit of the clump recognition process is that it provides an intuitively simple description of a pattern by describing the size structure of clumps, and it does this without making assumptions about the underlying distribution.
We have outlined the difficulties of constructing Monte Carlo simulations with which to test association between species. Randomizing the positions of clump centers has been shown as an effective method of preserving intraspecific patterns while randomizing one pattern with respect to another [ILLUSTRATION FOR FIGURE 6 OMITTED]. Previous approaches to simulation include transposing one pattern relative to another, considering the quadrat as a torus so that plants shifted outside the quadrat on one side are reincorporated on the opposite side (Lotwick and Silverman 1982). Another approach is to reflect and rotate patterns (Palmer and van der Maarel 1995).
The clump recognition process reduces small-scale spatial autocorrelation, and the application of Dixon's test to the clump centers can be used to test for associations. The measure of association is different from Ripley's [K.sub.12]-function, because the test is based on the identity of nearest neighbors, rather than on counts within circles.
TABLE 4. Spatial interrelationships among four species of winter annuals growing on dunes in Norfolk, UK. We performed (A) tests of segregation between plants of each species, (B) tests of segregation among clump centers, thereby testing for association, and (C) test of association using Ripley's K-function approach. Valeria- Species Cerastium Phleum Saxifraga nella (A) [S.sub.ii] of plants Cerastium 0.168 0.329 0.153 Phleum 0.437 0.573 0.489 Saxifraga 0.672 0.498 0.549 Valerianella 0.614 0.539 0.78 (B) [S.sub.ii] of clump centers ([P.sub.stop] = 0.60) Cerastium -0.061 -0.034 -0.064 Phleum -0.204 -0.046 -0.032 Saxifraga -0.076 -0.033 -0.053 Valerianella -0.171 -0.203 -0.159 (C) [SI.sub.ij] using r = 30 mm(*) Phleum 1.00 Saxifraga 1.23 1.15 Valerianella 1.28 1.08 1.06 Note: Significant deviations from null expectations are shown in bold (P = 0.05), with allowances made for multiple comparisons. * Clumps used for Monte Carlo simulation were generated with [P.sub.stop] = 0.60. Only [SI.sub.ij] values are given because [SI.sub.ij] = [SI.sub.ji]. Association indices and significances were similar for r = 10, 30, and 50 mm.
What assumptions are made in the clump recognition process?
Intraspecific aggregation and interspecific associations depend on the scale at which a pattern is considered, so what scale is being considered in our method? The value of dw in Eq. 3 is the maximum difference between the observed cdf and the Poisson cdf and thereby tests for significant aggregation at any scale within the range of radii considered. It follows that the clump recognition process continues to "boil down" a pattern until it is indistinguishable from Poisson on all scales.
We have illustrated the effectiveness of the clump recognition process for analyzing tightly bound clusters that are Poisson distributed. In nature, such patterns are commonly observed among ramets of clonal species, and among annual plants restricted to pockets within a perennial matrix. The method is less reliable for analyzing intermingled clusters, such as those generated by annual plants that disperse many of their seeds outside the neighborhood in which they compete (Pacala and Silander 1990). The process produces a descriptive statistic, but it is unable to predict the underlying process from the pattern.
For patterns of intermingled clumps, the measures of aggregation estimated by the clump recognition process are dependent on the choice of stopping probability ([P.sub.stop]). The dune annual data were reanalyzed at a range of [P.sub.stop] values to test whether aggregation and association measures remained robust. All species showed a steady rise in the amalgamation index across the range ([P.sub.stop] = 0.15-0.90, and all species (except Cerastium) had similar slopes to the relationship between AI and [P.sub.stop]. Consequently, the process provided a reasonable method of ranking species across a range of [P.sub.stop] values. The [S.sub.ii] statistics between clump centers remained stable irrespective of the choice of stopping probabilities ([P.sub.stop] = 0.15, 0.30, 0.45, and 0.60 were tried), and none was significant. We recommend quantifying the robustness of aggregation and segregation statistics to changes in [P.sub.stop] whenever the process is used to compare species, sites, or years.
The method is also less effective when clumps are restricted to only a part of the quadrat, because its attempt to remove clumping at the large scale reduces the pattern to just a few cluster centers. The process tends to result in regularity at low radii, resulting in simulated aggregated patterns from clump centers with slightly too much aggregation on the fine scale [ILLUSTRATION FOR FIGURE 6 OMITTED].
Can processes be distinguished?
As shown in the final example, the plant's eye view of the world may be very different to that ascertained by assuming communities to be devoid of spatial structure. The spatial pattern of biological communities may result from a combination of processes, that include (1) spatial niches caused by heterogeneity in topography, resource supply, or background vegetation; (2) local dispersal; and (3) a survival template that restricts species to limited patches. lt has long been recognized that different processes can produce similar signatures, so pattern analysis can only provide inferences about the nature of biological processes (Pielou 1961). Processes may be distinguished when several species are found to co-occur at a given location. In the example of patterns among dune annual species, we saw that there were significant differences among species in the degree of aggregation and in the associations between clump centers. The processes that generated the pattern may be inferred by relating aspects of spatial patterning to plant characteristics such as seed size, growth form, and taxonomic similarity. For example, if niches are important we might expect clump centers of ecologically similar species to be positively associated, but if local dispersal is the principal factor in pattern formation we might expect ecologically similar species to be strongly segregated (Pacala 1997).
Many statistical methods are available for comparing patterns, and each highlights a different facet of a pattern that may have arisen by a complex interrelationship of processes. The clump recognition process is a flexible tool for making comparisons, both because it does not assume an underlying mechanism and because it allows individuals to be objectively grouped into clumps. The statistical properties of these can then be determined. The values of the various measures of aggregation are dependent on the choice of stopping probability, so the robustness of results needs to be assessed at a range of probabilities. In this way statistical descriptions of patterns can be quantified at a range of ecologically relevant spatial scales.
The research was supported by a grant from the Natural Environment Research Council. The detailed and careful comments by two anonymous referees were greatly appreciated.
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|Author:||Coomes, David A.; Rees, Mark; Turnbull, Lindsay|
|Date:||Mar 1, 1999|
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