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- Materials and methods
- Supporting Information
1. Attempts to infer underlying ecological process from observed patterns in ecology have been widespread, but have generally relied on first-order (non-spatial) community characteristics such as the species abundance distribution (SAD). This measure has become an important test of several theories of species coexistence, but has proved unsuccessful in distinguishing between them.
2. Spatially explicit data are increasingly available for a range of ecological communities, and analysis methods for these data are well developed. However, relatively little work has investigated the potential of these data for similar inference about mechanisms of coexistence.
3. In this study, we systematically investigate the spatial and non-spatial signals of simulated ecological processes. We include neutral, niche, lottery, Janzen–Connell and heteromyopia models, deriving and comparing first- and second-order measures for the patterns they generate.
4. We find that the SAD is unable to distinguish reliably between underlying models, with random variation in its shape concealing any systematic differences.
5. A new second-order summary measure of spatial structure derived in this paper, in contrast, proves highly sensitive to the type of ecological interaction being modelled, and is robust to random variation.
6.Synthesis. A simple summary measure of the spatial structure of plant communities is presented and found to be a more powerful indicator of underlying process in simulated data than a widely used first-order measure, the SAD. The potential for answering important ecological questions using spatial statistics, particularly concerning mechanisms of coexistence in diverse communities, appears to be great.
- Top of page
- Materials and methods
- Supporting Information
A great deal of research in ecology tries to infer ecological processes from patterns observed in nature. In community ecology, the species abundance distribution (SAD) has received particular attention (McGill et al. 2007). A SAD describes the absolute or relative abundances of species in a community and is found to conform to a near-universal ‘hollow curve’ shape, comprising a small number of common species and a large number of rare ones. There is no obvious a priori reason to expect this shape, and the detailed features of SADs have therefore been used to discriminate between underlying processes, such as those involved in species coexistence. This work began with theories of niche assembly (e.g. Motomura 1932; MacArthur 1957; Tokeshi 1990), and more recently comparing SADs has become a key tool for validating the neutral theory against observed data from ecological communities (Hubbell 1979, 1997, 2001).
Unfortunately, a single ecological process can produce rather variable SADs (Williamson & Gaston 2005; Magurran 2005; Volkov et al. 2005), making the detection of processes from empirically derived SADs difficult (McGill et al. 2007). This is not surprising: a SAD is, after all, just a description of species’ relative abundances averaged over space. Processes affecting coexistence rely on spatial proximity of individuals, especially in sessile organisms, and SADs convey no information on spatial structure. In the context of a spatial analysis of communities, a SAD would be said to be a first-order measure (Illian et al. 2008).
In principle, spatial correlations ought to provide a more sensitive indicator of ecological interactions among plant species because of the importance of interactions as drivers of spatial pattern in plant communities (Bolker & Pacala 1997; Murrell & Law 2003; Wiegand et al. 2007). There is a long history in plant ecology of using spatial pattern to gain insight into ecological processes (e.g. Watt 1947; Clark & Evans 1954; Sterner, Ribic & Schatz 1986), and indeed this was one motivation for the development of spatial point process methods (Matérn 1960; Ripley 1977; Stoyan & Penttinen 2000). It would be unrealistic to expect a unique mapping from a spatio-temporal process to a spatial pattern because of the array of biotic and abiotic factors at play (e.g. Baddeley & Silverman 1984; Lepš 1990), but it is reasonable to ask whether an analysis that makes use of spatial structure is a better discriminator among ecological mechanisms than one based on SADs that ignores this information.
Ecologists do often have far more information at their disposal than just that needed to construct SADs. For example, several complete spatial censuses exist for tropical rainforest trees on the 50-ha plot at Barro Colorado Island (BCI) in Panama (Hubbell, Condit & Foster 2005) and numerous other sites (Losos & Leigh 2004). In the search for evidence about underlying processes on these plots, it should be possible to go beyond SADs (e.g. Hubbell 2001; Volkov et al. 2003; Etienne & Olff 2005; He 2005) to second-order measures such as spatial correlations that make use of this spatial information. The potential of these has been recognized in studies of the roles of seed dispersal and habitat heterogeneity (Condit et al. 2000; John et al. 2007), the aggregations produced by neighbourhood recruitment and mortality (Hubbell et al. 2001; Uriarte et al. 2005), and spatial patterns in diversity (Wiegand, Gunatilleke & Gunatilleke 2007). Elsewhere, temporal patterns have been used to discriminate between neutral and non-neutral mechanisms for maintaining the structure and diversity of communities (e.g. Clark & McLachlan 2003; McGill, Hadly & Maurer 2005), as have comparisons between different spatial scales (Gilbert & Lechowicz 2004; Dornelas, Connolly & Hughes 2006; McGill, Maurer & Weiser 2006). However, a systematic analysis of the spatial signatures generated by different kinds of species interaction has not been attempted.
This study evaluates the effectiveness of SADs and measures of spatial correlations to discriminate between multispecies spatial patterns that make different underlying assumptions about ecological interactions. Our baseline was the neutral model, with its assumption of per capita ecological equivalence between species (Hubbell 2001). Two niche models were included: a conventional niche model in which species favour specific environmental conditions that are defined spatially (e.g. Grinnell 1917; Hutchinson 1958; Zillio & Condit 2007), and a lottery model in which temporal environmental variance favours different species at different times (Sale 1977; Chesson & Warner 1981; Chesson & Huntly 1988). The Janzen–Connell hypothesis, according to which young plants suffer increased mortality in the neighbourhood of their parents, was also implemented (Janzen 1970; Connell 1970), as was a purely spatial heteromyopia model in which interspecific competition occurs over shorter distances than intraspecific competition (Murrell & Law 2003).
We generated multispecies spatial patterns through realisations of spatio-temporal stochastic processes [stochastic individual-based models (IBMs)] using the different underlying models of ecological interactions. At first order, we computed SADs on these spatial patterns. At second order, we computed a new community-level measure of species segregation, built from spatial pair-correlation functions, referred to as the cross-pair overlap distribution (xPOD).
First-order signals of the modelled ecological interactions were expected to be limited, given the inherent variability of SADs. Some differences in community diversity and evenness were anticipated, but were difficult to predict because the relative strength of each form of interaction in promoting coexistence has not previously been assessed. Second-order spatial signals were expected to be substantially stronger, and to take a more predictable form for each model. In particular, the spatial niche model was expected to increase the segregation among species, while the Janzen–Connell model was predicted to constrain conspecific clumping and so have the opposite effect.
- Top of page
- Materials and methods
- Supporting Information
Attempts to infer underlying process from pattern in ecology have tended to rely on first-order community characteristics such as the SAD, which is the chief empirical test of many theories of species coexistence and the focus of a great deal of theoretical study in its own right (Fisher, Corbet & Williams 1943; Hubbell 2001; Volkov et al. 2003; McGill et al. 2007). Relatively little work has investigated the potential of second-order spatial information for similar inference, and we are aware of none that has systematically assessed the first- and second-order signals of modelled ecological processes.
The SADs produced by the models considered here show, on visual inspection, a great deal of overlap and considerable variation within those produced by any one model. There may be certain regions of the distributions where the models are distinct from one another, particularly in terms of community evenness. The lottery model produced the least even communities and the Janzen–Connell model the most even and diverse ones. These differences are small, however, and do not prove reliably distinguishable by the statistical tests used here. Differences in overall species diversity are attributable to model design, especially in the niche model where fecundity is depressed outside species’ optimum environment.
The Kolmogorov–Smirnov test indicates that random variations in species abundances outweigh similarities engendered by ecological process, as the test statistic takes very similar values for single-model pairs and cross-model pairs. On the basis of the values found here, SADs produced by the same model are, at best, only slightly more similar to one another than they are to those produced by different – or even contradictory – processes. The variations we find in SADs cannot, therefore, be classified as random or systematic by magnitude alone.
Truncating the SADs at an abundance of 500 slightly depresses within-model variation relative to between-model variation. Further analysis indicates that a threshold of 100 is better yet; this suggests that it may be possible to find an optimum section of the distribution in which random variation is minimized. Nevertheless, random and systematic differences between distributions, as captured by the Kolmogorov–Smirnov test statistic, remain of the same order in the threshold values that we have investigated (1, 100 and 500).
The xPODs of each model are more reliably different from one another than the SADs. This would be expected because an xPOD is essentially an accessible summary of the spatial relationships between species and should reflect different models of species interactions in so far as they generate different spatial patterns. There are large differences between results from each model, with the exception of the neutral and heteromyopia models, and prediction intervals demonstrate that random within-model variation does not erode the distinction between models. This holds true even for the lottery model, in which random variation is large. At these parameter settings, the ecological processes modelled here are eminently distinguishable by their spatial signals.
The neutral and heteromyopia models produce xPODs with slightly negative means and small variances, the lottery and niche models give more negative means and skew distributions with greater variances, and the Janzen–Connell model a smaller negative mean and smaller variance. These signals broadly conform to expectations. The negative mean of the xPOD under neutrality is because of separation of species by density-dependent mortality, which is balanced within species by local dispersal. In the niche model, a greater spread of (mainly negative) values is directly attributable to the introduction of spatial niches which separate species according to their response to the underlying environment. While the spatial extent of each niche is approximately equal, the strength of niche separation itself is weak; niches are randomly assigned and so may overlap or entirely coincide. Stronger niche separation, corresponding to competition between species, would be expected to increase the spread of the xPOD further (and has been found to do so in trial results).
The lottery model lacks a mechanism for such strong separation of species, but the temporal variations in species fecundity allow dominant species to mingle with others and force them apart, so producing the observed spread of values. This effect is likely to be dependent upon the frequency of variation in fecundity (O’Malley et al. 2010). The Janzen–Connell effect counteracts the tendency of species to separate slightly under neutrality by suppressing conspecific clumping. There remains no mechanism to encourage the mingling of species, however, so the distribution produced has a smaller spread, as expected. Although the Janzen–Connell effect modelled here is not explicitly overcompensating, it does effectively prevent the survival of individuals in the immediate vicinity of their parent at equilibrium density, as required by the original theory and observed in some tropical tree species (Janzen 1970; Freckleton & Lewis 2006; Bagchi et al. 2010).
The heteromyopia model produces distributions indistinguishable from those of the neutral model, with identical means and variances. This may be because density-dependent mortality within species is spread over a larger area, so allowing increased clumping at small scales which counterbalances any repression of clumping at medium scales. This conformity of spatial patterns between models illustrates the absence of unique links between process and pattern.
Nevertheless, goodness-of-fit tests show that xPODs have far more success in distinguishing models than SADs. xPODs produced by different models are highly dissimilar, except those produced by the neutral and heteromyopia models, while distributions drawn from the same model are far more alike, producing smaller and more restricted values of the test statistic (Fig. 5, Table 6). The sole exception is the lottery model, in which random fluctuations in fecundities give rise to substantial variation in spatial pattern. Despite this, the difference between these and the comparable results for SADs (Fig. 3) is striking; the fingerprint of each model emerges from the background noise clearly and consistently in the case of the xPOD.
Table 6. Mean and 90% limit values (in brackets) of the Kolmogorov–Smirnov test statistic comparing pairs of cross-pair overlap distributions within and between contrasting models of ecological interactions. Within-model means are in bold. J–C denotes Janzen–Connell
|Neutral||0.056||0.540 ||0.113 ||0.136 ||0.063 |
|Niche|| ||0.072||0.375 ||0.586 ||0.529 |
|Lottery|| || ||0.223||0.409 ||0.317 |
|J–C|| || || ||0.095||0.164 |
|Heteromyopia|| || || || ||0.056|
Our findings are subject to several caveats, the most important of which relates to the direct deduction of process from pattern. While we find that second-order patterns are significantly more informative than their first-order counterparts, it remains unreasonable to assume an inviolable causal link from underlying mechanisms (e.g. Baddeley & Silverman 1984; Lepš 1990). It must also be stressed that our results come from computer-based simulations, and that behaviour which we hold constant in the interests of clarity does not obey this requirement in real-world plant communities, potentially producing quite different outcomes (as similarly noted by e.g. Chave & Leigh 2002; Chave, Muller-Landau & Levin 2002; Levin et al. 2003). In particular, dispersal distances vary widely between species and have a dramatic impact on spatial structure (Levin et al. 2003). Changes in the relative scales of dispersal and environmental variation may have a similarly confounding effect (e.g. Lande, Engen & Saether 1999; Wiegand et al. 2007). Complex species-specific parameterization of dispersal kernels would be difficult to perform accurately and would undermine our attempts to identify the spatial signals of isolated processes, and so requires further research.
We also make assumptions about the environment and the expression of niche differentiation in our spatial and temporal niche models which may affect observed behaviour. While greater separation of niches is expected to strengthen the signal detected in our niche model, greater niche overlap would have the opposite effect because species remain effectively neutral within shared niches. We constrain our environment to a single circular distribution of values to ensure that differences in niche availability do not affect our results. However, the signals of niche differentiation in patchier environments would remain similar to those observed here, at similar relative scales, because of spatial separation between species with differing niche requirements. As in all models presented here, niche processes act through changes to the death term for the sake of consistency but are equivalent to the opposite adjustment to the birth term.
It is notable that the xPOD, as used here, still only makes use of a small fragment of the spatial information that is often available. For instance, marked point patterns from forests often contain information on spatial structure in relation to size, age and environment, and at different spatial scales. Moreover, there is spatial structure at third order and beyond. Nevertheless, second-order information evidently helps to distinguish underlying process, and is more informative than SADs, the well-established first-order measure. Together with the ready availability of spatially explicit data, this suggests that much of the power of spatial analysis in ecology remains to be exploited.
While bearing the above caveats in mind, the differences between the spatial signatures found here are large and intuitive, especially between the neutral model and the spatial niche model. We therefore predict, as a hypothesis for future empirical research, that plant communities in environments with more physical heterogeneity should have xPODs characterized by more negative means and larger variances than plant communities in environments with less physical heterogeneity. This hypothesis can be tested on tropical rainforest data currently available and provides a stronger test of the neutral theory than first-order measures. If the neutral model is correct, the hypothesis will not be supported.