http://orcid.org/0000-0001-9687-6178
Figures
Abstract
Patterns of trait distribution among competing species can potentially reveal the processes that allow them to coexist. It has been recently proposed that competition may drive the spontaneous emergence of niches comprising clusters of similar species, in contrast with the dominant paradigm of greater-than-chance species differences. However, current clustering theory relies largely on heuristic rather than mechanistic models. Furthermore, studies of models incorporating demographic stochasticity and immigration, two key players in community assembly, did not observe clusters. Here we demonstrate clustering under partitioning of resources, partitioning of environmental gradients, and a competition-colonization tradeoff. We show that clusters are robust to demographic stochasticity, and can persist under immigration. While immigration may sustain clusters that are otherwise transient, too much dilutes the pattern. In order to detect and quantify clusters in nature, we introduce and validate metrics which have no free parameters nor require arbitrary trait binning, and weigh species by their abundances rather than relying on a presence-absence count. By generalizing beyond the circumstances where clusters have been observed, our study contributes to establishing them as an update to classical trait patterning theory.
Author summary
Species traits determine how they compete with each other. As such, patterns in the distributions of traits in a community of competing species may reveal the processes responsible for coexistence. One central idea in theoretical ecology is that the strength of competition relates to similarity in species needs and strategies, and therefore if competition plays out at short timescales, coexisting species should be more different than expected by chance. However, recent theory suggests that competition may lead species to temporarily self-organize into groups with similar traits. Here we show that this clustering is a generic feature of competitive dynamics, which is robust to demographic stochasticity and can be indefinitely maintained by immigration. We show that clustering arises whether species coexist by partitioning resources, environmental preferences, or through tradeoffs in life-history strategies. We introduce and validate metrics that, given species traits and abundances, determine whether they are clustered, and if so, how many clusters occur. By showing the generality of self-organized species clusters and providing tools for their detection, our study contributes to updating classical ideas about how competition shapes communities, and motivates searches for them in nature.
Citation: D’Andrea R, Riolo M, Ostling AM (2019) Generalizing clusters of similar species as a signature of coexistence under competition. PLoS Comput Biol 15(1): e1006688. https://doi.org/10.1371/journal.pcbi.1006688
Editor: David Vasseur, Yale University, UNITED STATES
Received: June 2, 2018; Accepted: November 30, 2018; Published: January 22, 2019
Copyright: © 2019 D’Andrea et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: The R code for running all simulations analyzed in the manuscript can be found at https://github.com/rafaeldandrea/Generalizing-clusters-code. The code for the metrics can be found at https://github.com/rafaeldandrea/Clustering-metric.
Funding: AO was supported by the National Science Foundation under grant 1038678, ‘`Niche versus neutral structure in populations and communities’’ funded by the Advancing Theory in Biology program, as well as by the Danish National Research Foundation through support of the Center of Macroecology, Evolution and Climate (grant DNRF96), and by the Miller Institute for Basic Research in Science at the University of California, Berkeley. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Introduction
Competition is a driving force in nature, and a central question in ecology is how it shapes community structure. Species traits mediate their interactions with the environment and each other, and therefore determine how they compete. As such, patterns of trait distribution among co-occurring species can give insights into the underlying coexistence mechanisms [1–5]. Theory shows that coexistence is only stable if species differ in their ecological needs and impacts—i.e. they must display niche differences (see Glossary in S1 Text) [6, 7]. These differences reduce competition between species compared to competition within species, thus allowing each species to grow from low abundance in the presence of the others. If species are arranged on niche axes so that proximity on those axes indicates similarity in their niche strategies, they are expected to display limiting similarity–greater-than-chance differences on those axes [8]. Ecologists use trait axes as proxies for niche axes, and look for similar patterns therein, usually in the form of overdispersion or even spacing between species on these axes [9–17]. This classical idea remains the dominant paradigm, despite mixed empirical support [2, 3, 5].
In contrast, recent studies suggest that competition causes the spontaneous emergence of transient clusters of species with similar traits [18–20]. They find that limiting similarity results only if competition plays out to the final stage, whereby all species are excluded except for those with optimal niche strategies (see Glossary in S1 Text). Those species are niche-differentiated enough from one another (i.e. have enough space between them on the niche axis) to all stably coexist together, having emerged as dominant over other strategies through the competitive process. When species outnumber optimal niches, e.g. phytoplankton communities with more species than resources [21], competitive exclusion will ensue. However, species near the optimal niche strategies are excluded more slowly than those further away from these optimal strategies [22–24]. As a result, the community temporarily self-organizes into clusters of similar species near optimal strategies, with gaps in between.
Clustering has traditionally been associated with environmental filters [25], and more recently with one-sided competition without a balancing tradeoff, such as competition for light where taller is better [26]. However, those will tend to produce a single big cluster around the favored trait, whereas partitioning of a niche axis will lead to multiple clusters, one per optimal niche strategy. Patterns of multiple clusters have in fact been reported in empirical studies of body size in phytoplankton communities and certain animal taxa [18, 27–29]. This phenomenon has been interpreted to bridge coexistence through differences (niche theory) and similarity (neutral theory) [30], and as such potentially represents a unification of classical ideas and a generalization of limiting similarity.
Scheffer and van Nes [18] first demonstrated the emergence of transient clusters in Lotka-Volterra dynamics. Later work further suggested that clusters are a generic outcome of Lotka-Volterra dynamics [20] except for special shapes of the function connecting competition to traits [31, 32]. Several mechanisms can make these transient clusters persist, such as specialist enemies [18] and periodic environments [33, 34].
However, the generality of clusters as a signature of competition cannot be established without showing that they emerge in communities subject to stochastic processes. In particular, immigration and ecological drift are intrinsic to most communities, and have been amply demonstrated to be important players in nature [30, 35–37]. In fact, models ignoring all of biology but drift and immigration successfully describe observed macroecological patterns [38] (which of course does not mean deterministic forces are unimportant). Yet, clustering remains unseen in competition models incorporating these processes [39–41]. In plankton models where clustering occurs, immigration has as a negative impact [21].
Moreover, clusters have not been widely demonstrated beyond Lotka-Volterra dynamics. Lotka-Volterra competition equations are a heuristic description which does not specify a niche mechanism. They are a special limit [42] of MacArthur’s consumer-resource dynamics model [43], and cannot describe all types of competition [44]. While species clusters have been shown to emerge in explicit consumer-resource dynamics [21, 34], these studies ignored the possibility of resource depletion effects. The latter have been shown to greatly affect competitive relations between consumers, for example by violating the assumption that competition always decreases with trait differences [45]. It is also not clear that clustering should emerge among species competing along environmental gradients; indeed, early studies of stochastic niche dynamics [39–41] focused on competition of this type and found no clustering pattern (although these studies, which predate [18], were not specifically looking for clustering). Finally, it is not known whether clusters emerge in communities characterized by hierarchical competition among species that coexist via life-history tradeoffs. These mechanisms, such as the competition-colonization tradeoff [46] and the tolerance-fecundity tradeoff [47], have been chiefly studied in terrestrial plants, but may enable coexistence among other sessile organisms with a dispersive stage, such as coral, coral fishes, and microbes [47].
To determine whether clusters are a general outcome of competition as opposed to an artifact of specific models, and to verify their robustness to stochastic forces, here we use a stochastic niche simulation approach to investigate the emergence of clustering by traits in species assemblages undergoing competitive dynamics and open to immigration.
We start with Lotka-Volterra dynamics, where clusters are known to emerge in the deterministic model [18], to address the question of their robustness to stochastic forces. We test this robustness with and without the confounding influence of environmental filters, which we hypothesize might mask any clustering caused by niche partitioning. We then see how regional diversity and immigration rate influence this robustness. Clustering intrinsically involves many species, and should in principle be stronger, or at least more detectable, under a high species-to-niches ratio. Therefore, we expect more clustering under higher regional diversity. As for immigration, we expect that while it may contribute to the persistence of clusters by keeping weak competitors from being excluded, too much immigration will drown any pattern caused by competition.
Next, to determine whether clusters are a general outcome of competition, we analyze models spanning three key niche mechanisms: resource partitioning, habitat partitioning, and a competition-colonization tradeoff. Under resource partitioning, we further explore the differences between scenarios with low and high resource depletion. Under competition for habitat, we also examine the impact of dispersal limitation as opposed to global dispersal. We expect that larger scale dispersal relative to the scale of the environmental gradient can influence cluster emergence by lessening the dominance of emergent optimal strategies, since their propagules often spread to locations with suboptimal habitat. If clusters are a signature of competition and niche differentiation, we expect clustering to emerge generally under all scenarios of all three niche mechanisms.
To determine whether communities are clustered beyond patterns that could arise by chance, we present two metrics: the first uses the k-means algorithm [48], which assigns species to clusters by minimizing trait differences within clusters. The second is based on Ripley’s K [49], and quantifies clustering based on the sparseness of the regions between clusters. Both metrics apply the gap statistic method for comparing data with null models [50]. These metrics improve over existing tools to detect clustering [18, 27–29, 51] as they contain no free parameters and do not arbitrarily bin traits. Furthermore, rather than reducing species to presence-absence counts, our metrics weigh them by their abundance. We validate the metrics by confirming that they detect clusters in Lotka-Volterra communities but not in neutral communities.
Results
All niche mechanisms (Fig 1) resulted in species clustering (Fig 2A–2G and Fig 3), demonstrating both the robustness to stochasticity and generality of clustering emergence under niche differentiation. In contrast, neutral communities (Fig 2H) were clustered only about as often as expected by chance, and communities under no niche mechanism but subject to environmental filters collapsed into a single cluster (S1 Fig). Our success at detecting clusters under niche differentiation, combined with this lack of false detection when niche mechanisms are absent, suggests our metrics are well-suited to quantifying the pattern arising from niche differentiation.
Panel A illustrates Lotka-Volterra competition, while panels B-D illustrate different niche mechanisms, with the niche axis represented on the abscissa. A. Lotka-Volterra: competition coefficients are tied to traits, here beak size. Curve shows competitive impact by each species on the focal species, increasing with trait similarity. B. Partitioning resources: We assume a continuum of resources (seeds), and species traits (beak size) determine their use of the different resources. Depending on the degree of specialization, resource depletion may occur (see text). C. Partitioning environment: species with different traits (seed size) are optimally adapted to different environments. This model is spatially explicit, and we implement it both with and without dispersal limitation; D. Competition-colonization tradeoff: species trade off fecundity (seed output) with ability to win sites against other species (mediated by seed size). Seed icons adapted from design by Brgfx—Freepik.com.
For each scenario (A-H) we show one representative replicate. All communities with a niche mechanism (A-G) are clustered at the p < 0.05 level, while the neutral community (H) is not (p = 0.1). Alternating black and red colors highlight the clusters. We truncate the y-axis in the competition-colonization tradeoff scenario (G) to better show abundance structure among rare species. (Immigration rate m = 0.08, regional diversity c. 400 species. All communities had 21,000 individuals, except in the partitioning environment scenario, which had 1,000 individuals. Parameters of niche models were tuned so that without stochasticity and immigration they would produce about 13 transient clusters, and eventually 13 species stably coexisting at equilibrium).
Bars show percentage of replicates that were significantly clustered at level p < 0.05 according to the k-means metric (blue) and the Ripley’s K metric (orange), out of 100 replicates. Numbers next to bars show average z-score. For comparison, 8% of neutral communities were clustered by the k-means metric (), and 7% by the Ripley’s K metric (
), thus close to the 5% background detection expected from a null model (z-scores and p-values were obtained by comparing each replicate against 100 null communities. m = 0.08, c. 400 regional species across scenarios).
Lotka-Volterra
We implemented a stochastic analogue of classical Lotka-Volterra competition, which assumes species influence each other directly via competition coefficients. We tied those coefficients to similarity in traits (Fig 1A), so that trait differences can enable stable coexistence (niche differentiation). In its deterministic form, this type of model is known to produce clusters [18] (except for special cases where competition coefficients form a positive-definite matrix [32, 52]). We thus used this model to consider the robustness of clusters to stochastic forces. We also used it to consider whether that robustness of clustering is influenced by the presence of environmental filtering, and by regional diversity and immigration.
Clustering was strong enough in the Lotka-Volterra communities to be easily distinguishable by eye (Fig 2A and 2B), with over 80% of all 100 replicates testing significant at the p < 0.05 level by either metric (Fig 3). We note that the transient clusters produced by deterministic Lotka-Volterra competition are being maintained in our stochastic model by immigration; when immigration is turned off, communities begin to lose species and tend towards a limiting similarity pattern [23].
Environmental filtering.
Environmental filters [25] or physiological constraints may cause some species to grow faster than others in the absence of competitors. For example, species with intermediate traits may grow faster because extreme trait values may have metabolic costs or are maladapted to the local environment, regardless of competitive interactions. This will affect abundances, and might interfere with the emergence of niche-driven clusters. In our simulations, environmental filters against extreme traits caused species near the center of the trait axis to be more abundant than their counterparts near the edges, as expected (Fig 2B), but this had minimal impact on clustering relative to no filtering (compare average z-scores, Fig 3). However, as we increased the strength of environmental filters, they ultimately eclipsed niche partitioning as a driver of pattern, and the community appeared as a single cluster (S2 Fig), consistent with results from environmental filters acting alone.
Regional diversity.
Clustering increased with regional diversity. As we increased the number of species in the regional pool while keeping the number of clusters fixed, the z-score of the clustering metric rose monotonically (Fig 4). This reflects the fact that clustered patterns in general are increasingly distinguishable from randomness as the number of items per cluster increases (S3 Fig).
Clustering increases with regional diversity (black), and has a modal relationship with immigration pressure (red). Bottom axis: number of species in the pool divided by the number of clusters in the community. Top axis: immigration rate. Points show the mean z-score of the Ripley’s K metric across 10 replicates. Error bars show 1.96 standard errors of the mean. (m = 0.08 when varying regional diversity, and the latter is fixed at c. 400 regional species when varying immigration).
Immigration rates.
Clustering had a modal relationship with immigration pressure: as we increased the immigration rate from low values, the clustering z-score first rose, then peaked and fell (Fig 4). This indicates that while immigration sustains the clusters by replenishing populations that would otherwise be excluded, too much immigration dilutes the effects of local dynamics and adds noise to the pattern (S3 Fig). While this modal shape is arguably general, the position of the mode is specific to the niche mechanism and other factors such as environmental filters. Communities in the other niche models under immigration rate m = 0.01 as opposed to 0.08 generally showed a similar number of clusters but fewer species per cluster, and gaps between clusters were more rarified (S4 Fig). In some cases, this increased clustering (S5 Fig), suggesting that m = 0.08 falls to the right of the mode in those niche models.
Resource partitioning
Assuming fast resource dynamics relative to consumers, one can derive the Lotka-Volterra model from resource-consumer dynamics [42, 53], with trait similarity indicating similar resource preferences [8, 43] (Fig 1B). However, this approach ignores the effects of stochastic fluctuations in resource availability, and particularly resource depletion. The latter has been shown to impact coexistence outcomes [54] and even competition-similarity relationships [45]. These relationships are drivers of pattern [31], and we therefore tested whether incorporating stochastic resource dynamics and depletion affected the emergence of clusters.
In the case of low resource depletion, we found weak clustering (Fig 3). This is because when resource depletion is low, consumers undergo approximate Lotka-Volterra dynamics with competition coefficients following a Gaussian function of trait separation [8]. This function is positive-definite, a property that has been shown to lead to weak or no pattern (see Fig S8 in [23] and [32, 52]). Species sorting under Gaussian competition is slow (see Fig S8 in [23]), and therefore easily overpowered by immigration. Indeed, clustering was stronger under lower immigration pressure (S4B and S5 Figs).
Resource depletion breaks the link to Gaussian Lotka-Volterra dynamics, thereby fundamentally changing the competitive interactions between consumers [45, 54]. That strengthened clustering in our model (compare low- and high-depletion scenarios in Fig 3). As resources were extirpated, species ended up clustering based on specialization to the remaining ones (S6 Fig). Resource depletion thus sets the optimal niche strategies (i.e. specialization onto remaining resources), thereby strengthening the pattern.
Habitat partitioning
Habitat is a critical resource for which species compete, and as such the environment is thought of as a key axis of niche differentiation [55–57]. Traits reflect adaptations to different environments, and hence competition for habitat must shape trait distribution.
To test whether niche-differentiation based on environmental preference leads to clustering, we used the model introduced by [40]. We assume a linear landscape on a habitat gradient, e.g. an elevation gradient. Different species are optimally adapted to different habitats, and competition arises from overlap in environmental preference (Fig 1C). Competition occurs at the recruitment stage, where the probability of recruiting is based on tolerance to the local environment. Because dispersal can play a central role in competition for space [57, 58], we consider two scenarios: global and local dispersal, which differ by whether individuals are more likely to disperse shorter distances from their parents. Since this is an individual-based model, we used a smaller community size (1,000 individuals rather than 21,000) for computational expedience.
We found that clusters also emerge under competition for habitat. Switching dispersal from global to local had a strong impact on cluster shape, and reduced the number of species per cluster (compare Fig 2E and 2F). This occurred as dispersal limitation effectively decreases immigration. Moreover, the attending reduction of the diluting effects of immigration substantially strengthened clustering (Fig 3).
Competition-colonization tradeoff
All models examined so far describe symmetric competition, whereby the competitive impact of species A on B is similar to that of B on A. Competitive hierarchies stand in contrast to this. That is the case of the competition-colonization tradeoff [46, 59], where propagule production, or colonization ability, trades off with the ability to displace individuals of other species, or competitive ability (Fig 1D). Even though in this case some species are better competitors than others, the tradeoff is a niche mechanism because it allows for the stable coexistence of multiple species (see Glossary in S1 Text, S7 Fig). In fact, the model can be cast in Lotka-Volterra form, and one can show that the net competitive impact, while asymmetric, is stronger between more similar species (S7 Fig).
Clusters also emerged under this niche mechanism. The asymmetry in species interactions was reflected in its asymmetric clusters (Fig 2G). The k-means metric picked up on just three clusters in most replicates, even though without stochasticity and immigration the model produces about 13 transient clusters (S7 Fig). This is perhaps because species in the first cluster so strongly dominated the community (e.g., the most abundant species in the first cluster in Fig 2G had 4,798 individuals, compared with 395 in the second and 392 in the third). This indicates that species adopting the high-competitiveness strategy (left side of the trait axis) outperform both those who invest in high fecundity and the intermediate group. Dispersal limitation could reduce the asymmetry by augmenting the benefits of high fecundity. We hypothesize that a spatially explicit formulation of this mechanism would produce more similarly sized clusters.
Metric performance
While our metrics performed equally well in the Lotka-Volterra communities, they differed in the other niche mechanisms. Our Ripley’s K metric fared better than the k-means metric in scenarios where species partition resources and habitat (Fig 3) (though performance was similar for resource partitioning under low immigration, see S5 Fig). The Ripley’s K metric focuses on identifying a scale of interspecific trait difference with particularly low representation (i.e. by which very few species pairs are separated). As such, it relies on regular spacing between clusters and intercluster gaps. Ripley’s K is good at identifying clusters when that spacing is regular, even in cases where the overall pattern is noisy (the resource partitioning cases), or involves low-occupancy clusters (the habitat partitioning cases). On the other hand, the k-means metric found the clusters in the competition-colonization tradeoff while Ripley’s K missed them (Fig 3). This is because the k-means algorithm is less sensitive to strong asymmetries between the clusters. In these cases, the k-means metric is a better choice.
Discussion
Ecologists have long sought to understand how competition shapes community structure. While competing species are usually expected to be more different than predicted by chance [25, 60], recent studies suggest that competition may cause species to cluster by traits, such that the community self-organizes into groups of similar species [18], a phenomenon which has been interpreted to bridge coexistence through differences–niche theory–and similarity–neutral theory. Our study verified that clustering transcends Lotka-Volterra dynamics, occurring under a number of niche mechanisms. Further, we showed that clustering is robust to stochastic drift, an intrinsic property of real-life communities. Immigration maintains clusters that are otherwise transient, and the strength of clustering has a modal relationship with immigration pressure. We showed that clustering may be detectable under the confounding influence of environmental filters, and is enhanced by regional diversity. Finally, we provided metrics for detecting and quantifying clusters in nature.
Why do clusters arise? Different niche mechanisms share the common property that competition is stronger between species with more similar strategies. It thus seems paradoxical that clusters should emerge. However, it is precisely because species with similar niches compete more strongly that clusters appear [61]. While similar pairs compete more strongly, they experience similar competitive pressure (or relief) from the rest of the community. If a given niche strategy is favored because it minimizes competition with the rest of the community or capitalizes on greater resource supplies, then similar strategies are similarly favored. This hilly fitness landscape causes exclusion to be slower near the center of the niches than in the gaps between them, making it easy for immigration to permanently maintain the clusters [23].
Modern coexistence theory [7] splits coexistence-promoting processes into those that reduce competition among species relative to competition within species (stabilizing mechanisms, here referred to as niche mechanisms), and those that reduce differences in average fitness between species (equalizing mechanisms). It is thus tempting to interpret clusters as reflecting a harmonious combination of stabilizing and equalizing forces: species within a cluster are equalized, while those in different clusters are stabilized. However, this interpretation is problematic. The equalization-stabilization dichotomy is based on applying invasibility criteria to closed communities regulated by a small number of limiting factors [62], an approach which does not extend easily to multispecies communities under immigration and a continuum of resources. In diverse communities with complex competitive interactions, it is difficult to calculate equalizing and stabilizing terms and tie them to trait differences, let alone interpret how specific patterns such as clustering connect with equalization and stabilization forces [62]. One approach is to assume all pairs of species compete with equal intensity [7, 63], but this assumption is strongly violated in all models where clustering has been observed so far.
While competition is responsible for clusters’ emergence, immigration is responsible for their persistence. Immigration joins other mechanisms that have been previously shown to sustain clusters, namely specialist enemies [18, 64] and environmental fluctuations [33, 34]. We found that clustering appears generally under different immigration regimes, especially if the number of species far exceeds the number of available niches. Studies of stochastic competitive preceding Scheffer and van Nes 2006 found little impact of immigration on resulting trait pattern [39], [40]. However, successful immigration was highly infrequent in those models. In [39], resources made available through deaths were assumed to be redistributed broadly, so that resource supply remained low everywhere, making recruitment of new individuals highly unlikely. In [40], immigration took the form of a single immigrant seed being added to a large pool of local seeds competing for the site. As such, immigration rates were effectively much lower than ours. [41] tested both very high and very low immigration, and also saw no clusters (in no small part because the authors were not looking for them!). This could be due to their use of a Gaussian competition kernel (i.e. competition coefficients are a Gaussian function of trait difference), which leads to weak niche sorting dynamics, easily overwhelmed under high immigration [32].
The fact that immigration maintains clusters seemingly defers the question of coexistence to the regional scale. The problem dissipates by considering mass effects in the metacommunity framework [65]. The regional pool is a combination of local communities, and species are selected for different traits at different sites due to their own local niche dynamics. Therefore, each community receives immigrants which may be dominant elsewhere despite being disfavored locally. However, this does not mean that pattern is expected regionally, as the sum total of heterogeneous communities, each with a different trait pattern, may result in no discernible pattern at a regional scale.
Clusters caused by partitioning of a niche axis are often distinct from clustering due to other processes. Environmental filters favoring a single best trait and one-sided competitive dynamics where a particular trait outperforms all others without a balancing tradeoff [26] will produce a single cluster as opposed to multiple clusters. Where evolutionary rates are commensurate with ecological dynamics, small mutations and sympatric speciation may also generate species or genotype clusters without niche differentiation [66, 67]. Ruling out these alternative sources of clustering could require sampling at a larger scale than applicable to the niche mechanism [5, 60], or directly verifying competitive effects and frequency dependence [4]. While clusters may have more than one source, observation of multiple clusters in specific functional traits can help identify potential drivers of niche differentiation in a community.
We presented and validated two nonparametric abundance-weighted tools for detecting and quantifying clusters. Our metrics successfully distinguished clustering in niche-differentiated communities from no clustering in neutral communities and a single cluster in communities under environmental filters without a niche mechanism. We note that none of these results would appear without considering species abundances, as presence-absence counts do not reveal clustering in our communities. Also, our metrics did not require arbitrarily binning traits, nor fitting parameters to the data.
Although our study focused on one-dimensional trait axes, competitive interactions may often be mediated by variation in multiple traits [68]. Theoretical work on simple models indicates that multidimensional niche space leads to multidimensional clustering [20]. Our metrics can quantify clusters in any dimension, using generalized measures of niche separation. The main challenge is to connect multidimensional phenotypes to competitive relations, in order to define the correct measure of distance in high-dimensional trait space [61]. However, even if species cannot be arranged on linear trait axes, our Ripley’s K metric can still detect clustering by similarity as long as a measure of species differences can be defined, e.g. Hamming distance in genetic sequences [66].
Clustering as a signature of coexistence under competition is an update to the still dominant paradigm that competing species will display greater-than-chance differences [3–5, 16, 17]. Our finding that clusters appear under various niche mechanisms and can be easily maintained by immigration, even when confounding forces are at play, suggests that clusters are a likely feature of nature beyond the instances where it is currently known to occur [18, 28, 29]. For example in tropical forests, where both competition and dispersal are recognized as key drivers of community assembly [69, 70], clustering could help explain the high diversity and seemingly continuous phenotypic variation.
Methods
Models and simulation design
We used a lottery model framework [71] to implement stochastic niche dynamics in a fixed-size community open to immigration from a regional pool. We start with a random draw of offspring from the regional pool, and then alternate death and recruitment events until species abundance distributions are stationary. A proportion m of deaths are replaced by immigrants, and the remainder by local offspring. This is analogous to Hubbell’s neutral model [38], except here the niche mechanism sets the probabilities of birth and death across different species. Schematic illustrations of our niche models are shown in Fig 1.
We used the 50-hectare plot of tropical forest on Barro Colorado Island, Panama, as a reference point for our community size (21,000 individuals >10 cm dbh [38]) and immigration rate (0.08 immigrant recruits per recruitment event [72]). Our regional pool is a fixed neutral metacommunity with biodiversity parameter θ = 50 [38] and 150,000 individuals, leading to c. 400 species (bigger metacommunities did not change results).
All simulations and statistical analyses were done in R [73].
Lotka-Volterra.
The probability that species i is selected for a death event is ∑j AijNiNj/∑k∑l AklNkNl, where Ni is the abundance of species i and the competition coefficient Aij quantifies the competition caused on species i by species j. The sums are over all species present in the local community, and the denominator normalizes the probabilities to 1. The probability that species i is chosen at the subsequent recruitment stage is mqi + (1 − m) riNi/∑jrjNj, where qi is the relative abundance of species i in the regional pool, and ri is its intrinsic growth rate. (We are placing competitive interactions on mortality, but results are unchanged when we place them on births instead).
We link competition to traits by setting Aij = exp[−(|xi − xj|/w)4], where xi, xj ∈ [0, 1] are the trait values of species i and j. Thus competition peaks at xi = xj (intraspecific competition), and declines with increasing trait difference (Fig 1A). This is a generic unimodal function commonly used in competition models [31]. Other unimodal shapes bring no qualitative changes (with a few exceptions, see [32]). The scaling parameter w sets the width of niche overlap, and therefore the number of niches available to species. We set w = 0.063, leading to c. 13 clusters. Where possible, we set parameters to keep the number of clusters consistent.
In the scenario with environmental filtering we set intrinsic growth rates ri = xi(1 − xi), such that species with intermediate traits grow faster than those with extreme traits. In all other Lotka-Volterra scenarios, we set all ri = 1 identical across species.
We set default values m = 0.08 for immigration rate, and S ≃ 400 for regional diversity (via θ = 50 in the regional pool). When examining the impact of immigration pressure, we used m = 0.005, 0.01, 0.05, 0.08, 0.15. When examining regional diversity, we used S ≃ 50, 100, 180, 250, 400 by setting θ = 5, 10, 20, 30, 50.
Partitioning resources.
Let Xa and Ni be the abundances of resource a and species i. We assume resource consumption affects species reproduction, and all species have equal mortality. As such, the probability that species i is chosen for a death event is simply Ni/∑j Nj, and the probability that it is chosen for a recruitment event is mqi + (1 − m) ∑a CaiXaNi/∑j∑b CbjXbNj, where Cai quantifies the preference of species i for resource a.
Resources simultaneously follow their own stochastic birth-death process. Consumption lowers resource populations, which would otherwise grow logistically. Therefore, resource a is selected for a death event with probability ∑i CaiXaNi/∑b∑j CbjXbNj, and for a birth event with probability (1 − Xa/K)Xa/∑b(1 − Xb/K)Xb, where K is the resource carrying capacity, set equal across all resources for simplicity. There is no resource immigration; if a resource population is depleted to zero, the resource is extirpated and cannot be replenished.
Resources are arranged on a line between 0 and 1. We define Cai = exp[−(dai/σ)2], where dai is the difference between resource a and the preferred resource of consumer i (Fig 1B). The width parameter σ controls degree of resource overlap and thus number of niches. We set σ = 0.03 as it leads to about 13 clusters, for consistency with the Lokta-Volterra scenarios. We control resource depletion via the resource carrying capacity. When K = 100, most resources are never severely depleted by consumption, whereas when K = 400 resource depletion is severe and many resources are extirpated. Both scenarios start with 500 resources drawn from a standard uniform distribution.
Partitioning habitat.
We used the model introduced by [40]. We assume a linear array of sites forming a habitat gradient. Different species are optimally adapted to different environments, and competition arises from overlap in environmental preference (Fig 1C). Here, a death event is followed by a lottery in the local seed bank, which is formed via dispersal from all individuals in the community, plus a proportion of immigrants. A seed’s probability to win the lottery is given by its tolerance to the local environment, which declines with differences from the optimal environment of that species.
All individuals have the same probability of dying. The probability that species i fills the vacancy is then , where Dab is the probability of dispersal between an individual at site b and the vacated site a, and Tai is species i’s tolerance to site a. We set Tai = exp[−(Δai/σ)2], where Δai is the difference between the local environment a and the niche optimum of species i. The width parameter σ controls niche overlap, and hence the number of niches. We set σ = 0.07 as it gives us c. 13 clusters.
We use a Gaussian dispersal kernel, Dab = exp[−0.5(dab/ddisp)2], where dab is the distance between vacated site a and a parent individual at site b. We use 1,000 sites on a line, with adjacent sites one unit distance apart (da, a+1 = 1), leading to a fixed community size of 1,000 individuals. The reduced size relative to 21,000 individuals in the other models was used to keep this agent-based model computationally expedient. We explore a scenario with local dispersal (ddisp = 50), and one global dispersal (ddisp = 105), i.e. where the probability of arrival is independent of distance to the parent.
Competition-colonization tradeoff.
Under this mechanism, changes to a population result from gains through recruitment and losses via death and displacement. Collecting the density-dependent and density-independent factors separately, one can show (see Supplementary Information) that the deterministic dynamics follow the equation
where fi is the fecundity of species i, μ is a constant, and Γij is the ability of a propagule of species j to displace an individual of species i.
The equation above is in Lotka-Volterra form. Therefore in parallel with our Lotka-Volterra scenarios, we place the density dependence on mortality. We thus set the probability that species i is chosen to die as ∑j(fi + fj)ΓijNiNj/∑k∑l(fk + fl)ΓklNkNl, while the probability it is chosen to recruit is mfiqi/∑j fjqj + (1 − m)(fi − μ)Ni/∑j(fj − μ)Nj. Notice that high-fecundity species immigrate more frequently. The difference fi − μ is an intrinsic growth rate, i.e. the net rate at which species grow in the absence of competition. Fecundities are log-uniform distributed between 1 and 6.2. We set μ = 1.
The tradeoff between fecundity and ability to displace is encoded in the displacement matrix Γij. If we arrange species in order of fecundity, Γij will be approximately bottom-triangular, with Γij ≈ 0 if i < j and Γij ≈ 1 if i > j. That is, species with higher fecundity rarely displace those with lower fecundity, while low-fecundity species commonly displace high-fecundity ones. In the limit of an absolute tradeoff, lower-ranking species never displace and are always displaced by higher-ranking species. This limit has been shown to drastically inflate coexistence under this mechanism, despite the fact that such absolute tradeoffs are unlikely to occur in nature [74–76]. As such, here we use a probabilistic tradeoff: Γij = 0.5 (1 − tanh[s(fj − fi)]), which has a sigmoidal shape, is equal to 0.5 when fi = fj, and asymptotes to 1 and 0 when fi ≫ fj and fi ≪ fj, respectively. Parameter s controls the steepness of the transition, and hence the number of clusters. We set s = 0.15, which in the absence of immigration and stochasticity leads to c. 13 coexisting species (S7 Fig).
Clustering metrics: The gap statistic
We apply a variation of the gap statistic method [50], using two different clustering measures: k-means dispersion [48] and Ripley’s K function [77]. In general terms, the metric takes in the list of species traits and abundances, and returns the number of clusters (k-means) or average trait separation between them (Ripley’s K), as well as a z-score and a p-value. In S1 Box we summarize the k-means version and give a step-by-step recipe for its implementation, and illustrate it at work on two example communities. Both versions are described in detail in S1 Appendix. The code for the k-means version is available on GitHub [78].
We assess statistical significance and degree of clustering by comparing a community against 100 null communities where we randomly shuffle local abundances across all species in the regional pool. This allows us to test specifically for a nonrandom association between traits and abundances, while keeping the observed abundance distribution fixed. One alternative null model is neutrality, whereby abundances follow a characteristic dispersal-limited multinomial distribution [79].
Because our models are stochastic, we run 100 replicates to account for variation within the same scenario. For each niche scenario we report the average z-score across 100 replicates, as well the percentage of replicates that were significantly clustered at level p < 0.05. To assess if a given run is significantly clustered we compared with the distribution across 100 nulls.
We check for false positives by testing our metric on neutral communities and communities where differences in species performance are due strictly to environmental filtering. From the former we expect significant clustering in circa 5% of runs when using a p = 0.05 cutoff, and from the latter we expect that species will cluster around the favored trait. Thus we distinguish niche differentiation from neutrality and pure environmental filtering by the presence of multiple clusters as opposed to none or a single one.
Supporting information
S1 Box. Detecting clusters with k-means and the gap statistic.
https://doi.org/10.1371/journal.pcbi.1006688.s002
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S1 Appendix. Mathematical formulation of the competition-colonization model.
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S2 Appendix. Detailed description of the clustering metrics.
https://doi.org/10.1371/journal.pcbi.1006688.s004
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S1 Fig. Community under environmental filters but no niche mechanism shows a single cluster.
A: Example simulation outcome of Lotka-Volterra stochastic dynamics with neutral competition coefficients, αij = 1, such that there is no niche mechanism, and intrinsic growth rates given by ri = xi(1 − xi), where xi is the trait of species i. The latter represents environmental filtering for species with intermediate traits. B: Corresponding gap curve, showing gap index for each number of clusters between 1 and 20, has a clear maximum at 1 cluster. The estimated number of clusters is therefore K = 1. The gap statistic is well above the 95% quantile of the null distribution (red line), indicating significance at P < 0.05. All 100 replicates of this scenario were clustered, with a single cluster detected by the k-means metric in all cases.
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S2 Fig. Lotka-Volterra niche scenarios with environmental filtering.
A: We implemented environmental filters via a modal relationship between intrinsic growth rate r and species traits xi as follows: r = 0.5 (no filtering, black); ri = xi(1 − xi) (red); ri = exp(−(xi − 0.5)2/σ2) with σ = 0.5 (green), 0.2 (blue), 0.1 (magenta). B: Under no filtering, the community shows no overarching abundance trend. C-F: Under increasing filter intensity, communities show increasingly steeper abundance trends. Our metrics detected multiple clusters in 10/10 replicates of B-D, but only 2/10 in E and 1/10 in F, with the remaining replicates having a single cluster. When no niche mechanism is at play, such that species compete neutrally but still differ by intrinsic growth rates, all replicates result in a single cluster centered on the species with the highest intrinsic growth rate.
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S3 Fig. Effects of immigration rates and regional diversity.
Example Lokta-Volterra communities with increasing immigration rate m (left) and regional diversity parameter θ (right).
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S4 Fig. Lotka-Volterra and other niche communities under lower immigration, m = 0.01.
Comparing panels A-F against their respective m = 0.08 counterparts in Fig 2 (panels A, C-G with matching titles), we see that communities show a similar number of clusters but fewer species per cluster, and gaps between clusters are more rarified.
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S5 Fig. Clustering results under lower immigration, m = 0.01.
Results are similar to m = 0.08 (compare with Fig 3), although z-scores and significance are often higher, particularly for resource-partitioning communities, and habitat-partitioning communities under global dispersal. This indicates that higher immigration is drowning the pattern in these niche scenarios.
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S6 Fig. Effects of resource depletion.
Example communities where consumers partition resources, under low and high resource depletion (left and right columns, respectively). Resources are shown on top, consumers at the bottom. Under high resource depletion, gaps left by resource extirpation cause corresponding gaps among consumer species. These gaps strengthen the clustering pattern relative to the low depletion scenario. (m = 0.08, c. 400 regional species).
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S7 Fig. Competition-colonization tradeoff, deterministic implementation without immigration.
A. Transient state shows visible clustering. B. Equilibrium state, showing abundances of the coexisting species, which are those that dominate their respective clusters in the transient state (marked with red dots in A). Legend shows maximum eigenvalue of the Jacobian of the equilibrium, indicating dynamical stability of the equilibrium. C. Competition kernel shows the strongest competitors on focal species, whose traits are shown in legend, are species with similar traits, despite the competitive hierarchy. This contributes to stabilization of the community. D. For each species (traits plotted on x-axis), the position of the peak of the competition kernel (i.e. the species with the strongest net competitive impact on it) is plotted on the y-axis. The proximity of the curve to the one-to-one line (dotted line) throughout the trait axis shows that the competition-colonization tradeoff stabilizes the community, thus acting as a niche mechanism. The kernel maxima plotted here also explain the absence of coexisting species below trait ≃ 0.2, as the strongest competitors of those species have higher fecundity than themselves (solid curve is above dotted line), thus being both more competitive and more fecund. The wide gap between the first cluster and the other clusters is also reflected in the relatively larger distance from the one-to-one curve in that region of the trait axis.
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Acknowledgments
We thank David Ackerly for helpful discussions, and James O’Dwyer, Brian Sedio, Stephen Smith, and Lacey Knowles for comments on previous versions of the manuscript.
References
- 1. Simberloff D, Boecklen W. Santa Rosalia Reconsidered: Size Ratios and Competition. Evolution. 1981;35(6):1206–1228. pmid:28563391
- 2.
Chase JM, Leibold M. Ecological Niches: Linking Classical and Contemporary Approaches. The University of Chicago Press; 2003.
- 3. Götzenberger L, de Bello F, Bråthen KA, Davison J, Dubuis A, Guisan A, et al. Ecological assembly rules in plant communities-approaches, patterns and prospects. Biological Reviews. 2012;87(1):111–127. pmid:21692965
- 4. HilleRisLambers J, Adler PB, Harpole WS, Levine JM, Mayfield MM. Rethinking Community Assembly through the Lens of Coexistence Theory. Annual Review of Ecology, Evolution, and Systematics. 2012;43(1):227–248.
- 5. D’Andrea R, Ostling AM. Challenges in linking trait patterns to niche differentiation. Oikos. 2016;125(10):1369–1385.
- 6. Chesson P. A Need for Niches? Trends in ecology & evolution. 1991;6(1):26–28.
- 7. Chesson P. Mechanisms of maitenance of species diversity. Annual Review of Ecology and Systematics. 2000;31(2000):343–358.
- 8. MacArthur RH, Levins R. The Limiting Similarity, Convergence, and Divergence of Coexisting Species. The American Naturalist. 1967;101(921):377–385.
- 9. Grant PR. Colonization of islands by ecologically dissimilar species of birds bill length feeding. Canadian Journal of Zoology. 1969;47(1):41–43.
- 10. Botta-Dukát Z. Rao’s quadratic entropy as a measure of functional diversity based on multiple traits. Journal of Vegetation Science. 2005;16(5):533–540.
- 11. Laliberté E, Legendre P. A distance-based framework for measuring functional diversity from multiple traits. Ecology. 2010;91(1):299–305. pmid:20380219
- 12. de Bello F. The quest for trait convergence and divergence in community assembly: are null-models the magic wand? Global Ecology and Biogeography. 2011; p. no–no.
- 13. Stubbs WJ, Wilson JB. Evidence for limiting similarity in a sand dune community. Journal of Ecology. 2004;92(4):557–567.
- 14. Cornwell WK, Schwilk DW, Ackerly DD. A trait-based test for habitat filtering: Convex hull volume. Ecology. 2006;87(6):1465–1471. pmid:16869422
- 15. Ingram T, Shurin JB. Trait-based assembly and phylogenetic structure in northeast Pacific rockfish assemblages. Ecology. 2009;90(9):2444–53. pmid:19769123
- 16. Astor T, Strengbom J, Berg MP, Lenoir L, Marteinsdottir B, Bengtsson J. Underdispersion and overdispersion of traits in terrestrial snail communities on islands. Ecology and Evolution. 2014;4(11):2090–2102. pmid:25360251
- 17. Pontarp M, Petchey OL. Community trait overdispersion due to trophic interactions: concerns for assembly process inference. Proc Roy Soc Lond B. 2016;283:20161729.
- 18. Scheffer M, van Nes EH. Self-organized similarity, the evolutionary emergence of groups of similar species. Proceedings of the National Academy of Sciences of the United States of America. 2006;103(16):6230–6235. pmid:16585519
- 19. Fort H, Scheffer M, van Nes EH. The paradox of the clumps mathematically explained. Theoretical Ecology. 2009;2(3):171–176.
- 20. Fort H, Scheffer M, van Nes EH. The clumping transition in niche competition: a robust critical phenomenon. Journal of Statistical Mechanics: Theory and Experiment. 2010;2010(05):P05005.
- 21. Roelke DL, Eldridge PM. Mixing of supersaturated assemblages and the precipitous loss of species. The American Naturalist. 2008;171(2):162–75. pmid:18197769
- 22. Scranton K, Vasseur DA. Coexistence and emergent neutrality generate synchrony among competitors in fluctuating environments. Theoretical Ecology. 2016;9(3):353–363.
- 23. D’Andrea R, Ostling A. Biodiversity maintenance may be lower under partial niche differentiation than under neutrality. Ecology. 2017;98(12):3211–3218. pmid:28898396
- 24. Scheffer M, van Nes EH, Vergnon R. Toward a unifying theory of biodiversity. Proceedings of the National Academy of Sciences. 2018;115(4):201721114.
- 25. Keddy PA. Assembly and response rules: two goals for predictive community ecology. Journal of Vegetation Science. 1992;3(2):157–164.
- 26. Mayfield MM, Levine JM. Opposing effects of competitive exclusion on the phylogenetic structure of communities. Ecology letters. 2010;13(9):1085–93. pmid:20576030
- 27. Vergnon R, Dulvy NK, Freckleton RP. Niches versus neutrality: uncovering the drivers of diversity in a species-rich community. Ecology letters. 2009;12(10):1079–90. pmid:19747181
- 28. Segura AM, Calliari D, Kruk C, Conde D, Bonilla S, Fort H. Emergent neutrality drives phytoplankton species coexistence. Proceedings Biological sciences / The Royal Society. 2011;278(1716):2355–61.
- 29. Segura AM, Kruk C, Calliari D, García-Rodriguez F, Conde D, Widdicombe CE, et al. Competition Drives Clumpy Species Coexistence in Estuarine Phytoplankton. Scientific Reports. 2013;3(1):1037. pmid:23301158
- 30. Holt RD. Emergent neutrality. Trends in Ecology & Evolution. 2006;21(10):531–3.
- 31. Leimar O, Sasaki A, Doebeli M, Dieckmann U. Limiting similarity, species packing, and the shape of competition kernels. Journal of Theoretical Biology. 2013;339:3–13. pmid:23954548
- 32. Hernández-García E, López C, Pigolotti S, Andersen KH. Species competition: coexistence, exclusion and clustering. Philosophical Transactions of the Royal Society A. 2009;367(1901):3183–95.
- 33. Ernebjerg M, Kishony R. Dynamic phenotypic clustering in noisy ecosystems. PLoS computational biology. 2011;7(3):e1002017. pmid:21445229
- 34. Sakavara A, Tsirtsis G, Roelke DL, Mancy R, Spatharis S. Lumpy species coexistence arises robustly in fluctuating resource environments. Proceedings of the National Academy of Sciences. 2018;
- 35. Gaston KJ, Chown SL. Neutrality and the niche. Functional Ecology. 2005;19(1):1–6.
- 36. Vellend M. Conceptual synthesis in community ecology. The Quarterly review of biology. 2010;85(2):183–206. pmid:20565040
- 37. Rosindell J, Hunter MO, Etienne RS. The unified neutral theory of biodiversity and biogeography at age ten. Trends in ecology & evolution. 2011;26(7):340–8.
- 38.
Hubbell SP. The Unified Neutral Theory of Biodiversity and Biogeography. Princeton, NJ: Princeton University Press; 2001.
- 39. Tilman D. Niche tradeoffs, neutrality, and community structure: a stochastic theory of resource competition, invasion, and community assembly. Proceedings of the National Academy of Sciences of the United States of America. 2004;101(30):10854–61. pmid:15243158
- 40. Schwilk DW, Ackerly DD. Limiting similarity and functional diversity along environmental gradients. Ecology Letters. 2005;8(3):272–281.
- 41. Gravel D, Canham CD, Beaudet M, Messier C. Reconciling niche and neutrality: the continuum hypothesis. Ecology Letters. 2006;9(4):399–409. pmid:16623725
- 42. O’Dwyer JP. Whence Lotka-Volterra? Theoretical Ecology. 2018; p. 1–12.
- 43.
MacArthur RH. Geographical ecology; patterns in the distribution of species. Harper & Row; 1972.
- 44. Abrams P. Limiting Similarity and the Form of the Competition Coefficient. Theoretical Population Biology. 1975;8:356–375. pmid:1220046
- 45. Abrams P, Rueffler C, Dinnage R. Competition-similarity relationships and the nonlinearity of competitive effects in consumer-resource systems. The American Naturalist. 2008;172(4):463–74. pmid:18717636
- 46. Levins R, Culver D. Regional Coexistence of Species and Competition between Rare Species. Proceedings of the National Academy of Sciences of the United States of America. 1971;68(6):1246–1248. pmid:16591932
- 47. Muller-Landau HC. The tolerance-fecundity trade-off and the maintenance of diversity in seed size. Proceedings of the National Academy of Sciences of the United States of America. 2010;107(9):4242–7. pmid:20160078
- 48.
MacQueen JB. Some Methods for classification and Analysis of Multivariate Observations. 5th Berkeley Symposium on Mathematical Statistics and Probability 1967. 1967;1(233):281–297.
- 49. Ripley BD. The Second-Order Analysis of Stationary Point Processes. Journal of Applied Probability. 1976;13(2):255–266.
- 50. Tibshirani R, Walther G, Hastie T. Estimating the number of clusters in a data set via the gap statistic. Journal of the Royal Statistical Society: Series B (Statistical Methodology). 2001;63:411–423.
- 51. Yang Z, Powell J, Zhang C, Du G. The effect of environmental and phylogenetic drivers on community assembly in an alpine meadow community. Ecology. 2012;93(11):2321–2328. pmid:23236903
- 52. Pigolotti S, López C, Hernández-García E. Species Clustering in Competitive Lotka-Volterra Models. Physical Review Letters. 2007;98(25):1–4.
- 53. Rosenzweig ML, MacArthur RH. Graphical Representation and Stability Conditions of Predator-Prey Interactions. American Naturalist. 1963;97(895):209–223.
- 54. Abrams P, Rueffler C. Coexistence and limiting similarity of consumer species competing for a linear array of resources. Ecology. 2009;90(3):812–22. pmid:19341150
- 55. Harner RF, Harper KT. The Role of Area, Heterogeneity, and Favorability in Plant Species Diversity of Pinyon- Juniper Ecosystems. Ecology. 1976;57:1254–1263.
- 56. Grubb PJ. the Maintenance of Species-Richness in Plant Communities: the Importance of the Regeneration Niche. Biological Reviews. 1977;52(1):107–145.
- 57. Comins HN, Noble IR. Dispersal, Variability, and Transient Niches: Species Coexistence in a Uniformly Variable Environment. The American Naturalist. 1985;126(5):617–640.
- 58. Amarasekare P. Competitive coexistence in spatially structured environments: A synthesis. Ecology Letters. 2003;6(12):1109–1122.
- 59.
Tilman D. Resource competition and community structure. Monographs ed. Princeton University Press; 1982.
- 60. Kraft NJB, Ackerly DD. Functional trait and phylogenetic tests of community assembly across spatial scales in an Amazonian forest. Ecological Monographs. 2010;80(3):401–422.
- 61. D’Andrea R, Ostling A, O’Dwyer JP. Translucent windows: How uncertainty in competitive interactions impacts detection of community pattern. Ecology Letters. 2018; p. 826–835. pmid:29601655
- 62. Barabás G, D’Andrea R, Stump SM. Chesson’s coexistence theory. Ecological Monographs. 2018;
- 63. Chesson P. Multispecies Competition in Variable Environments. Theoretical Population Biology. 1994;45(3):227–276.
- 64. Barabás G, D’Andrea R, Rael R, Meszéna G, Ostling AM. Emergent Neutrality or Hidden Niches? Oikos. 2013;122:1565–1572.
- 65. Leibold MA, Holyoak M, Mouquet N, Amarasekare P, Chase JM, Hoopes MF, et al. The metacommunity concept: A framework for multi-scale community ecology. Ecology Letters. 2004;7(7):601–613.
- 66. D’Andrea R, Ostling AM. Can Clustering in Genotype Space Reveal “Niches”? The American Naturalist. 2016;187(1):130–135. pmid:27277409
- 67. Houchmandzadeh B. Neutral aggregation in finite-length genotype space. Physical Review E. 2017;95(1):1–12.
- 68. Clark JS, Dietze M, Chakraborty S, Agarwal PK, Ibanez I, LaDeau S, et al. Resolving the biodiversity paradox. Ecology letters. 2007;10(8):647–59; discussion 659–62. pmid:17594418
- 69. Leigh EG, Davidar P, Dick CW, Puyravaud JP, Terborgh J, ter Steege H, et al. Why do some tropical forests have so many species of trees? Biotropica. 2004;36(March 2003):447–473.
- 70. Seidler TG, Plotkin JB. Seed dispersal and spatial pattern in tropical trees. PLoS Biology. 2006;4(11):2132–2137.
- 71. Chesson P, Warner RR. Environmental Variability Promotes Coexistence in Lottery Competitive Systems. The American Naturalist. 1981;117(6):923–943.
- 72. Chisholm RA, Lichstein JW. Linking dispersal, immigration and scale in the neutral theory of biodiversity. Ecology Letters. 2009;12(12):1385–93. pmid:19807774
- 73.
R Core Team. R: A Language and Environment for Statistical Computing; 2018. Available from: https://www.r-project.org/.
- 74. Barabás G, D’Andrea R, Ostling AM. Species packing in nonsmooth competition models. Theoretical Ecology. 2013;6(1):1–19.
- 75. D’Andrea R, Barabás G, Ostling AM. Revising the tolerance-fecundity trade-off; or, on the consequences of discontinuous resource use for limiting similarity, species diversity, and trait dispersion. The American Naturalist. 2013;181(4):E91–101. pmid:23535625
- 76. Haegeman B, Sari T, Etienne RS. Predicting coexistence of plants subject to a tolerance-competition trade-off. Journal of Mathematical Biology. 2014;68(7):1815–47. pmid:23728210
- 77.
Dixon PM. Ripley’s K function. In: El-shaarawi AH, Piegorsch WW, editors. Encyclopedia of Environmetrics. vol. 3. Chichester: John Wiley & Sons; 2002. p. 1796–1803.
- 78.
D’Andrea R. Code for clustering metrics; 2018. Available from: https://github.com/rafaeldandrea/Clustering-metric.
- 79. Etienne RS, Alonso D, McKane AJ. The zero-sum assumption in neutral biodiversity theory. Journal of theoretical biology. 2007;248(3):522–36. pmid:17640675