The dimensionality of plant–plant competition

Much of our current understanding of species coexistence derives from studying exploitation competition—competition between similar species for a shared pool of finite, limiting resources such as water, nutrients, light, or space^1,2,3,4,5,6. In relatively constant environments, the coexistence of many species is strongly influenced by two conditions above all others: there should be at least as many resources available as coexisting species^7,8,9 and those species should have different niches (e.g., resource-use requirements)^10,11,12,13. When species have distinct niches—for example, by specializing on unique resources^14,15—intraspecific competition should exceed interspecific competition, and this can in turn help prevent communities from becoming overrun by the most dominant competitor or competitors^16,17. When species are functionally similar, in contrast, their coexistence may benefit from the presence of additional, external sources of variability to buffer against otherwise unfavorable years, habitats, or environmental conditions^18,19. Indeed, ecological drift alone can be sufficient to exclude all but one species from any finite community whose constituent species show no niche differentiation^20,21.

The theoretical basis linking niche differentiation to coexistence is quite clear^22,23,24,25. The importance of niche differences for maintaining species diversity also has strong empirical support^26,27,28. The link between theory and empirical observations is, however, still unclear since there is no clear picture of the number of niche dimensions that are actually exploited within diverse natural communities. This is partly due to the fact that researchers traditionally select a priori the relevant resources or limiting factors over which species are thought to compete^8,29,30,31. Unfortunately, such approaches cannot guarantee that all relevant dimensions have been taken into account, or that those selected are even the most relevant ones in practice; even controlled scenarios can overlook previously-unknown, but equally-important, mechanisms. To manage these complications, some researchers have taken a more indirect approach of relating the strength of competition and resource-use variation to differences in functional traits¹³ or evolutionary histories^32,33, based on the assumption that these species characteristics are reasonable summaries of the multiple dimensions that compose a species’ niche. Yet indirect, correlative approaches also cannot conclusively identify which resources are limiting⁸ or, more importantly, how many niche dimensions are realized³⁴. It therefore remains unclear whether or not the interactions between species in any given ecological assemblage are indicative of highly differentiated or highly similar species competing over a large or a small resource pool (Supplementary Note S1).

As tantalizing as it might be, exhaustive inference of niche dimensionality in diverse empirical communities—as achieved by direct manipulation of all potential niche axes (e.g., light, water, space, nutrients, pollinators, natural enemies, etc.)—is next to impossible. With this in mind, we introduce here an alternative perspective on this long-standing question. Rather than characterize and compare resource use explicitly or select proxies from species’ characteristics, we develop an approach that uses information about the strengths and signs of interactions between species to infer their implied niche basis (see “Mathematical framework for estimating niche dimensionality”). In particular, our method provides a means to quantify the “niche dimensionality” of any interacting species assemblage. Conceptually, niche dimensionality can be thought of as a proxy for the number of limiting factors over which those species show niche variation (Supplementary Note S2). In addition, our framework implies that it is also a data-informed and mathematical property of the emergent interactions between species.

In addition to estimating niche dimensionality, our approach decomposes pairwise interactions between species into (i) species’ response traits, (ii) species’ effect traits, and (iii) the relative strength of these responses and effects across the underlying niche dimensions (see “Mathematical framework for estimating niche dimensionality”). This decomposition helps us identify species’ competitive strategies across the niche dimensions shared among the assemblage (Supplementary Note S2). Specifically, effect traits modulate the impact of each species on all others while response traits modulate the impact of all other species on each species^17,35. When combined, the effect traits of one species and the response traits of another determine whether their directed, pairwise interactions will be strong or weak, competitive or facilitative^32,36,37,38. Given that the net strength of an interaction between two species should be proportional to their niche overlap^7,39, our framework quantifies species’ niche differences in terms of the extent to which they have an ability to withstand neighbour effects (their response) as well as an ability to generate neighbour effects (their effect). Along each niche dimension, each set of response and effect traits thus relates to a comparable (albeit latent and unobservable) set of “effective resources” such that a species with a large effect trait can be thought of as one that depletes the corresponding resource to the detriment of others, and a species with a large response trait can be thought of as one that is particularly sensitive to scarcity of that same resource. As such, response and effect traits allow us to capture the multi-dimensional strategies employed by species to outcompete each other or to avoid being outcompeted^34,35 based directly on the observed outcomes of species–species interactions (as opposed to indirectly; e.g., via their phenotypic traits⁴⁰).

In this work, we apply our approach to data from 12 empirical assemblages drawn from plant communities across the globe (see “Methods”). These assemblages cover a broad array of habitat types and plant life-history strategies, from deserts to forests and annual plants to trees (Table 1). Each dataset includes information regarding the strength and sign of species interactions between three to ten different plant species, and allows us to determine the extent to which each species performs better or worse in the presence of all others. Moreover, the data come from field or common garden settings—in which we might expect extrinsic factors could give rise to greater realized variation—and greenhouses—in which we might expect reduced environmental heterogeneity and hence realized variation should be driven almost entirely by intrinsic species properties. Studying these data allows us to investigate whether there are common patterns of niche dimensionality across a broad diversity of ecological communities. In general, we observe that niche dimensionality of each assemblage is much lower than the number of species therein. We also find that changing environmental conditions act to increase an assemblage’s effective niche dimensionality, and using a toy model, we illustrate how this increase can lead to improved prospects of coexistence. Overall, our study challenges a key assumption regarding the structure underpinning interactions in diverse communities while also demonstrating the importance of the abiotic environment in shaping species’ realized niches.

Mathematical framework for estimating niche dimensionality

For each assemblage of S plant species, at the core of our approach sits an S × S pairwise interaction matrix A. In this interaction matrix, the element α_ij in row i and column j gives the per capita strength of the effect of species j on species i. Rather than assume that each of these values is unique, we instead aim to provide as accurate an approximation of them as possible^32,38. To do so, our mathematical framework provides a principled way to split every pairwise interaction α_ij into d separate components, which we refer to as “niche dimensions”.

For any integer value of 1 ≤ d ≤ S, we define the per capita strength of the effect of species j on species i as

The sum here is across the d different niche dimensions, and each of these is weighted by the dimension’s average strength of interactions (σ_k > 0). For the simplest case of d = 1, the per capita strength of the effect of any species j on any species i is given by α_{ij∣d = 1} = σ₁ r_i,1 e_j,1; that is, the product of the average strength of interactions in the first dimension (σ₁), the “response trait” of species i in the first dimension (r_i,1), and the “effect trait” of species j in the first dimension (e_j,1). We refer to the r_i,k parameters as response traits because they influence how the performance of the same focal species i “responds” to the presence of different neighbour plants; similarly, we refer to the e_j,k parameters as effect traits because they influence how the same neighbour plant “effects” the performances of different focal plants. Importantly, the separation of responses from effects allows us to generate asymmetric interaction matrices—where the effect of species i on j differs from the effect of species j on i—which is consistent with patterns observed in empirical data^38,41,42.

Estimating best-fit reduced-dimensionality model parameters

When we had data consisting of a previously inferred empirical interaction matrix A for a given assemblage, we determined the maximum-likelihood parameters for Eq. (1) using singular value decomposition (SVD)⁴³. Specifically, SVD can be used to factorize A into three separate matrices such that A ≡ R Σ E^T. Here, R and E are S × S orthogonal matrices (i.e., matrices whose columns and rows are all orthogonal unit vectors), and Σ is an S × S matrix with the decreasing singular values of A along the diagonal (i.e., σ₁ > σ₂ > ⋯ > σ_S ≥ 0) and zeroes elsewhere. We show in the Supplementary Note S3 why SVD is mathematically equivalent to the additive partition scheme provided by Eq. (1). At every niche dimensionality 1 ≤ d < S, SVD works in such a way that ({hat{{{{bf{A}}}}}}_{d}={hat{{{{bf{R}}}}}}_{d},{hat{{{{mathbf{Sigma }}}}}}_{d},{hat{{{{bf{E}}}}}}_{d}^{T}) is the best least-squares approximation of A when: ({hat{{{{bf{R}}}}}}_{d}) is the S × d sub-matrix given by the first d columns of R, ({hat{{{{bf{E}}}}}}_{d}) is the S × d sub-matrix given by the first d columns of E, and ({hat{{{{mathbf{Sigma }}}}}}_{d}) contains the d largest singular values of A.

When we had raw empirical data for which the values composing the interaction matrix A were a priori unknown, we inferred them by fitting a model of density-dependent plant performance to the empirical data (see “Methods”). For any niche dimensionality 1 ≤ d < S, this required estimating the d positively constrained values along the diagonal of ({hat{{{{mathbf{Sigma }}}}}}_{d}) and the values within the two S × d matrices ({hat{{{{bf{R}}}}}}_{d}) and ({hat{{{{bf{E}}}}}}_{d}). To avoid over-parameterization (and to maintain coherence with SVD), we constrained the columns of both ({hat{{{{bf{R}}}}}}_{d}) and ({hat{{{{bf{E}}}}}}_{d}) to be orthogonal unit vectors. For each of these two matrices, this implies that their first column has S−1 free parameters, their second column has S−2 free parameters, and so on, up to a maximum of (Sleft(S-1right)/2) free parameters per matrix when d = S. In total, we must therefore perform inference on (dleft(2S-dright)) parameters. Though our specific orthogonality constraints are not the only possible solution to this problem, it is important to note that it will always be necessary to impose constraints on the matrices ({hat{{{{bf{R}}}}}}_{d}), ({hat{{{{bf{E}}}}}}_{d}), and ({hat{{{{mathbf{Sigma }}}}}}_{d}) to avoid issues of unidentifiability. This is because the matrix product ({hat{{{{bf{R}}}}}}_{d},{hat{{{{mathbf{Sigma }}}}}}_{d},{hat{{{{bf{E}}}}}}_{d}^{T}) will always produce a rank-d matrix ({hat{{{{bf{A}}}}}}_{d}) of size S × S which has exactly (dleft(2S-dright)) degrees of freedom.

There are at least three salient points regarding our inference process that are worth highlighting explicitly. First, even if species interactions were the result of real-world response and effect traits that were not orthonormal, our inference process built around SVD will always return estimated ({hat{{{{bf{R}}}}}}_{d}) and ({hat{{{{bf{E}}}}}}_{d}) that are. Otherwise, we risk over-parameterizing the problem and suffer from the aforementioned identifiability problems^44,45. This is also why we noted above that the dimensions of those matrices relate to latent, “effective resources” (see also Supplementary Notes S1 and S2). Second (and building from the first point), the orthogonal nature of ({hat{{{{bf{R}}}}}}_{d}) and ({hat{{{{bf{E}}}}}}_{d}) implies that they should be thought of as providing detail about where the different species sit relative to each other in the assemblage’s overall competitive hierarchy. Third (and building from the second point), ({hat{{{{bf{R}}}}}}_{d}) and ({hat{{{{bf{E}}}}}}_{d}) are best thought of as plastic species traits that are indicative of the conditions under which the interactions took place, both biotic (e.g., the set of species that were interacting) and abiotic (e.g., the types of resources that are available). Consequently, realized niche dimensionality of an assemblage will always reflect both the fundamental niche variation between species and which resources are locally most limiting (Supplementary Notes S1 and S2).

Niche dimensionality and species coexistence

Mathematically, an S × S interaction matrix A_d that is exactly rank-d is “rank deficient”. This means that, when embedded in typical population-dynamics models, the matrix A_d imposes a strict limit such that d is the maximum number of species that could coexist in the absence of exogenous variation^7,9,10,46. We can therefore get an estimate of the ease with which species from an empirical assemblage can achieve multi-species coexistence by examining the extent to which the d-dimensional approximation A_d captures the variation in their un-approximated interaction matrix A. As we scan across the values ({hat{sigma }}_{1}), ({hat{sigma }}_{2}), …, ({hat{sigma }}_{S}), we expect their magnitudes to decrease much more quickly at first before leveling off close to zero^47,48. For each assemblage, there should therefore be a point (hat{d}) beyond which the inclusion of additional dimensions will only weakly change the interaction strengths. We refer to this value (hat{d}) as the assemblage’s “niche dimensionality”, where (hat{d}) is number of niche dimensions required to explain at least 95% of the variation observed within the data. Though all values (d > hat{d}) will give a more exact description of species’ fundamental niches, we argue that (hat{d}) is a better reflection of the niche dimensions that underpin the realized species–species interactions. Even when the matrix A has not been approximated (i.e., there is no strict mathematical limit to the maximum number of species that could potentially coexist), the value of (hat{d}) inferred from that matrix implies that it should be possible for at least some n-species subsets from the assemblage to coexist (when (nle hat{d}) and species’ fitness differences allow it²³) but will be increasingly challenging for all n-species subsets comprising more than (hat{d}) species.

Inferred niche dimensionality of empirical datasets

We started by determining the niche dimensionality (hat{d}) of all 12 plant assemblages in our data (see “Methods”). As noted above, niche dimensionality (hat{d}) gives an indication of the effective number of resources over which the species are likely competing. For every dataset and regardless of the experimental context, we observed that three or fewer niche dimensions ((hat{d}le 3)) were sufficient to accurately capture the pairwise interactions between plant species (Table 1). In fact, the first niche dimension alone explained on average 86.7% of the variation in the plant–plant interactions, and this ranged from 59.6 to 99.3% across the 12 datasets (Fig. 1 and Supplementary Notes S4 and S5). To verify that these low values of (hat{d}) are an ecological feature of these species assemblages and not merely an artefact of our methodology, we also estimated the niche dimensionality for simulated data with randomly-assigned interaction coefficients and within randomizations of the empirical interaction matrices. As expected, randomization tended to create unstructured data with niche dimensionality greater than observed in the natural and experimental assemblages (Supplementary Note S6).

We next examined species’ inferred response and effect traits to get an idea about the forces underpinning species interactions. As representative examples, the best-fit parameters for Dataset 1-Wet and Dataset 1-Dry indicate that the 10 constituent species play distinct ecological roles across their leading niche dimension (Fig. 2). In both cases, species appear to be sorted into classic competitive hierarchies within which species with strong effects tended to have weak responses and those with weak effects tended to have strong responses. However, realized niche differentiation in Dataset 1-Wet was driven by greater variation in species’ effects than their responses, whereas species had heterogeneous responses and heterogeneous effects in Dataset 1-Dry. We observed similar patterns across each of the empirical datasets (Supplementary Note S7), indicating that the leading dimensions of niche differentiation are those that create variation in species’ responses to and effects on other species in their community.

We consistently found that estimates of niche dimensionality (hat{d}) were lower than the total number of species S in each empirical assemblage. Furthermore, this did not depend on whether or not data came from experiments in the field, garden, or greenhouse (Table 1). Though our estimates of niche dimensionality increased with increasing species richness (Fig. 3), the rate of increase was much lower than one dimension per species as expected from theory^7,8,9,11 and lower than what we observed in randomizations of the empirical data (Supplementary Note S6). When niche dimensionality is less than the number of species in an assemblage, the interactions of two or more species directly mirror each other¹⁰. Furthermore, should any among a set of functionally-similar species have even the slightest competitive advantage, it will tend to dominate the others in that set over the long run²⁰. Our results, therefore, are indicative of communities for which many-species coexistence should pose a serious challenge. Though this observation runs contrary to high local diversities found in nature, it may provide an explanation for why numerous previous studies have found that models parameterized with empirical data very rarely predict coexistence^13,49,50.

Inferred niche dimensionality under different environmental conditions

To investigate this idea more deeply, we next determined whether changing environmental conditions impact plant species’ roles within their assemblage’s competitive hierarchy. To do so, we explored two of our datasets in greater detail as they each comprise the same underlying species assemblages, but with interactions between those species taking place in two distinct environments. Specifically, one pair of datasets (Dataset 1-Wet & Dataset 1-Dry) consists of ten species in control and simulated-drought conditions⁵¹, designed to mimic the impact of extreme climatic events of the sort expected with ongoing global change⁵²; the second pair of datasets (Dataset 2-Sun & Dataset 2-Shade) consists of eight species in control and artificially-shaded conditions, since light availability is known to structure local diversity in that system⁵³. Viewed independently, Dataset 1 was captured by two and three niche dimensions in the Wet and Dry environments, respectively; Dataset 2 was captured by one and two niche dimensions in the Sun and Shade environments, respectively. Nevertheless, the question remains whether the species in each Dataset sorted themselves along the same leading axes of niche variation in the contrasting environments. To check this, we measured the correlation between species’ inferred response and effect traits as realized under the two environmental conditions (see “Methods”). These correlations were weak and non-significant (Procrustes ρ = 0.52, p = 0.37 for Dataset 1 when compared to n = 999 random permutations; Procrustes ρ = 0.40, p = 0.06 for Dataset 2 when compared to n = 999 random permutations), indicating that the niche dimensions being exploited by both sets of species likely differ following changes in environmental conditions. Environmental variation, therefore, leads to significant shifts in species’ niches (Figs. 2 and S7), in contrast to niche theories based on environmental tolerance⁵⁴ or functional traits⁵⁵, which generally assume that niches are immutable species’ attributes.

Low-dimensional coexistence is achievable in varying environments

Motivated in particular by our results for Dataset 1, we lastly put together a toy model to explore whether environmentally driven variation of species’ response and effect traits can potentially increase the likelihood of coexistence (Supplementary Note S8). Starting from a widely studied model for competing annual-plant species^56,57, we parameterized the model for two species (i and j) such that species i is favoured in Wet conditions and species j is favoured in Dry conditions. Within our response–effect parameterization, this was achieved by species i having a weaker response to the presence of neighbours in Wet conditions than did species j, and the converse in Dry conditions. In all other facets of the model, the two species are ecologically equivalent⁵⁸. As a result, the species’ dominance hierarchy flips when shifting between the two types of environments.

Because we constrained the interactions between these two species in our toy model to occur at niche dimensionality d = 1, it is mathematically impossible for the two species to coexist in the long term under fixed environmental conditions because they lack niche differences^23,46, and simulations of the model in perpetually Wet or Dry environments bear out this result (Fig. S15). Indeed, in either constant environment no amount of fitness differences will ever be able to overcome the barrier presented by a lack of niche differences between these two species²⁰. If, on the other hand, those same species compete in an environment that changes from Wet to Dry conditions (and back) over time, both species may be able to avoid competitive exclusion in the long run (Fig. S15). Since each species remains vulnerable to prolonged exposure to its less preferred environment, environmental variation is not a sufficient condition for long-term coexistence¹⁷. Nevertheless, environmentally driven changes in species’ interaction patterns—as modulated by variation in their response and/or effect traits changing from one set of conditions to another—can provide a plausible means through which they may maintain a firmer foothold.

We provide strong evidence that plant–plant interactions are organized over a small number of niche dimensions. Intriguingly, this observation holds despite the differences between the plant communities represented in our study—such as their biogeographical provenance, evolutionary histories, or plant growth form. Less clear is whether the emergent pattern is more a consequence of the plant assemblages exhibiting a striking lack of fundamental niche differences or competing over a small number of truly limiting resources. Our exploration of a common consumer–resource model emphasizes that the number of distinct features limiting plant populations, such as the number of limiting resources or species’ degrees of resource specialization, can differ substantially from the number of dimensions realized in their interaction matrix (Supplementary Note S2). To convincingly tease these two apart in the future, we suspect ecologists may require a clever combination of manipulative and observational experiments beyond those that exclusively serve to estimate one set of interaction strengths. These could include invasibility experiments conducted in different environments¹⁷ or manipulating as many potential limiting factors as possible and seeing how interaction strengths—even for only a subset of species of interest—respond as a result⁵⁹.

Regardless of its root cause, low niche dimensionality would appear to imply limited prospects for many-species coexistence and, notably, belies common theoretical expectations. This notwithstanding, limited variation in terms of species’ interactive responses and effects is actually in strong agreement with other assessments of global plant diversity being captured by a small number of life histories³⁴ and limited phenotypic trait combinations^55,60. Though disentangling the underlying drivers of species’ response and effect traits inferred across our empirical datasets was beyond the scope of this particular study, at least one immediate extension would be to explore whether or not these latent traits can be related to measurable phenotypic characters across the species in question, such as root or leaf morphology.

In addition to demonstrating that low-dimensional competition is pervasive in plant assemblages, our approach provides a new lens through which to interrogate coexistence in varying environments. Previous studies of the species found in Datasets 1 & 2 also predicted that the number of species expected to coexist in a single environment is generally small^51,61,62; those studies’ predictions closely align with our own estimates of both assemblages’ niche dimensionality. Likewise, results from those same two systems found that which species are expected to coexist changes from one set of environmental conditions to another^51,61,62,63. If we combine these previous observations with our own results, it appears that the environment can leave such a strong imprint on interactions that its variability behaves conceptually like an additional niche dimension and thus provides a de facto landscape across which species can evolve novel strategies to avoid being outcompeted^27,34,64. While this can make it harder to identify consistent structural patterns in empirical interaction matrices^41,42,65, it also highlights a critical link between changing environmental conditions and species’ realized niches^64,66.

Much research is dedicated to exploring how the abiotic environment impacts species’ performance in an interaction-free context^67,68. Indeed, most predictions about how communities will change in the face of external disturbance, such as climate variability, hinge on this relationship^52,54. Comparatively less is known about how environmental conditions impact pairwise interactions in multi-species assemblages and therefore about species’ ability to persist under changing environmental conditions^{42,51,69,70,71,72}. One possible reason is that it is difficult to relate changes across multivariate interactions between species to changes in univariate single-species outcomes^25,50. Here, we show how to decompose pairwise interactions into species-level response and effect traits, thereby providing a potential simplification for such inquiries.

Moreover, the impact of the environment on species’ response and effect traits generates multiple ecologically-relevant ways for pairwise interactions to vary along environmental gradients. All else being equal, any species whose response traits decrease in value from one environment to another will suffer less from competition; in contrast, any species whose effect traits increase in value from one environment to another should become increasingly dominant. Both pathways can assist in species avoiding being competitively excluded. Going forward, we therefore expect it will be particularly exciting to determine which one of these tendencies predominates in nature. Of particular utility here would be more studies like Datasets 1 & 2, but with interactions measured across multiple points along continuous, as opposed to categorical, environmental gradients.

Empirical data

We analyzed data from 12 empirical assemblages to test our core hypothesis regarding variation in the structure of pairwise interactions between co-occurring plants (Table 1). For Datasets 1-Wet, 1-Dry, 2-Sun, and 2-Shade, we directly analyzed raw empirical data from experimental studies of competition between annual plants in order to statistically infer the interaction strengths that make up the pairwise interaction matrix A. For Datasets 3–12, we analyzed previously-estimated pairwise interaction matrices A that were available in the literature. All such interactions were inferred based on the performance of plant individuals growing in isolation relative to individuals growing in the presence of conspecific and/or heterospecific neighbours⁴¹.

Inferring fixed-dimensionality interaction strengths from empirical data

Each of Datasets 1-Wet, 1-Dry, 2-Sun, and 2-Shade consists of estimates of individual plant performance and the abundance or density of co-occurring plants within interaction neighbourhoods^51,63. To infer interaction strengths that make up their corresponding matrix A, we therefore first had to define a mathematical model for how performance varies as a function of neighbour composition and abundance. In line with current best practice³⁷, we estimated the per capita effects of neighbouring species on the performance of focal individuals of each of the datasets using a model of the form

where F_i is the observed estimate of the performance of a focal individual from species i, λ_i captures intrinsic performance of these individuals in the absence of competition, α_ij∣d is the per capita impact of species j on species i, N_j is the abundance of species j in the focal individual’s interaction neighbourhood, and the sum is across all species in that neighbourhood (which could potentially include conspecifics of species i). Given data of focal-plant performance and those plants’ interaction neighbourhoods, inference of both intrinsic performance and the strength of pairwise interactions can usually be achieved with standard regression approaches⁷³. For example, Eq. (2) with count responses (as in Dataset 1 and Dataset 2), can be fit as a Poisson regression with inverse link function⁷⁴. However, these regression approaches are no longer feasible when requiring interactions to occur at a fixed niche dimensionality due to our orthgonality constraints. We therefore used the mle2 function from the bbmle package⁷⁵ in the statistical programming language R⁷⁶ to identify the best-fit (dleft(2S-dright)) total parameters that make up the optimal values of ({hat{{{{mathbf{Sigma }}}}}}_{d}), ({hat{{{{bf{R}}}}}}_{d}), and ({hat{{{{bf{E}}}}}}_{d}) at each niche dimensionality 1 ≤ d < S. In order to ensure that the parameters of ({hat{{{{bf{R}}}}}}_{d}) and ({hat{{{{bf{E}}}}}}_{d}) maintain our orthogonality constraints and uniqueness during optimization, we followed the Rational Cayley Transform method of refs. ^44,45. To facilitate comparison to the approach with which the matrix A is composed entirely of free parameters (i.e., when d = S), we used mle2 to minimize the exact same deviance function as used for Poisson regression rather than to maximize the data’s log-likelihood.

Statistical analysis

For Datasets 1-Wet, 1-Dry, 2-Sun, and 2-Shade, we inferred the best-fit parameters for interactions constrained to occur when dimensionality d = S. We then used the inferred values of ({{hat{sigma }}_{1},{hat{sigma }}_{2},ldots,{hat{sigma }}_{S}}) and identified the niche dimensionality (hat{d}) that was sufficient to capture 95% of the variation in the inferred interaction matrix. For Datasets 3–12—for which we only had an estimated interaction matrix A—we determined the niche dimensionality (hat{d}) based directly on the singular values of A with an equivalent threshold of 95%. In both instances, the variance explained by each dimension j can be calculated as ({sigma }_{j}^{2}/mathop{sum }_{k=1}^{d}{sigma }_{k}^{2}). For all dataset types, values (d < hat{d}) fail to capture biologically meaningful variation in the observed plant–plant interactions; values of (d > hat{d}) require the use of an over-parameterized statistical model with an excess of resource dimensions. We note that our 95% threshold may be a conservative test of our theory. However, to the best of our knowledge, there is no other rule of thumb with which to decide precisely how much variation is biologically meaningful.

For the paired Datasets 1 & 2, we tested whether the positions of species in their underlying (hat{d})-dimensional trait spaces were correlated using a Procrustes analysis. Since the inferred niche dimensionalities differed across environmental conditions (Table 1), we first converted the two (Stimes hat{d}) matrices ({hat{{{{bf{R}}}}}}_{d}) and ({hat{{{{bf{E}}}}}}_{d}) (for each environmental condition in each Dataset) into a single S × S distance matrix whose elements were the Euclidean distances between species’ vectors of response and effect traits. We then used the Procrustes analysis to estimate the similarity between the distance matrices computed using the best-fit parameters in each environmental condition⁷⁷. A significant Procrustes statistic indicates that the effective response and effect hierarchies between species in both environments are positively related (e.g., the same species always tend to be the strongest or weakest effectors or responders), whereas a non-significant result implies that species likely play different roles when the environmental conditions change. We used the protest function from the vegan package⁷⁸ in the statistical programming language R⁷⁶ to perform our Procrustes tests.