Abstract

Ecological theory suggests that coexistence of many species within communities requires negative frequency-dependent feedbacks to prevent exclusion of the least fit species. For plant communities, empirical evidence of negative frequency dependence driving species coexistence and diversity patterns is rapidly accumulating, but connecting these findings to theory has been difficult as corresponding theoretical frameworks only consider small numbers of species. Here, we show how frequency-dependent feedback constrains community coexistence, regardless of the number of species and inherent fitness inequalities between them. Any interaction network can be characterized by a single community interaction coefficient, IC, which determines whether community-level feedback is positive or negative. Negative feedback is a necessary (but not sufficient) condition for persistence of the entire community. Even in cases where the coexistence equilibrium state cannot recover from perturbations, IC < 0 can enable species persistence via cyclic succession. The number of coexisting species is predicted to increase with the average strength of negative feedback. This prediction is supported by patterns of tree species diversity in more than 200,000 deciduous forest plots in the eastern United States, which can be reproduced in simulations that span the observed range of community feedback. By providing a quantitative metric for the strength of negative feedback needed for coexistence, we can now integrate theory and empirical data to test whether observed feedback–diversity correlations are strong enough to infer causality.

Main

Despite decades of research, it remains surprisingly challenging to explain how species competing for the same resources can coexist in the large numbers observed in plant communities1,2,3,4,5,6,7. Ecological theory states that stabilizing mechanisms creating negative frequency-dependent feedbacks are needed to prevent competitive exclusion due to species differences in fitness6,7,8. Recent empirical studies of species-rich plant communities are in qualitative agreement with the expected connection between negative frequency-dependent feedback and patterns of species abundance9,10 and species richness11,12,13. Until now, utilization of these data for more direct tests of ecological theory is hampered by the lack of a theoretical analytical framework linking species interactions to community diversity and stability. In contrast with studies of evolutionary game dynamics14, theoretical ecological studies have focused either on coexistence of small numbers of species3 or on sufficiently large communities, enabling a distributional approach based on probabilities (of particular species interactions15). Alternatively, coexistence has been studied for equivalent species16, species exerting equal effects on all competitors17, (anti-)symmetric species interactions5,7 and species experiencing strictly hierarchical competition2. These assumptions enable a detailed description of potential plant community dynamics, but do not consider asymmetric competitive interactions. In addition, the above approaches do not represent the forces driving negative frequency-dependent feedbacks at the community level.

Focusing on a specific mechanism through which frequency-dependent feedbacks may arise from asymmetric species interactions, plant–soil feedback theory has been developed by means of models that can be directly parameterized with data from pot or field experiments, providing a promising means to integrate experiments with theory18,19,20. Yet, theoretical development and empirical testing of plant–soil feedback theory has been limited to small numbers of interacting species18,19,20,21,22,23. Here, we generalize the plant–soil feedback model18 (see Supplementary Information Section 1), yielding a generic phenomenological model of frequency-dependent interactions. When rewriting the species’ density dynamics into frequency dynamics, this generalized model projects onto the well-known replicator equation originating from evolutionary game theory14. The replicator equation can be solved analytically for communities containing any number of species. This procedure does not require an assumption of species equivalence, meaning that equilibrium solutions retain species-specific parameters describing the growth for each species. We also derive a fundamental criterion for species persistence. Our results provide a framework to assess the importance of empirically observed frequency-dependent interactions in maintaining coexistence of highly diverse plant communities (Fig. 1).

Fig. 1: Conceptual representation of how the frequency-dependent feedback framework can be extended to the community level for any number of plant species (here, n = 5).
Fig. 1

a, Plant species (Pi) are modelled as frequencies within a community. The diagram includes trees from Laubwerk Plant Kit 1 (www.laubwerk.com). b, Plant species exert conspecific (σii) and heterospecific (σij) effects, the magnitude of which (per capita) depends on the frequency of the exerting species. c, The resulting interaction network determines whether the community coexistence equilibrium is feasible; equilibrium frequencies are calculated by reduced determinants of the interaction matrix, Ai, the sum of which yields the community interaction coefficient, IC. d, IC < 0 generates negative community-level feedback and is a necessary condition for persistence of all plant species within the community. e, IC > 0 generates positive community-level feedback and leads to competitive exclusion.

Results and discussion

The proposed framework consists of a set of ordinary differential equations describing plant frequencies and hence community dynamics over time (Supplementary Information Section 1). The only parameters in this model are the coefficients σij, which quantify the growth rate of species i in an environment dominated by species j (Fig. 1). Thus, when these coefficients are set to non-neutral values, species fitness becomes frequency dependent, creating a feedback between community composition and the growth of each species. For a system of two species, net feedback is described by the two-species interaction coefficient IS18,19:

$$I_{\rm{S}} = \sigma _{11} - \sigma _{12} - \sigma _{21} + \sigma _{22}$$
(1)

Provided that the coexistence equilibrium is feasible (that is, both plant species have a positive frequency at equilibrium), IS< 0 will generate negative feedback, and the equilibrium is stable. Alternatively, IS > 0 will generate positive feedback and the equilibrium is unstable, meaning that one of the plant species will be excluded18,19 (Supplementary Information Section 2). IS can thus be interpreted as a pairwise feedback: the sum of two species-specific feedbacks, which are described by the difference between a species’ conspecific and its heterospecific responses (thus, the species-specific feedback of plant species 1, for example, is given by24: σ11 – σ21). For the two-species case, the equilibrium frequency for each plant species is then given by the ratio between its competitor’s species-specific feedback and the pairwise feedback18, yielding:

$$\hat P_1 = \frac{{\sigma _{22} - \sigma _{12}}}{{I_{\rm{S}}}},\,\hat P_2 = \frac{{\sigma _{11} - \sigma _{21}}}{{I_{\rm{S}}}}$$
(2)

in which \(\hat P_i\) indicates the species’ equilibrium frequencies. These results can be generalized from two to any number of species (Supplementary Information Section 3). General results can be obtained using the interaction matrix:

$${A} = \left[ {\begin{array}{*{20}{c}} {\sigma _{11}} & {\sigma _{12}} & {..} & {\sigma _{1n}} \cr {\sigma _{21}} & {..} & {..} & {..} \cr {..} & {..} & {..} & {..} \cr {\sigma _{n1}} & {..} & {..} & {\sigma _{nn}} \end{array}} \right]$$
(3)

As in the two-species case, the frequency of a given species at the coexistence equilibrium is the ratio between the feedback effects that this species experiences from all competitors and the sum of all feedback effects within the community:

$$\hat P_i = \frac{{{\rm{det}}\,{A}_i}}{{\mathop {\sum}\limits_{j = 1}^n {{\rm{det}}\,{A}_j} }}$$
(4)

where Ai indicates the interaction matrix A, but with the ith column being replaced with a column vector of ones (Fig. 1). Thus, based on the interaction structure within the plant community (equation (3)), the equilibrium frequencies for species at coexistence can be calculated (equation (4)). Stability of this equilibrium can then be assessed by standard eigenvalue analysis of the system’s Jacobian matrix. We used this approach to study community assembly and the resulting plant species richness (quantified as the maximum number of species that were found to stably coexist) under various regimes of negative feedback (Fig. 2; see also Methods and Supplementary Information Section 6). We found that under more negative feedback, indicated by the system’s average pairwise feedback10,19,23, communities with higher species richness could be assembled (Fig. 2). For a plant community in which the number of species was fixed, more negative feedback increased the resilience of the coexistence equilibrium (Fig. 3a, in which resilience is measured as the return time after perturbation25). In addition, more negative feedback increased community evenness (Fig. 3b). The fact that average pairwise feedback correlates well with these community-level properties is surprising because pairwise descriptors provide no formal (that is, mathematically formulated) conditions for coexistence in communities of more than two species7. Therefore, to establish whether pairwise descriptors can be used as an indicator of community-level feedback, a formal measure of community-level feedback is also needed.

Fig. 2: Two hundred runs of plant community assembly show that more species can stably coexist under stronger negative feedback.
Fig. 2

The solid line shows a quadratic fit assuming a minimum community size of two plant species (y = 39.00x2 + 12.52x; Radj2 = 0.86).

Fig. 3: Effects of frequency-dependent feedback on plant community structure.
Fig. 3

a, Increasingly negative feedback increases the resilience of the coexistence equilibrium (measured as the absolute real part of the leading (and negative) eigenvalue27). b, Increasingly negative feedback increases species evenness within the stably coexisting community measured with Pielou’s evenness index (that is, the ratio between the actual and the maximum value of the Shannon–Wiener index). c, Even though community-level feedback is determined by an analytical expression of the order \(\sigma _{ij}^{n - 1}\), the strength of average pairwise feedback (that is, of the order \(\sigma _{ij}^1\)) correlates well with the (rescaled) strength of this community-level feedback.

We found that for the model framework analysed, the two-species interaction coefficient IS can be generalized to the community level, yielding an exact measure for community-level feedback. More specifically, the generalized community interaction coefficient, IC, is given by (Supplementary Information Section 3):

$$I_{\rm{C}} = \left( { - 1} \right)^n\mathop {\sum}\limits_{j = 1}^n {{\rm{det}}\,{A}_j}$$
(5)

A pattern generally observed in empirical studies is that conspecific effects (and variation therein) are relatively strong compared with heterospecific effects (and variation therein)11,12,13. Under these conditions, our measure of community-level feedback correlated well with the average strength of pairwise feedback (equation (1); Fig. 3c and Supplementary Information Section 6). This result did not only hold for asymmetric interaction matrices (Fig. 3c), but also for (anti)-symmetric interaction matrices (Supplementary Information Section 3). This monotonic relation between average pairwise feedback and community-level feedback thus provides an explanation for recent empirical observations of average pairwise feedback correlating well with community diversity11,12,13.

We could prove that negative community-level feedback, IC < 0, presents a necessary condition for persistence of all species (Supplementary Information Section 3). Interestingly, the value of IC is not affected by inherent fitness inequalities6 between species (Supplementary Information Section 7). Following the community assembly approach (see Methods), we found that negative feedback could still be present in communities for which the real part of the leading eigenvalue of the coexistence equilibrium was positive (Supplementary Information Section 4). Thus, even when small perturbations would drive the community away from the coexistence equilibrium point, negative community-level feedback may still prevent species extinction. In this case, species abundances fluctuate, but dominance is passed on from one species to another in ways that create repeating loops (for example, A → B → C → A), referred to as cyclic succession22,26 (Supplementary Information Section 4). However, IC < 0 is not a sufficient condition for persistence of all species within a community (Supplementary Information Section 3). These observations highlight the importance of knowing the exact structure of the right-hand side of equation (3), as the loops creating the cyclic succession dynamics are not conserved when all the heterospecific effects that a species experiences have been averaged (for example, Supplementary Information Section 4).

Parameterizing the right-hand side of equation (3) requires full-factorial experimental designs, the size of which increases rapidly with increasing plant community size. However, even in cases where information is incomplete, our theoretical framework can be used to infer plant community characteristics by constraining the right-hand side of equation (3) within a realistic range of parameter values (rather than providing the exact value of each parameter). Observational data, as recently gathered in diverse forest ecosystems9,11,12,13, can be used for this aim. To illustrate this potential, we analysed data from North American deciduous forests east of the hundredth meridian, aggregating surveys from 207,444 plots to 2° × 2° grid cells11 (Fig. 4a). Previous measures of conspecific and averaged heterospecific frequency dependence11 can be translated into the model parameters used in the current framework (equations (3) and (4), and Supplementary Information Section 5). Focusing on grid cells containing sufficient data for five or more tree species11 (Supplementary Information Section 5), we distinguished four community diversity classes (5–15, 15–25, 25–35 and ≥35 species occurring in a grid cell; Fig. 4a). We used the previously derived measures for conspecific and averaged heterospecific density dependence11 to calculate for each of these cells the average pairwise feedback experienced by the tree species present. Consistent with previous observations of feedback–diversity correlations11,12, the average pairwise feedback became more strongly negative with increasing community diversity (see the coloured bars in Fig. 4b). A hitherto unanswered question, however, is whether these observed differences in feedback are indeed strong enough to drive (rather than correlate with) the variation in community diversity. Therefore, we compared the pattern obtained from these observations with independently assembled theoretical communities that spanned the same range of diversity classes. In these theoretical communities, the differences in species diversity were entirely driven by variation in the strength of community-level feedback. The conspecific and heterospecific coefficients in these theoretical communities were varied over the same range as observed in the field data (Supplementary Information Section 5). The community assembly process was carried out as before (Fig. 2; see Methods). We found that the variation in pairwise feedback strength between these theoretical communities (grey bars in Fig. 4b) was similar to the variation empirically observed (Fig. 4b). Although a correlation between feedback and diversity has been previously observed in northern American deciduous forests11,12, application of the framework developed here verifies that the observed variation in feedback is in theory strong enough to be a causal factor driving the observed variation in community diversity (Fig. 4).

Fig. 4: Application of the theoretical framework reveals that observed patterns of negative frequency dependence in over 200,000 forest plots in the eastern United States are consistent with predictions from theoretical communities in which community-level feedback determines species richness.
Fig. 4

a, Map showing the locations of the 2° × 2° grid cells and the community diversity observed in each cell (divided into four diversity classes: 5–15 species (n = 24), 15–25 species (n = 36), 25–35 species (n = 22) and ≥ 35 species (n = 31)). b, Grid cells with higher community diversity are characterized by stronger negative feedback (coloured bars; Kruskal–Wallis H test, \({X}_{3,109}^2\)= 15.34, P = 0.0015), the variation in average pairwise feedback strength being consistent with the range occurring in theoretical communities (grey bars: n5–15 = 1,059; n15–25 = 426; n25–35 = 272; n≥35 = 355). Bars indicate means ± 1 s.e.m., and different letters indicate differences at the P = 0.05 significance level (Kruskal–Wallis H test, \({X}_{7,2217}^2 = 1,574\), P < 1.10−323, followed by Dunn–Sidak post hoc tests).

Our results further suggest that in the absence of inherent fitness differences between species, a species’ experienced feedback will positively correlate with its relative abundance (equation (4)), as has been commonly observed in the field10,11,13,27. Although inherent fitness differences may decouple this link between experienced feedback and relative abundance, we found that such differences would not undermine the stabilizing force of community-level negative feedback (Supplementary Information Section 7). Together, these results provide an explanation for recent observations of consistent signatures of negative feedback in tropical and temperate forest communities but declining slopes of feedback–abundance relationships with increasing latitude13.

Quantitative studies of ecological networks have mainly focused on food webs and mutualistic networks15,28, or have focused on competition within a trophic level by studying networks that are symmetric (for example, mimicking overlap in resource use7) or anti-symmetric (where there is a winner and a loser in each two-way interaction5; see also ref. 3). Frequency-dependent feedback, as studied here, considers a community in which plants interact through a fully connected network but where interactions between species can be asymmetric (for example, species A strongly affects species B, but B only weakly affects A). Empirical evidence for specific mechanisms creating this kind of interaction structure is rapidly increasing. Examples include density-dependent seedling mortality9,10, plant–soil feedbacks initiated by invertebrate fauna29, soil pathogens21 or mycorrhizal associations27, and autotoxic effects of extracellular self-DNA originating from plant litter30. Understanding the contribution of these mechanisms to plant community coexistence will be contingent on moving beyond the study of interactions between small numbers of species18,19,20. The theoretical framework presented here provides a crucial link between empirical estimates of frequency-dependent feedback and its consequences for community stability and species diversity patterns.

Methods

Strength of negative frequency-dependent feedback and species richness of communities

A random interaction matrix for 1,000 species was generated. From this matrix, communities were assembled, starting with two randomly selected species. Analytical calculation of the coexistence equilibrium (see equations (3) and (4)) and the relevant eigenvalues of the corresponding Jacobian matrix (Supplementary Information Section 3) were then calculated to assess whether this two-species coexistence equilibrium was stable. If coexistence was feasible and stable, another species was randomly selected and added to the community, after which the procedure was repeated. This process continued until the community was no longer locally stable, or until a predetermined species number was reached, at which point the community resilience (by means of the magnitude of the relevant eigenvalue with the maximal real part25) and evenness (using Pielou’s index) were assessed. Average feedback strengths in the 1,000-species matrix were varied by manipulating the strengths of conspecific effects relative to the strengths of heterospecific effects. In the simulations, conspecific coefficients ranged between 0.05 and 0.3, and heterospecific coefficients ranged between 0.3 and 0.9 (refs 10,23). The subsets considered within these ranges varied along the feedback gradient (see Supplementary Information Section 6 for details). However, we verified that alternative parameterizations led to qualitatively similar feedback–community diversity relations (Supplementary Information Section 6).

Plant–soil feedbacks and the Forest Inventory and Analysis dataset

We analysed tree species diversity of forest plots located east of the hundredth meridian, which are part of the US Forest Service’s Forest Inventory and Analysis (FIA) dataset. These data were previously extracted and preprocessed11. In this previous work, the strength of density dependence was inferred from (maximum likelihood) fitting a negative exponential relation between observed seedling densities and tree density in the surrounding plot. Here, we used the dataset described in ref. 31 to obtain exponential coefficients, using cumulative basal area as the predictor variable for adult density. The dataset analysed in ref. 31 differed from ref. 11 in that it also included ecologically relevant joint absences of seedlings and adult trees within forest plots, which tends to increase coefficient values32. The exponential coefficients were used to calculate values for the conspecific frequency-dependent effect parameters, σii, and for the average heterospecific effect, σij, on each species (Supplementary Information Section 5). Theoretical results showed that average pairwise feedback (equation (1)) correlates well with community-level feedback (see Fig. 3c and Supplementary Information Sections 3 and 6). Average pairwise feedback could also be calculated for the FIA data selection, as it only requires information on the average of heterospecific effects that a tree species experiences11 (Supplementary Information Section 5). We calculated average pairwise feedback strengths aggregating observations from different plots at the level of grid cells spanning 2° × 2° (latitude × longitude)11 (Supplementary Information Section 5). In total, average pairwise feedback was calculated for 113 grid cells in which the average pairwise feedback could be calculated for five or more tree species (up to 51 tree species). These calculated feedbacks were compared with a null prediction in which the associations between seedlings and adults within each grid cell were randomized (Supplementary Information Section 5). Patterns of feedback strength and observed community diversity in the corresponding grid cells were then compared with patterns in theoretical communities in which community diversity was determined by the strength of community-level feedback. As the data provided row-averaged values of heterospecific effects (that is, the off-diagonal entries of the community matrix; see equation (3)), the model parameters in simulations could be constrained; that is, varied over the same range as observed in the data, including the ranges assumed for within-row and between-row variation in heterospecific effects (Supplementary Information Section 5).

Reporting Summary

Further information on experimental design is available in the Nature Research Reporting Summary linked to this article.

Code availability

The MATLAB and R scripts written for this study only contain standard functions and analysis methods. Scripts are available from the corresponding author upon reasonable request.

Data availability

The US Forest Service’s FIA data that support the findings of this study are available from www.fia.fs.fed.us/tools-data/ under FIA Data Mart. The database subset that was used in the current study is available from the corresponding author upon reasonable request.

Additional information

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Acknowledgements

This work was conducted as part of the ‘Plant–soil feedback theory’ Working Group at the National Institute for Mathematical and Biological Synthesis, sponsored by the National Science Foundation through NSF Award DBI-1300426, with additional support from The University of Tennessee, Knoxville. The work was further supported by NSF grant DEB-1738041, awarded to J.D.B. We thank K. C. Abbott, J. Bauer, L. S. Comita, K. M. Crawford, S. A. Mangan, J. Umbanhowar and K. N. Suding for discussions of this work during working group meetings, J. Aning for assistance with the FIA data analysis, and T. Markus for assistance with the preparation of the figures. We also thank F. May for comments that improved the manuscript. Finally, we thank all of the FIA employees who collected the data that enabled us to test model predictions.

Author information

Affiliations

  1. Copernicus Institute of Sustainable Development, Environmental Sciences, Faculty of Geosciences, Utrecht University, Utrecht, the Netherlands

    • Maarten B. Eppinga
    •  & Mara Baudena
  2. Department of Earth and Environmental Sciences, Los Alamos National Laboratory, Los Alamos, NM, USA

    • Daniel J. Johnson
  3. Department of Soil and Water Conservation, Nanjing Forestry University, Nanjing, China

    • Jiang Jiang
  4. Department of Biology, Illinois College, Jacksonville, IL, USA

    • Keenan M. L. Mack
  5. Department of Biology, Grice Marine Laboratory, College of Charleston, Charleston, SC, USA

    • Allan E. Strand
  6. Department of Ecology and Evolutionary Biology and Kansas Biological Survey, University of Kansas, Lawrence, KS, USA

    • James D. Bever

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Contributions

M.B.E., D.J.J., K.M.L.M., A.E.S. and J.D.B. designed the study. M.B.E. and M.B. performed the analytical model analyses. M.B.E., M.B., J.J., K.M.L.M. and A.E.S. developed the code, and performed and analysed the model simulations. M.B.E., M.B., D.J.J. and J.D.B. analysed the field observation data. M.B.E. and J.D.B. wrote the first draft, after which all authors contributed to improving the manuscript.

Competing interests

The authors declare no competing interests.

Corresponding author

Correspondence to Maarten B. Eppinga.

Supplementary information

  1. Supplementary Information

    Supplementary Information sections 1–7 and Supplementary Figures 1–13

  2. Reporting Summary

  3. Supplementary Table 1

    A spreadsheet containing the values of average pair-wise feedback summarized in Fig. 4b of the main text

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https://doi.org/10.1038/s41559-018-0622-3