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Higher-order interactions capture unexplained complexity in diverse communities


Natural communities are well known to be maintained by many complex processes. Despite this, the practical aspects of studying them often require some simplification, such as the widespread assumption that direct, additive competition captures the important details about how interactions between species impact community diversity. More complex non-additive ‘higher-order’ interactions are assumed to be negligible or absent. Notably, these assumptions are poorly supported and have major consequences for the accuracy with which patterns of natural diversity are modelled and explained. We present a mathematically simple framework for incorporating biologically meaningful complexity into models of diversity by including non-additive higher-order interactions. We further provide empirical evidence that such higher-order interactions strongly influence species’ performance in natural plant communities, with variation in seed production (as a proxy for per capita fitness) explained dramatically better when at least some higher-order interactions are considered. Our study lays the groundwork for a long-overdue shift in how species interactions are used to study the diversity of natural communities.

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Figure 1: Effects of direct interactions and HOIs on individual fecundity.
Figure 2: Examples of predicted fecundities for three species pairs using direct-interactions-only (dashed line) and the full HOI-inclusive model (solid line).
Figure 3: Decomposition of the observed impacts of direct and higher-order effects on the fecundity of six focal species.


  1. 1

    Chesson, P. in Encyclopedia of Sustainability Science and Technology (ed. Meyers, R. A. ) Ch. 13, 223–256 (Springer, 2012).

    Google Scholar 

  2. 2

    Allesina, S. & Levine, J. M. A competitive network theory for species diversity. Proc. Natl Acad. Sci. USA 108, 5638–5642 (2011).

    CAS  Article  Google Scholar 

  3. 3

    Volterra, V. Fluctuations in the abundance of a species considered mathematically. Nature 118, 558–560 (1926).

    Article  Google Scholar 

  4. 4

    May, R. Will a large complex system be stable? Nature 238, 413–414 (1972).

    CAS  Article  Google Scholar 

  5. 5

    Thuiller, W. et al. A road map for integrating eco-evolutionary processes into biodiversity models. Ecol. Lett. 16, 94–105 (2013).

    Article  Google Scholar 

  6. 6

    Sih, A., Englund, G. & Wooster, D. Emergent impacts of multiple predators on prey. Trends Ecol. Evol. 13, 350–355 (1998).

    CAS  Article  Google Scholar 

  7. 7

    HilleRisLambers, J., Adler, P. B., Harpole, W. S., Levine, J. M. & Mayfield, M. M. Rethinking community assembly through the lens of coexistence theory. Annu. Rev. Ecol. Evol. S. 43, 227–248 (2012).

    Article  Google Scholar 

  8. 8

    Chesson, P. Mechanisms of maintenance of species diversity. Annu. Rev. Ecol. Syst. 31, 343–366 (2000).

    Article  Google Scholar 

  9. 9

    Kunstler, G. et al. Plant functional traits have globally consistent effects on competition. Nature 529, 204–207 (2016).

    CAS  Article  Google Scholar 

  10. 10

    Levine, J. M. & HilleRisLambers, J. The importance of niches for the maintenance of species diversity. Nature 461, 254–257 (2009).

    CAS  Article  Google Scholar 

  11. 11

    Falster, D. S., FitzJohn, R. G., Brännstrom, Å., Diekmann, U. & Westoby, M. Plant: a package for modelling forest trait ecology and evolution. Methods Ecol. Evol. 7, 136–146 (2016).

    Article  Google Scholar 

  12. 12

    Madin, J. S., Hoogenboom, M. O. & Connolly, S. R. Integrating physiological and biomechanical drivers of population growth over environmental gradients on coral reefs. J. Exp. Biol. 215, 968–976 (2012).

    Article  Google Scholar 

  13. 13

    Connolly, S. R. & Roughgarden, J. Theory of marine communities: competition, predation, and recruitment-dependent interaction strength. Ecol. Monogr. 69, 277–296 (1999).

    Article  Google Scholar 

  14. 14

    Smith-Gill, S. J. & Gill, D. E. Curvilinearities in the competition equations: an experiment with ranid tadpoles. Am. Nat. 112, 557–570 (1978).

    Article  Google Scholar 

  15. 15

    Wootton, T. J. The nature and consequences of indirect effects in ecological communities. Annu. Rev. Ecol. Syst. 25, 443–466 (1994).

    Article  Google Scholar 

  16. 16

    White, E. M., Wilson, J. C. & Clarke, A. R. Biotic indirect effects: a neglected concept in invasion biology. Divers. Distrib. 12, 443–455 (2006).

    Article  Google Scholar 

  17. 17

    Roughgarden, J. & Diamond, J. in Community Ecology (eds Diamond, J. & Case, T. J. ) 333–343 (Harper and Row, 1986).

    Google Scholar 

  18. 18

    Schoener, T. W. Some methods for calculating competition coefficients for resource-utilization spectra. Am. Nat. 108, 332–340 (1974).

    Article  Google Scholar 

  19. 19

    Freckleton, R. P. & Watkinson, A. R. Predicting competition coefficients for plant mixtures: reciprocity, transitivity and correlations with life-history traits. Ecol. Lett. 4, 348–357 (2001).

    Article  Google Scholar 

  20. 20

    Billick, I. & Case, T. J. Higher order interactions in ecological communities: what are they and how can they be detected? Ecology 75, 1529–1543 (1994).

    Article  Google Scholar 

  21. 21

    Werner, E. E. & Peacor, S. D. A review of trait-mediated indirect interactions in ecological communities. Ecology 84, 1083–1100 (2003).

    Article  Google Scholar 

  22. 22

    Abrams, P. A., Menge, B. A., Mittlebach, G. G., Spiller, D. & Yodzis, P. in Food Webs: Integration of Patterns and Dynamics (eds Polis, G. & Winemiller, K. ) 371–395 (Chapman and Hall, 1996).

    Book  Google Scholar 

  23. 23

    Peacor, S. D. & Werner, E. E. The contribution of trait-mediated indirect effects to the net effects of a predator. Proc. Natl Acad. Sci. USA 98, 3904–3908 (2001).

    CAS  Article  Google Scholar 

  24. 24

    Trussell, G. C., Ewanchuk, P. J. & Matassa, C. M. Habitat effects on the relative importance of trait- and density-mediated indirect interactions. Ecol. Lett. 9, 1245–1252 (2006).

    Article  Google Scholar 

  25. 25

    Wootton, J. T. Indirect effects and habitat use in an intertidal community: interaction chains and interaction modifications. Am. Nat. 75, 1544–1551 (1993).

    Google Scholar 

  26. 26

    Vandermeer, J. H. The competitive structure of communities: an experimental approach with protozoa. Ecology 50, 362–371 (1969).

    Article  Google Scholar 

  27. 27

    Bairey, E., Kelsic, E. D. & Kishony, R. High-order species interactions shape ecosystem diversity. Nat. Commun. 7, 12285 (2016).

    CAS  Article  Google Scholar 

  28. 28

    Godoy, O. & Levine, J. M. Phenology effects on invasion success: insights from coupling field experiments to coexistence theory. Ecology 95, 726–736 (2014).

    Article  Google Scholar 

  29. 29

    Goldberg, D. E. & Werner, P. A. Equivalence of competitors in plant communities: a null hypothesis and a field experiment. Am. J. Bot. 170, 1098–1104 (1983).

    Article  Google Scholar 

  30. 30

    Case, T. J. & Bender, E. A. Testing for higher order interactions. Am. Nat. 118, 920–929 (1981).

    Article  Google Scholar 

  31. 31

    Anderson, T. L. & Whiteman, H. H. Non-additive effects of intra- and interspecific competition between two larval salamanders. J. Anim. Ecol. 84, 765–772 (2015).

    Article  Google Scholar 

  32. 32

    Wilbur, H. M. Competition, predation, and the structure of the Ambystoma–Rana sylvatica community. Ecology 53, 3–21 (1972).

    Article  Google Scholar 

  33. 33

    Shmueli, G. To explain or to predict? Stat. Sci. 25, 289–310 (2010).

    Article  Google Scholar 

  34. 34

    Burnham, K. P. & Anderson, D. R. Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach (Springer, 2002).

    Google Scholar 

  35. 35

    Adler, P. B., HilleRisLambers, J. & Levine, J. A niche for neutrality. Ecol. Lett. 10, 95–104 (2007).

    Article  Google Scholar 

  36. 36

    Okuyama, T. & Holland, J. N. Network structural properties mediate the stability of mutualistic communities. Ecol. Lett. 11, 208–216 (2008).

    Article  Google Scholar 

  37. 37

    Novak, M. & Wootton, J. T. Estimating nonlinear interaction strengths: an observation-based method for species-rich food webs. Ecology 89, 2083–2089 (2008).

    Article  Google Scholar 

  38. 38

    Abrams, P. A. Implications of dynamically variable traits for identifying, classifying, and measuring direct and indirect effects in ecological communities. Am. Nat. 146, 112–134 (1995).

    Article  Google Scholar 

  39. 39

    Dwyer, J. M., Hobbs, R. J., Wainwright, C. E. & Mayfield, M. M. Climate moderates release from nutrient limitation in natural annual plant communities. Global Ecol. Biogeogr. 24, 549–561 (2015).

    Article  Google Scholar 

  40. 40

    Law, R. & Watkinson, A. R. Response-surface analysis of two-species competition: an experiment of Phleum arenarium and Vulpia fasciculata . J. Ecol. 75, 871–886 (1987).

    Article  Google Scholar 

  41. 41

    Angert, A. L., Huxman, T. E., Chesson, P. & Venable, D. L. Functional tradeoffs determine species coexistence via the storage effect. Proc. Natl Acad. Sci. USA 106, 11641–11645 (2009).

    CAS  Article  Google Scholar 

  42. 42

    Quinn, G. P. & Keough, M. J. Experimental Design and Data Analysis for Biologists (Cambridge Univ. Press, 2002).

    Book  Google Scholar 

  43. 43

    Rao, C. R., Toutenburg, H. & Shalabh, H. C. Linear Models and Generalizations (Springer, 2008).

    Google Scholar 

  44. 44

    Venables, W. N. & Ripley, B. D. Modern Applied Statistics with S 4th edn (Springer, 2002).

    Book  Google Scholar 

  45. 45

    Larsen, W. A. & McCleary, S. J. The use of partial residual plots in regression analysis. Technometrics 14, 781–790 (1972).

    Article  Google Scholar 

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This project was made possible by funding awarded to M.M.M. by the Australian Research Council (DP140100574 and FT140100498) and to D.B.S. from the Royal Society of New Zealand (UOC-1101 and a Rutherford Discovery Fellowship). We thank H. R. Lai, X. Loy, C. Wainwright and J. HilleRisLambers for help with data collection and J. HilleRisLambers, J. Dwyer, J. Tylianakis and the Mayfield and Stouffer labs for constructive comments. We also thank X. Loy for the art used to create Supplementary Fig. 1.

Author information




Authors are equal contributors to this paper. M.M.M. and D.B.S. conceived of the project and the framework together. M.M.M. collected and provided all data (with acknowledged help) and D.B.S. conducted all analyses. M.M.M. led the joint effort of writing the manuscript.

Corresponding authors

Correspondence to Margaret M. Mayfield or Daniel B. Stouffer.

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Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Information

Supplementary Methods; Supplementary Figs 1–3; Supplementary Tables 1–4; Supplementary References. (PDF 2093 kb)

Supplementary Data 1

Full dataset analysed in this study. This file contains an R list object, called, made up of six data frames (one per focal species). Within these data frames, each row (with unique identifier) contains the data for a single focal plant. The first four columns provide the following information: “Seed”: the number of seeds produced by focal plant; “focal”: focal species name; “site”: site code (B = Bendering, K = Kunjin); and “quadrat”: quadrat number. All remaining columns report densities of each potential competitor species, listed by name. Values in these “competitor” columns are abundances within the 7.5 cm radius “neighbourhood” around the focal plant. This file is also accessible from Dryad. (ZIP 0 kb)

Supplementary Data 2

This Rdata file contains an example dataset usable with Supplementary Code 2. Note that this file does not include all of the data used to generate the results presented in this paper but is simplified to better help users become familiar with our model-fitting R code. Our full dataset is available through Dryad and as Supplementary Data 1. (ZIP 14 kb)

Supplementary Code 1

This file includes the model-fitting code for our regression framework. It includes the function needed to fit fecundity models based on the following model formulations: the full and intermediate forms of the negative binomial model (Table 1 and Supplementary Table 3), a Poissonian form (not included in results), the negative binomial with quadrat included as a random effect, and the linear form and the inverse forms all presented in Supplementary Table 2. Notes for running this code with our full dataset (Supplementary Data 1) or the simpler and smaller example dataset (Supplementary Data 2) are provided at the beginning of the files. (TXT 6 kb)

Supplementary Code 2

This file provides a sample workflow for data analysis from this paper. The provided code fits the null-no competition, direct-competitive only and the full HOI-inclusive negative binomial models (equation (1)). It is designed to run with the example data file (Supplementary Data 2). This code runs much faster (when using the example data) than Supplementary Code 1 with the full dataset. (TXT 0 kb)

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Mayfield, M., Stouffer, D. Higher-order interactions capture unexplained complexity in diverse communities. Nat Ecol Evol 1, 0062 (2017).

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