Higher-order interactions capture unexplained complexity in diverse communities

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1 In the format provided by the authors and unedited. VOLUME: 1 ARTICLE NUMBER: 0062 Higher-order interactions capture unexplained complexity in diverse communities Authors: Margaret M Mayfield and Daniel B Stouffer Contents: 1. Supplementary Methods. Methods used to compare model fit results presented in Supplementary Table Supplementary Figure 1. A hypothetical example illustrating how direct and higherorder interactions contribute to fecundity outcomes. 3. Supplementary Figure 2. York-Gum woodland annual plant communities. 4. Supplementary Figure 3. Distribution of AIC values for a random sample of 99,997 intermediate models and the three main models from Table 1 (null, directinteractions-only and HOI-inclusive). 5. Supplementary Table 1. Quadrat and focal species sample size details. 6. Supplementary Table 2. R 2 and Log-Likelihood goodness-of-fit assessments of three alternative FF "# models. 7. Supplementary Table 3. Comparison of full and intermediate negative binomial models, using R 2 and LL goodness-of-fit assessments and an AIC parsimony test. 8. Supplementary Table 4. Median and upper and lower confidence intervals (CI) for results presented in Figure Supplementary References NATURE ECOLOGY & EVOLUTION DOI: /s

2 Supplementary Methods. Methods used to compare model fit results presented in Supplementary Table 2. In order to check the robustness of our results, we compared the negative-binomial framework presented in the main text to three alternative models. The first alternative model was a direct extension of the model from the main text that also captures quadrat-to-quadrat variation by introducing a quadrat random effect. Mathematically, this is equivalent to: FF "# {NN = λλ ( ee * + # {-} ee / + # {-} ee 0 + #, where λ i, DD "# and HH "# are defined as in Eqs. 1-3 in the main text and qq "# is a Gaussian distributed random variate that captures the effect of the quadrat in which each focal species was found. We estimated the best-fit coefficients in this generalised linear mixed model separately for each focal species using the function glmmadmb from the R package of the same name, with a negative-binomial error distribution and log link function 1. The second alternative model was constructed around the most-common, linear form of interspecies competition. Mathematically, this model looks like: FF "# {NN = λλ ( 1 + DD "# + HH "#, where λ i, DD "# and HH "# are again defined as in Eqs. 1-3 in the main text. We estimated the best-fit coefficients in this model separately for each focal species using the function lm in the statistical program R 1. The third and final alternative model takes an inverse form that has also been found to be a good descriptor of species fecundities in competitive environments 2-4. Mathematically, this model takes the form: NATURE ECOLOGY & EVOLUTION DOI: /s

3 λλ ( FF "# {NN =, 1 DD "# HH "# where λ i, DD "# and HH "# are again defined as in Eqs. 1-3 in the main text. We estimated the best-fit coefficients in this model separately for each focal species using the function glm in the statistical program R, with a Gaussian error distribution and inverse link function 1. Note that the best-fit, maximum likelihood parameters of the second and third models could potentially make biologically unrealistic predictions, such as negative fecundities (FF "# ), unless the parameter-estimation procedure is constrained in some fashion (e.g. that DD "# +HH "# -1 for the linear form). Indeed, our analyses indicate that this is often the case in the unconstrained form (Supplementary Table 2), which means that any comparison based around these models should be regarded with caution (since, technically, the log-likelihood of a model that predicts negative fecundities should be near minus infinity). Previously, researchers have sought to overcome this problem by imposing a different set of constraints, namely that all coefficients α > 0 2,3. Unfortunately, this form of the constraint further reduces the biological realism of the models by implying that no species could ever exert a positive, facilitative influence on a neighbour, and there are currently no reliable methods to perform the appropriately constrained optimisation without adding additional trade-offs. Code for fitting all alternative models is provided in the SI R Code File 1. NATURE ECOLOGY & EVOLUTION DOI: /s

4 Supplementary Figure 1. A hypothetical example illustrating how direct and higherorder interactions contribute to fecundity outcomes. Panel a, illustrates the direct (grey and black lines) and higher-order interactions (coloured arrows) that impact fecundity of an individual m of species i surrounded by neighbours from two species (i and j), and in which N i = 2 and N j = 3. In panel b, λ i is the fecundity of the m individual of species i when no competitors are present (intrinsic fecundity); for our worked example, λ i = 100. In the remainder of panel b, we illustrate how direct interactions have additive effects on the fecundity of individual m i. The subpanel surrounded by a dashed line illustrates the direct effect of one competitor individual. In panel c, we show how higher-order interactions can flip the sign and/or change the magnitude of the cumulative effect the competitor has on the fecundity of the focal individual. The focal plants in panels b and c are scaled to reflect the impacts of direct and higher-order interactions on fecundity relative to the λ i pictured in the oval. Arrows in these panels are scaled to the magnitude of each interaction type, with the direction of the effect written in each arrow. The magnitude and direction of the effect of each α and β-interaction term is also shown mathematically under each arrow. Panel d, illustrates how the fecundity model presented in this study combines the effects of direct and higher-order interactions to provide a complete explanation of the influence of competition on fecundity. Colours match those used in Figure 1. (Cartoons by Xingwen Loy). NATURE ECOLOGY & EVOLUTION DOI: /s

5 Supplementary Figure 2. York-Gum woodland annual plant communities. The large photo is a York Gum community in Bendering reserve at peak biomass. The inlay shows a focal Waitzia acuminata plant approximately 3 weeks prior to peak biomass with the circle showing a 7.5 cm radius circle centred on the focal plant. Only plants rooted within this neighbourhood plot were included in the interaction neighbourhoods used for this study (Photo Source: M.M. Mayfield). NATURE ECOLOGY & EVOLUTION DOI: /s

6 Supplementary Figure 3. Distribution of AIC values for a random sample of 99,997 intermediate models and the three main models from Table 1 (null, directinteractions-only and HOI-inclusive). Results are presented by focal species (Aica, Hygl, Pogn, Tror, Uran, and Waac). The coloured vertical lines indicate where the AIC values for the full direct-interactions-only model (blue line) and the full HOI-inclusive model (red line) fall along the distribution of intermediate models. Smaller AIC values represent higher model parsimony, with any model with 2 AIC points less than another model considered to be a meaningful improvement 5. Summary statistics for best models (those in the extreme left tail) and better models (those left of the blue line) are provided in Table 2. NATURE ECOLOGY & EVOLUTION DOI: /s

7 Supplementary Tables Supplementary Table 1. Quadrat and focal species sample size details. For No. Focal Plants and No. Quadrats, data is broken down by site. The first number in each cell is for Kunjin reserve and the numbers in parentheses are for Bendering reserve. No. Competitor Species is the number of different competitor species found within plots of each focal species. No. Solo Plants is the number of plots for each focal species containing a single focal plant. No. Possible HOIs is the maximum number of HOIs that could hypothetically exist for each focal species given the diversity of competitors found across their neighbourhood plots in this study. Note that this number is much larger than the number of HOIs that were actually calculable (see No. estimated parameters in Table 1) given neighbourhood variation available in our dataset. Focal species are: Aica = Aira caryophyllea, Hygl = Hypochaeris glabra, Pogn = Podotheca gnaphalioides, Tror = Trachymene ornata, Uran = Ursinia anthemoides, and Waac = Waitzia acuminata. Aica Hygl Pogn Tror Uran Waac No. Focal Plants 93 (0) 85 (84) 64 (0) 95 (94) 75 (0) 98 (85) No. Quadrats 32 (0) 35 (35) 32 (0) 35 (35) 31 (0) 35 (34) No. Competitor Species No. Solo Plants No. Possible HOIs No. plots containing other individuals of focal species Aira caryophyllea Hypochaeris glabra Podotheca gnaphalioides Trachymene ornata Ursinia anthemoides Waitzia acuminata No. plots containing each non-focal competitor species Arctotheca calendula Austrostipa elegantissima Avena barbata Blennospora drummondii Brachyscome iberidifolia Brassica tournefortii Briza maxima Bromus rubens Calandrinia eremaea Ceratogyne obionoides Crassula sp Ehrharta longiflora Gnephosis tenuissima Gonocarpus nodulosus Goodenia sp Hyalosperma demissum Hydrocotyle pilifera NATURE ECOLOGY & EVOLUTION DOI: /s

8 Lawrencella rosea Lobelia gibbosa Lysimachia arvensis Neurachne alopecuroidea Nicotiana rotundifolia Oxalis sp Parentucellia latifolia Pentaschistis airoides Phyllangium sulcatum Podolepis lessonii Podotheca angustifolia Poranthera microphylla Rhodanthe citrina Rhodanthe laevis Rhodanthe manglesii Thysanotus rectantherus Trachymene cyanopetala Trachymene pilosa Vulpia bromoides Vulpia sp Wahlenbergia gracilenta Zaluzianskya divaricata NATURE ECOLOGY & EVOLUTION DOI: /s

9 Supplementary Table 2. R 2 and Log-Likelihood goodness-of-fit assessments of three alternative FF mmii models 3,4. Focal species are: Aica = Aira caryophyllea, Hygl = Hypochaeris glabra, Pogn = Podotheca gnaphalioides, Tror = Trachymene ornata, Uran = Ursinia anthemoides, and Waac = Waitzia acuminata. Full model-selection methods provided in Supplementary Methods. The negative binomial model results here (FF "# = λ i ee * + # ee / + # ee 0 + # ) differ from those in Table 1 in that here we have included quadrat (e 0 + # ) as a random effect for all species. Aica Hygl Pogn Tror Uran Waac FF "# = λ i ee * + # ee / + # ee 0 + # No. of estimated parameters λ i ee 0 + # λ i ee * + # ee 0 + # λ i ee * + # ee / + # ee 0 + # Log Likelihood Estimates λ i ee 0 + # λ i ee * + # ee 0 + # λ i ee * + # ee / + # ee 0 + # P-values for pairwise model comparisons λ i ee 0 + # vs. λ i ee * + # ee 0 + # < λ i ee * + # ee 0 + # vs. λ i ee * + # ee / + # ee 0 + # < < < FF "# = λλ ( 1 + DD "# + HH "# No. of estimated parameters λ i λ i (1+DD "# ) λ i (1+DD "# +HH "# ) Coefficient of determination, R 2 λ i λ i (1+DD "# ) λ i (1+DD "# +HH "# ) Log Likelihood λ i λ i (1+DD "# ) λ i (1+DD "# +HH "# ) P-values for pairwise model comparisons λ i vs. λ i (1+DD "# ) λ i (1+DD "# ) vs. λ i (1+DD "# +HH "# ) No. of fecundity estimates (FF "# ) predicted to be negative λ i λ i (1+DD "# ) λ i (1+DD "# +HH "# ) NATURE ECOLOGY & EVOLUTION DOI: /s

10 FF "# = 1 DD "# HH "# No. of estimated parameters λ i λ i /(1-DD "# ) λ i /(1-DD "# -HH "# ) Coefficient of determination, R 2 λ i λ i /(1-DD "# ) λ i /(1-DD "# -HH "# ) Log Likelihood λ i λ i /(1-DD "# ) λ i /(1-DD "# -HH "# ) P-values for pairwise model comparisons λ i vs. λ i /(1-DD "# ) < < < λ i /(1-DD "# ) vs. λ i /(1-DD "# -HH "# ) < < No. of fecundity estimates (FF "# ) predicted to be negative λ i λ i /(1-DD "# ) λ i /(1-DD "# -HH "# ) λλ ( NATURE ECOLOGY & EVOLUTION DOI: /s

11 Supplementary Table 3. Comparison of full and intermediate negative binomial models, using R 2 and LL goodness-of-fit assessments and an AIC parsimony test. Here we compare model fit (R 2 and Log likelihood) and model parsimony (AIC) between the null, direct-interactions only and three variations on the HOI-inclusive negative binomial models. Results for the null, direct interactions-only and full HOI-inclusive models are the same as in Table 1 and are reproduced for easy comparison with the intermediate models. The two intermediate models are: an intraspecific-only HOI model (H intra ) containing all calculable intraspecific HOIs (β iii, β ijj ) only (no interspecific HOIs), and an interspecific-only HOI model (H inter ) including all calculable interspecific HOIs (β iij, β ijk ) with no intraspecific HOIs. The over-dispersion parameter for the negative binomial model is included in the No. of estimated parameters. Bold AIC values correspond to the most parsimonious model for each species. Aica Hygl Pogn Tror Uran Waac No. of estimated parameters λ i λ i ee * + # λ i ee * + # ee / #>?@A λ i ee * + # ee / #>?B@ λ i ee * + # ee / + # Coefficient of Determination, R 2 λ i λ i ee * + # λ i ee * + # ee / #>?@A λ i ee * + # ee / #>?B@ λ i ee * + # ee / + # Log Likelihood Estimates λ i λ i ee * + # λ i ee * + # ee / #>?@A λ i ee * + # ee / #>?B@ λ i ee * + # ee / + # P-values for pairwise model comparisons (LL) λ i ee * + # vs. λ i ee * + # ee / #>?@A λ i ee * + # vs. λ i ee * + # ee / #>?B@ < λ i ee * + # vs. λ i ee * + # ee / + # < < < AIC parsimony test λ i λ i ee * + # λ i ee * + # ee / #>?@A λ i ee * + # ee / #>?B@ λ i ee * + # ee / + # NATURE ECOLOGY & EVOLUTION DOI: /s

12 Supplementary Table 4. Median and upper and lower confidence intervals (CI) for results presented in Figure 3. Note that the full confidence intervals for the three bars shown in part in Fig. 3 are provided here in full. Colours match bars in Fig. 3. Median Low CI Upper CI Aira caryophyllea FF "# (green) α-cumulative (black) β-cumulative (purple) α ii (grey) α ij (white) β iii (orange) β ijj (yellow) β iij (light blue) β ijk (dark blue) Hypochaeris glabra FF "# (green) α-cumulative (black) α ii (grey) α ij (white) β-cumulative (purple) β iii (orange) β ijj (yellow) β iij (light blue) β ijk (dark blue) Podotheca gnaphalioides FF "# (green) α-cumulative (black) β-cumulative (purple) α ii (grey) α ij (white) β iii (orange) β ijj (yellow) β iij (light blue) β ijk (dark blue) Trachymene ornata FF "# (green) α-cumulative (black) β-cumulative (purple) α ii (grey) α ij (white) β iii (orange) β ijj (yellow) NATURE ECOLOGY & EVOLUTION DOI: /s

13 β iij (light blue) β ijk (dark blue) Ursinia anthemoides FF "# (green) α-cumulative (black) β-cumulative (purple) α ii (grey) α ij (white) β iii (orange) β ijj (yellow) β iij (light blue) β ijk (dark blue) Waitzia acuminata FF "# (green) α-cumulative (black) β-cumulative (purple) α ii (grey) α ij (white) β iii (orange) β ijj (yellow) β iij (light blue) β ijk (dark blue) NATURE ECOLOGY & EVOLUTION DOI: /s

14 Supplementary References 1 Quinn, G. P. & Keough, M. J. Experimental Design and Data Analysis for Biologists. (Cambridge University Press, 2002). 2 Godoy, O. & Levine, J. M. Phenology effects on invasion success: insights from coupling field experiments to coexistence theory. Ecology 95, (2014). 3 Levine, J. M. & HilleRisLambers, J. The importance of niches for the maintenance of species diversity. Nature 461, (2009). 4 Law, R. & Watkinson, A. R. Response-surface analysis of two-species competition: an experiment of Phleum arenarium and Vulpia fasciculata. Journal of Ecology 75, (1987). 5 Burnham, K. P. & Anderson, D. R. Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach. (Springer, 2002). NATURE ECOLOGY & EVOLUTION DOI: /s