Code Files

All simulations were run in R version 4.2.1.

• Functions_PSF.R contains all functions used for contructing the plant-soil feedback model.
• GenerateData.R contains code for generating data using the mechanistic plant-soil feedback model.
• GenerateData_MultipleParams.R contains code for generating data using randomly selected parameters; this is required for the sensitivity analysis (see Appendix I).
• ExamineModels.R contains code for comparing the performance of several dynamic models and generate Figure S1.
• FeedbackRatios.R contains code for calculating the ratio of intra- vs. interspecific effect from microbiomes using empirical data (Yan et al. 2022 PNAS).
• ModelFitting.R contains all code for fitting the simulated data to phenomenological models of higher-order interactions.
• Analysis_Tidy.Rmd contains code for creating all figures in the main text and Appendix.

Data

The Data directory contains six subdirectories that are categorized below.

The name of raw data (i.e., data simulated from the plant-soil feedback model) has three components:

• ij or ijk: data generated from two- or three-plant models;
• al, nc, or co: representing different degrees of overlap between microbiomes; al represents low overlap, nc represents no overlap, and co represents high overlap (see also Figure 1);
• ts or rs: ts represents data generated from a time series, rs represents data generated by a response surface design for one year (see Methods).

All data are .csv files.

Population Dynamics

• Standard is the default directory that contains population dynamics used to produce all results in the main text.
• m_ii_small contains population dynamics with a set of parameters that specifies a smaller intraspecific microbial effect.
• m_ij_small contains population dynamics with a set of parameters that specifies a larger intraspecific microbial effect.

Group denotes the type of population (plants, seeds, or microbiomes), and init1, init2, and init3 denotes the initial population of plants, seeds, or microbiomes $i$, $j$, and $k$. pi, pj, and pk denotes germination phenology of plants $i$, $j$, and $k$; pij is the phenological difference calculated by $p_j-p_i$ (see Methods and Figure 1). For datasets ending in rs (response surface), Time ranges from 1 to 12 because we only data of one year to estimate the interaction coefficients (see Methods).

Interaction Coefficients

• Repetition contains fitted interaction coefficients and their standard error reported by the model from simulations with random parameters. This set of simulations is only performed with three-plant models.
• RealisticProp contains fitted interaction coefficients and their standard error reported by the model from simulations with random parameters drawn from distribution of ratios of intra- vs. interspecific effect from microbiomes using empirical data (see below). This set of simulations is only performed with three-plant models.

Coefficients starting with alpha are pairwise, coefficients starting with beta are higher-order (see Methods). pi, pj, and pk denotes germination phenology of plants $i$, $j$, and $k$; pij is the phenological difference calculated by $p_j-p_i$ (see Methods and Figure 1). rep denotes the repetition (from 1 to 100).

Empirical Data

• Metaanalysis_Yan contains data used to calculate the ratio of intra- vs. interspecific effect from microbiomes using empirical data.

See Yan et al. 2022 for the original data and formatting.Time dependent interaction modification generated from plant-soil feedback