Time dependent interaction modification generated from plantsoil feedback
Abstract
Pairwise interactions between species can be modified by other community members, leading to emergent dynamics contingent on community composition. Despite the prevalence of such higherorder interactions, little is known about how they are linked to the timing and order of species’ arrival. We generate population dynamics from a mechanistic plantsoil feedback model, then apply a general theoretical framework to show that the modification of a pairwise interaction by a third plant depends on its germination phenology. These timedependent interaction modifications emerge from concurrent changes in plant and microbe populations and are strengthened by higher overlap between plants’ associated microbiomes. The interaction between this overlap and the specificity of microbiomes further determines plant coexistence. Our framework is widely applicable to mechanisms in other systems from which similar timedependent interaction modifications can emerge, highlighting the need to integrate temporal shifts of species interactions to predict the emergent dynamics of natural communities.
README: Time dependent interaction modification generated from plantsoil feedback
Code Files
All simulations were run in R version 4.2.1.

Functions_PSF.R
contains all functions used for contructing the plantsoil feedback model. 
GenerateData.R
contains code for generating data using the mechanistic plantsoil 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 higherorder 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 plantsoil feedback model) has three components:

ij
orijk
: data generated from two or threeplant models; 
al
,nc
, orco
: representing different degrees of overlap between microbiomes;al
represents low overlap,nc
represents no overlap, andco
represents high overlap (see also Figure 1); 
ts
orrs
: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 $p_jp_i$ (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 threeplant 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 threeplant models.
Coefficients starting with alpha
are pairwise, coefficients starting with beta
are higherorder (see Methods). pi
, pj
, and pk
denotes germination phenology of plants $i$, $j$, and $k$; pij
is the phenological difference calculated by $p_jp_i$ (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 plantsoil feedback