Population growth rate
We calculated population growth rates by incorporating vital rates into the following classic equation for an annual plant population model (Cohen 1966), with an additional plant survival parameter:
\(\frac{N_{i,\ t+1}}{N_{i,\ t}}=s_{i}\left(1-g_{i}\right)+f_{i}z_{i}g_{i}\)Equation 1
where \(\frac{N_{i,\ t+1}}{N_{i,\ t}}\) is the per capita growth rate of a given population i , si is seed survival probability (proportion of seeds that remain viable in the soil over summer, see Supplementary Methods), gi  is emergence probability, fi  is the number of viable seeds produced per emergent and zi is survival probability of emerged plants (the proportion of emerged plants that produced at least one viable or inviable seed).
Population growth rate was calculated twice for each plot; with and without neighbours (the latter being intrinsic growth rate, λ). For these calculations, species-specific seed survival and plot-level emergence probabilities were held constant, while plant survival and seed production were calculated with and without neighbours. Plot-level seed production (fi  ) was calculated as the exponentiated average of log-transformed viable seed production per focal plant that produced at least one viable seed.