Common garden experiment
In the second year of our common garden experiment, we tested the effect of nutrient treatments on mineral nitrogen available to plants, biomass production, and understory light availability by performing ANOVA-type generalized linear models (McCullagh and Nelder, 1989) since our response include variables with normal and non-normal error distributions. Data that were analysed using normal error distribution included nitrogen available to plants and biomass production. Data with non-normal error distribution included the percentage of understory light availability, which was analysed with a quasibinomial error distribution to control for overdispersion.
To model plant growth, we fitted a four-parameter logistic curve to species biomass data through time (Pinheiro and Bates, 2000, Paine et al., 2012) using a non-linear mixed-effects regression model with the nlme function (Pinheiro and Bates, 2000). This model best fitted plant growth through the season which initially increases, stabilizes and then decreases over time but not necessarily in a symmetric way. Species, nutrient treatments and their interaction were treated as fixed effects and the four parameters of the logistic growth model (\(K,\ xmid,M_{0}\), and \(r\)) were treated as random effects allowing them to vary between species and nutrient treatments. To improve homoscedasticity of the residuals, aboveground biomass was natural log-transformed before analyses giving:
\(\log\left(M_{t}\right)=M_{0}+\frac{(K-M_{0})}{\frac{1+exp((xmid-t)}{r)}}\)eqn 1
where \(t\) is time in days of the year, \(M_{t}\) is aboveground plant biomass at time \(t\); \(M_{0}\) is the asymptotic mass as\(t\rightarrow-\infty\); \(K\) is the asymptotic mass as\(t\rightarrow\infty;\) xmid is the mass at the inflection point, the time at which RGR is maximized and \(r\) is a scale parameter.
RGR is given by \(\frac{d(log(M_{t}))}{\text{dt}}\), thus we estimated daily RGR during the growing season for each species as:
\(\text{RGR}_{t}=\frac{\frac{\left(K-M_{0}\right)exp((xmid-t)}{r})}{\frac{r(1+exp((xmid-t)}{r))}^{2}}\)eqn 2
Thus, for each species in each nutrient treatment combination, one value for \(\text{RGR}_{t}\) was generated for each day between the first and last day of the sequential harvests, yielding 119 values of\(\text{RGR}_{t}\) between day 53 and 171.
To assess whether early differences in growth rate between species in monocultures predict short-term competitive dominance at harvest in pairwise and in five-species mixtures under both productive and unproductive conditions, we related the relative differences in species biomass of the harvest of June 2008 for each pairwise mixture and for each combination of pairs in the five-species mixtures to the daily relative differences in growth rates of the respective species and nutrient treatment combination in monoculture, thus generating 119 regressions for each of the pairwise and five-species mixtures, one for each day between day 53 and 171.
Relative difference in abundance at harvest (day 171) in mixtures\(\left(\text{ΔB}_{\text{ij}}\right)\) between species \(i\) and \(j\)was calculated as the natural log ratio of differences in biomass as:
\(\left(\text{ΔB}_{\text{ij}}\right)=Ln\left(\frac{B_{i}}{B_{j}}\right)\)eqn 3
A positive value of relative difference in abundance means that the biomass of species \(i\) at harvest is higher than that of species \(j\)when growing together, i.e. species \(i\) has a greater relative abundance when growing with species \(j\), and vice versa. Ten values of relative difference in abundance (\(\text{ΔB}_{\text{ij}}\)) were calculated for each of the pairwise and five-species mixtures, one for each of the ten combination of pairs of species.
Daily relative differences in growth rates\(\left({\text{ΔRGR}_{t}}_{\text{ij}}\right)\) between species \(i\)and \(j\) were calculated for each day between day 53 and 171 as the natural log ratio of difference in RGR in monoculture as:
\(\left({\text{ΔRGR}_{t}}_{\text{ij}}\right)=Ln\left(\frac{{\text{RGR}_{t}}_{i}}{{\text{RGR}_{t}}_{j}}\right)\)eqn 4
A positive value of daily relative differences in growth rates means that the relative growth rate in monoculture at time \(t\) of species\(i\) is higher than that of species \(j\), i.e. species \(i\) grow relatively faster than species \(j\) at a given day in the year, and vice versa. For each of the ten species pairs, daily relative differences in growth rates (\({\text{ΔRGR}_{t}}_{\text{ij}}\)) were calculated for each day between day 53 and 171 for the pairwise and five-species mixtures.
We assessed the relationship between the relative abundance in mixture and daily relative differences in growth rates using generalized linear models with a normal error distribution. The relative abundance in mixture was the response variable and relative differences in growth rates, nutrient treatments and their interaction were the explanatory variables. A positive relationship would indicate that species with a higher RGR at time \(t\) have greater competitive ability and aboveground biomass at harvest. For each regression, we extracted the slope and 95% CI as well as the percentage of variance explained (R2 value).