Analysis:
Model of colony growth
We modelled within-colony dynamics as colony weight gain over time (see, e.g., (Westphal et al. 2009; Rundlöf et al. 2015; Crone & Williams 2016; Spiesman et al. 2017), which is much less invasive to measure than worker production over time (cf. Kerr et al.2019; Malfi et al. 2019). However, unlike most past studies, we analyzed weight gain using a change-point model in which the weight of each colony increases exponentially over the colony growth phase, then switches to decline during production of sexuals. The approach of fitting growth curves to each colony is more powerful than simply analyzing average weight through time because different colonies switch at different times, so information is lost by averaging (see Crone & Williams 2016). In this case, we tested whether the colony growth rate depended on the resource status, leading to the following equation for colony dynamics:
\(W_{t}=\left\{\par \begin{matrix}W_{0}{\lambda_{p}}^{t_{p}}{\lambda_{o}}^{t_{o}}&if\ t\ \leq\tau\\ W_{0}{\lambda_{p}}^{t_{p}}{\lambda_{o}}^{t_{o}}\delta^{t-\tau}&if\ t>\ \tau\\ \end{matrix}\right.\ \text{\ \ }\) eq (1)
where Wt is colony weight at time t(measured in days); W 0 is the initial colony weight (estimated as a model parameter to account for observation error, e.g., changes in relative humidity and subsequent weight); λp and λ0 are the colony growth rates during resource supplementation (pulse) and the period of ambient resources (off-pulse), respectively; tp andt 0 are the amounts of time spend in supplemented and ambient conditions (respectively), up to time t ; δ is change in colony mass after decline starts (λ0δ is the rate of weight loss after the switch to decline, and τ is the time at which colonies switch from growth to decline. Assuming λ0δ< 1 (i.e., the colony actually declines), then the colony reaches its peak mass at time τ. We constrained eq (1) such that the pulse growth rate was not influenced once the decline started (post-τ) (see Supporting Information, Appendix 5 and Appendix 6 ).
If carryover effects of resource conditions on subsequent growth are negligible, and colonies switch from growth to decline at the same time (regardless of treatment), then we would expect colony growth rates to be the same regardless of the order of treatments. In this case, colonies would reach the same mass at the end of the experiment; this is the familiar commutative property of multiplication. However, if colonies experience carry-over effects, then we might expect growth rates during the pulse and off-pulse periods to differ among treatments, and/or colonies to switch to reproduction at different times, leading to different peak weights.
  1. Testing effects of resource pulse on colony growth parameters We used change-point regressions to estimate parameters for eq (1) for each colony (following Crone & Williams 2016) (see Supporting information, Appendix 6 ), and then tested whether parameters differed among colonies in different treatments. Regressions were custom-coded in R (R Core Team 2018); a version of this code is available on GitHub for use by others (https://github.com/Aariq/bumbl). Visual inspection of fits for each colony are shown in Supporting Information (Fig S5,Appendix 5 ). We then tested for effects of the resource supplementation (early vs. late) on pulse (λp ) and off-pulse (λ0) growth rates, the timing of the switch point to reproduction (τ), and peak mass (\({\lambda_{p}}^{t_{p}}{\lambda_{o}}^{t_{o}}\), evaluated at time τ). Treatment effects were assessed with linear mixed models implemented using the lme4 package in R (Bates et al. 2015); within each model, colony was a replicate, and a random effect of site was included to account for non-independence of paired colonies placed at the same location. We analyzed switch points and log-transformed colony growth rates and peak mass using Gaussian family models.
  2. Testing effects of resource pulse on queen production
We evaluated whether receiving a food resource pulse early or late in colony development influenced the count of queens produced by colonies using a negative binomial generalized linear mixed model (GLMM). Given the high number of colonies that did not produce gynes (19/28), we compared the fit of this model to one that also accounted for zero-inflation. Accounting for zero-inflation did not improve model fit, so we interpreted the results of the original negative binomial GLMM model. The GLMM was fit using the lme4 package (Bates et al. 2014), and the 0-inflated model was fit using the pscl package (Zeileis et al. 2008) in R.