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.
- 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.
- 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.