The summed effects of demographic process variability measured by stochastic elasticities
Current evidence for demographic buffering has primarily been assessed using Matrix Population Models (MPMs , hereafter) (Pfister 1998; Rotella et al. 2012). However, Integral Projection Models (IPMs ) (Easterling et al. 2000; Ellner et al. 2016; Gascoigne et al. 2023b; Rodríguez-Caro et al. 2021; Wanget al. 2023) can also identify demographic buffering. MPMs and IPMs are structured, discrete-time demographic models (Caswell 2001; Ellner et al. 2016). For simplicity, here we focus on MPMs, but note that the same proposed approach applies to IPMs (Doak et al. 2021; Griffith 2017). Hereafter, we refer to demographic processes in the MPM A as its entries aij (i.e., upper-level parameters sensu Zuidema & Franco 2001) and the vital rates composing those matrix elements (i.e., lower-level parameters, ditto ). The conversion between matrix elements and vital rates is straightforward (Franco & Silvertown 2004).
We obtain the stochastic elasticities (Haridas & Tuljapurkar 2005) ofλ s to place species on a variance continuum. The variance continuum represents the summed effects of proportional increases in temporal variability across all demographic processes (aij ) of the MPM A on the population growth rate λs , operating at thebetween-populations level. The \(\text{ΣE}_{a_{\text{ij}}}^{S}\)can be partitioned into two components: i) the sum of stochastic elasticities with respect to variability11Standard deviation (σ) stands for a measure of variability.*(\({\Sigma E}_{a_{\text{ij}}}^{S^{\sigma}}\)) — assessing how variability in aij affectsλs —and ii) the sum of stochastic elasticities with respect to the arithmetic mean of demographic processes (\({\Sigma E}_{a_{\text{ij}}}^{S^{\mu}}\)) — assessing the impact of a change in mean values of demographic processes onλs (Haridas & Tuljapurkar 2005). A weak (i.e., near zero) summed effect of variability means that the population growth rate is relatively unaffected by the variability in demographic processes (Haridas & Tuljapurkar 2005), and this lack of effect by demographic process variability is consistent with demographic buffering. As such, a summed effect of variability offers a good proxy to evidence demographic buffering (Gascoigne et al. 2024b; Haridas & Tuljapurkar 2005) and enables the classification of populations along a continuum.
Species or populations are positioned along the variance continuum based on the impact of demographic process variance onλs . Species highly sensitive to environmental variability are on the left (potentially unbuffered221Increased variance does not necessarily imply demographic lability, defined as an increase in mean value of a demographic process in response to improved environmental conditions (Le Coeur et al. 2022). By examining stochastic elasticities, we can assess changes in the contribution of demographic process variability toλs , while mean values remain unchanged.), while species less sensitive are on the right (potentially buffered) end (Fig. 1A). Although the position on the continuum provides insight into how environmental variation affects λs ,\({\Sigma E}_{a_{\text{ij}}}^{S^{\sigma}}\)does not consider covariances between demographic processes and serial correlations, crucial for a full comprehension of demographic buffering (Haridas & Tuljapurkar 2005). Thus, species’ position at the buffered end of the variance continuum is a necessary but not sufficient condition for evidence of demographic buffering. To address this second criterion, as well as to test our hypothesis, we use second derivatives of population growth rate with respect to demographic processes to elucidate the impact of selection on the temporal variability of said demographic processes.