Discussion
We explore demographic buffering patterns through the integration of established demographic techniques. Our framework merges insights from both stochastic and deterministic demographic approaches, which revealed only limited support for our hypothesis. Specifically, we had anticipated that species exhibiting minimal influence from temporal variability in demographic processes on their stochastic growth rates would demonstrate concave selection affecting the demographic processes with the highest deterministic elasticities. However, using stochastic elasticities alongside the first- and second- order perturbation analysis of the deterministic population growth rate and applying these analyses to mammal species, we found that only the Columbian ground squirrel fully supported our hypothesis; humans showed partial support; other species did not.
Evidencing demographic buffering is not straightforward. Indeed, through the analysis of stochastic population growth rate (λs ) in our application of the framework to 43 populations of 37 mammal species, we identify the highest density of natural populations near the buffered end of the variance continuum. However, we show that most of the species then fail to exhibit signs of concave (∩-shaped) selection on impacting demographic parameters, opposed to our hypothesis. Such results suggest discordance between two features of demographic buffering, namely: 1) the stochastic population growth rate having a low sensitivity to temporal variability in demographic processes, and 2) demographic processes having their temporal variability reduced by selection.
The lack of association between the non-linear selection patterns (concave/convex) and species positioning on the variance continuum for the studied mammal species may have several explanations. Firstly, non-linear selection on demographic process variability isdynamic (Kajin et al. 2023). Within a life cycle, even minor changes in key demographic processes can trigger a domino effect, affecting not only the process itself but also the sensitivity ofλ1 to changes in said process (Stearns 1992). Consequently, correlations between demographic processes (negative correlations known as trade-offs) are influenced by minor alterations in the governing demographic processes (Doak et al. 2005). Because of these characteristics, second-order derivatives reveal “fine scale” fitness behaviour compared to sums of stochastic elasticities. Evolutionary demography still requires new tools to connect second-order fitness effects with stochastic elasticities in a biologically interpretable manner similar as in Tuljapurkar et al. 2023.
The stochastic elasticities explicitly account for the demographic process variation in time, while the first- and second- order effects on fitness are obtained from temporally averaged population matrices. Because a mean environment rarely characterizes the natural variation in demographic process typical of stochastic environments (Boyce et al. 2006), any metric derived from averaged matrix population models represent only an averaged realisation and could only rarely be representative of a pattern emerged from explicitly accounting for temporal variation.
Our original assumptions regarding demographically buffered populations, however, remain valid. We assumed that: 1) a buffered population is one with a weak summed effect of temporal variability on the long-term stochastic population growth rate, and 2) if a population is buffered, there should be signs of concave selection acting on the demographic process with the highest deterministic elasticity. The lack of support for our hypothesis supports the idea that the patterns of first- and second-order effects of demographic process variation on fitness are dynamic and can change rapidly in natural environments. Even if a given demographic process is primarily governing the population growth rate in one year, a different one might take over next year (Evers et al. 2021).
When placing our study species along a variance continuum, primates tend to be located on the buffered end. However, most primates displayed convex – instead of the expected concave – selection on adult survival. Similar results, where the key demographic process failed to display reduced temporal variability, have been reported for long-lived seabirds (Doherty et al. 2004). One explanation for the unexpected convex selection on adult survival involves trade-offs, as suggested by (Doak et al. 2005). When two demographic parameters are negatively correlated, the variance of population growth rate can be increased or decreased (Compagnoni et al. 2016; Evans & Holsinger 2012).
Correlations among demographic processes (positive and negative) inherently influence the biological limits of variance (Haridas & Tuljapurkar 2005). This is because the magnitude of variation in a particular demographic process is restrained by the variation of other demographic processes. Not surprisingly, correlations among demographic processes have been shown to be strongly subjected to ecological factors (Fay et al. 2022). Therefore, future studies may benefit from deeper insights using cross -second derivatives (Caswell 1996, 2001) to investigate correlations among demographic processes.
Biological variance estimates are inevitably subjected to several sources of bias (Simmonds & Jones 2024). To minimise bias, we randomly sampled the available matrices before obtaining the estimates. Despite the significant correlation between\(\Sigma E_{a_{\text{ij}}}^{S^{\sigma}}\) and the number of available matrices per species, the relative positioning of species remains meaningful for between-population level comparisons, as the correlation is very weak (-0.002). Still, researchers carrying out macroecological comparisons of demographic buffering might want to be even more stringent than we have been here with their datasets, as these grow longer with time (Compagnoni et al. 2021; Salguero-Gómezet al. 2021).
Regarding phylogenetic effects, our tests revealed a mild signal, but we note that future work regressing \(\Sigma E_{a_{\text{ij}}}^{S^{\sigma}}\) values against potential independent variables (e.g., climate values) may want to correct for this phylogenetic dependence. By having carefully chosen studies from a database that contains >400 species and retained only those that passed through a set of selection criteria (Che-Castaldo et al. 2020; Gascoigne et al. 2023b; Kendall et al. 2019; Römer et al. 2024; Simmonds & Jones 2024), we mitigate those biases a priori . Furthermore, we are using an elasticity-based approach, meaning we are comparing proportional variances. At present, the available methods still do not account for constraints in variance nor performing a perturbation approach disproportionately.
The analyses at both between- and within-populations levels are fundamentally interconnected. This connection is grounded on the fact that large summed elasticities to variability are intrinsically linked to high elasticity values, as demonstrated in equation 6 in (Haridas & Tuljapurkar 2005). This finding robustly endorses the perspective that species’ positions along the variance continuum should be interpreted with consideration of first and second-order effects, and additionally, in the context of selection pressures acting on the variability of demographic processes, as revealed by second-order derivatives.
Demographic processes within our study populations often face a mix of convex and concave selection. This mix of selection patterns was suggested by Doak et al. (2005), who noted that dramatic changes in population growth rate sensitivities are influenced by correlations among demographic processes. Here, only two of the 16 mammal species revealed concave selection on the key demographic processes: Columbian ground squirrel (Urocitellus columbianus ), and humans (Homo sapiens ). These two species were placed near (or relatively near) the buffered end of the variance continuum, supporting (partially) our hypothesis. Evidence of buffering has been reported across 22 ungulate species (Gaillard & Yoccoz 2003). However, in the one ungulate we examined, the moose (Alces alces ), we found only partial support for our hypothesis, as it is near the buffered end of the variance continuum but lacks concave selection pressures on the most important demographic process.
Our overall findings reveal varying levels of support for the notion that adult survival in long-lived species tends to be buffered. Indeed, Gaillard et al. (1998) found that adult female survival varied considerably less than juvenile survival in large herbivores. This finding was also supported by further studies in ungulates (Gaillard & Yoccoz 2003), turtles (Heppell 1998), vertebrates and plants (Pfister 1998), and more recently across nine species of plants (McDonaldet al. 2017). Gaillard and Yoccoz (2003) reported unexpectedly high adult survival in small mammals, even though the studied small mammals were annual, and as such, comparable to large mammal model. Seasonality, frequency and method of sampling all influence survival estimates and their estimated variability, thus, when comparing multiple species/studies, all the latter characteristics should be taken into account when interpreting the results.
Examining the drivers of demographic buffering has become an important piece of the ecological and evolutionary puzzle of demography. As such, understanding buffering can help us better predict population responses to environmental variability, climate change, and direct anthropogenic disturbances (Boyce et al. 2006; Gascoigne et al. 2024a; McDonald et al. 2017; Pfister 1998; Vázquez et al. 2017). By setting demographic buffering into a broader and more integrated frameworks, we hope to enhance comprehension and prediction of the implications of heightened environmental stochasticity on the evolution of life history traits. This understanding is crucial in mitigating the risk of extinction for the most vulnerable species.