Results
We ranked 43 populations from 37 mammal species into a continuum of variance according to the summed impact of variation in demographic processes on λs (Fig. 2). Most of the analysed taxonomic orders were placed on the low or zero variance end of the variance continuum (Fig. 2), coinciding with demographically buffered populations. The smallest contributions of variability in demographic processes (note that \(\Sigma E_{a_{\text{ij}}}^{S^{\sigma}}\) ranges from 0 to -1), suggesting buffered populations, were assigned to Primates: northern muriqui (Brachyteles hyphoxantus ,\(\Sigma E_{a_{\text{ij}}}^{S^{\sigma}}\) = -5.31 × 10-5 ± 2.09 × 10-5) (mean ± S.E.) (Fig. 2 silhouette a), mountain gorilla (Gorilla beringei ,\(\Sigma E_{a_{\text{ij}}}^{S^{\sigma}}\) = -1.28 × 10-5 ± 1.32 × 10-5) (Fig. 2 silhouette b), followed by the blue monkey (Cercopithecus mitis ,\(\Sigma E_{a_{\text{ij}}}^{S^{\sigma}}\) = -4.43 × 10-5 ± 1.18 × 10-5) (Fig. 2 silhouette c). The first non-primate species placed near the buffered end of the continuum was the Columbian ground squirrel (Urocitellus columbianus , Rodentia,\(\Sigma E_{a_{\text{ij}}}^{S^{\sigma}}\) = -3.38 × 10-3 ± 6.96 × 10-4) (Fig. 2 silhouette d). On the other opposite, the species with the highest contribution of variation in demographic processes – placed at the high-variance end of the continuum – was the stoat (Mustela erminea , Carnivora, \(\Sigma E_{a_{\text{ij}}}^{S^{\sigma}}\) = -0.310 ± 0.0162) (Fig. 2 silhouette e). All the 14 primate populations occupied the buffered side of the variance continuum, with the exception of the Patas monkey (Erythrocebus patas , Primates,\(\Sigma E_{a_{\text{ij}}}^{S^{\sigma}}\) = -0.0521 ± 5.38 × 10-3) (Fig. 2 silhouette f). The snowshoe hare (Lepus americanus , Lagomorpha,\(\Sigma E_{a_{\text{ij}}}^{S^{\sigma}}\) = -0.262 ± 0.0233) (Fig. 2 silhouette g) and the Bush rat (Rattus fuscipes , Rodentia,\(\Sigma E_{a_{\text{ij}}}^{S^{\sigma}}\) = -0.245 ± 4.29 × 10-3) (Fig. 2 silhouette h) were positioned on the non-buffered end of the variance continuum. Additional information (including standard errors of the elasticity estimates) is provided in Table S1. A posteriori , we quantified the impact of phylogenetic relatedness on the estimates of the sum of stochastic elasticities (Fig. 2), and then for the correlation between those estimates and the number of MPMs available per species. For the former, we estimated Blomberg’s K, a measure of phylogenetic signal that ranges between 0 (weak signal) to positive values 1 (strong) (Münkemüller et al. 2012).Blomberg’s K in our analyses was 0.23. The correlation between the number of available MPMs per study and the sum of stochastic elasticities (post jack-knifing) raised a weakly negative coefficient (-0.002), though significant (P = 0.017).
We found evidence in support of our hypothesis in only one of the studied species, the Columbian ground squirrel (Urocitellus columbianus ). This species is placed near the buffered end of the variance continuum (Fig. 2, silhouette d) and its most impactful demographic process shows signs of concave selection. The strongly negative self-second derivative with respect to growth from first to the second stage (Fig. 3B, MPM element a2,1) indicates thata2,1 is both important, and at the same time, kept constant through time in this population of U. columbianus .
In humans, the support for our hypothesis was present, but weaker, as humans are placed further away from the buffered end of the variance continuum (Fig. 2, silhouette j). However, the demographic parameters representing growth from the first to second age class and growth from second to third age class (matrix elements a2,1 and a3,2 , respectively) displayed high elasticities alongside negative self-second derivatives (Fig. 3D), corroborating with demographically buffered population.
For the remaining studied species the demographic processes with the highest elasticity values did not display strong negative self-second derivatives (Fig. 3). Particularly for the majority of primates, placed on the buffered end of the variance continuum, demographic processes with high elasticities had positive values for the self-second derivatives (indicated by yellow squares with white dots in Figure 3). Examples of primate species exhibiting high elasticities and positive values for their self-second derivatives include northern muriqui (Brachyteles hypoxanthus ), mountain gorilla (Gorilla beringei ), white-faced capuchin monkey (Cebus capucinus ), rhesus monkey (Macaca mulatta ), blue monkey (Cercopithecus mitis ), Verreaux’s sifaka (Propithecus verreauxi ) and olive baboon (Papio cynocephalus ) (Fig. 3). This implies that the key demographic processes influencing λ1 do not show evidence of selective pressure for reducing their variability.
The killer whale (Orcinus orca ) showed similar lack of support for our hypothesis as primates. Indeed, O. orca was positioned at the buffered end of the variance continuum (Cetacea,\(\Sigma E_{a_{\text{ij}}}^{S^{\sigma}}\) = -4.72 × 10-4 ± 1.53 × 10-4) (Fig. 2 silhouette not shown). However, the first- and second-order effects show that the governing three demographic processes in the killer whale life cycle (namely, matrix elements a2,2 ,a3,3 , and a4,4 ) are not under selection pressures for reducing their temporal variance, but the opposite (yellow and green squares with white dots, Fig. 3).
The primary governing demographic process for Soay sheep (Ovis aries ) displayed convex selection signatures. For O. aries (Fig. 2, silhouette i), remaining in the third age class (Fig. 3,a3,3 ,) impacts λt most and is under selection pressure to have its variance increased. These characteristics suggest potential conditions for lability, despite the species being positioned closer to the buffered end of the variance continuum.
Adding the second-order effect of variation on fitness to the toolbox for demographic buffering is an important addition. The high absolute values of self-second derivatives (large dots, either black or white, Fig. 3) suggest λt is sensitive to autororrelation in those demographic processes. This pattern also means that if, for example, the mean value of a5,4 forU. maritimus increased, the sensitivity ofλt to a5,4 would decrease because the self-second derivative of a5,4 is highly negative (depicted by the largest black dot in polar bear, Fig. 3 silhouette j). The opposite holds for the a4,4 , where an increase in the value of a4,4 would increase the sensitivity of λt toa4,4 , because the self-second derivative ofa4,4 is highly positive (the largest white dot in the polar bear MPM).