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).