Figure legends
Figure 1 . A) The variance continuum for 37
hypothetical species based on the summed stochastic elasticities
(\(\Sigma E_{a_{\text{ij}}}^{S^{\sigma}}\)) at the between populations
hierarchical level. The closer the\(\Sigma E_{a_{\text{ij}}}^{S^{\sigma}}\) is to zero, the weaker the
impact of variation in demographic processes on the stochastic
population growth rate, λs . The variance
continuum ranges from potentially buffered (right-hand side) to less
buffered (left-hand side) populations. The yellow-dotted populations can
be classified as having potentially buffered life cycles . The
left-hand side of the graph represents populations where variability in
demographic processes results in strong impact onλs (blue dots). Thus, the blue-dotted populations
can be classified as having potentially unbuffered life cycles .
The vertical axis delineates the values of the probability density
function, indicating the frequency of populations at each value of\({\Sigma E}_{a_{\text{ij}}}^{S^{\sigma}}\). The placement of data
points (species/populations) along the horizontal axis corresponds to
their calculated values of \({\Sigma E}_{a_{\text{ij}}}^{S^{\sigma}}\)and is arranged linearly, while the placement along the y-axis is random
for improved visual comprehension. B) First-order effects or
elasticities for separate populations at within-species level. Shown are
the elasticities of the deterministic population growth rate
(λ1 ) for a hypothetical population of wolves and
revealing the most important demographic process(es) in the life cycle
(yellow cells: high elasticity, blue cells: low elasticity). C) Combined
results for first- (yellow and blue cells) and second-order effects
(black dots), where the latter reveals the nonlinear selection pressures
at the within-species level.
Figure 2 . The variance continuum for 43 populations
from 37 species of mammals from the COMADRE database based on the summed
stochastic elasticities of λs to temporal
variability in demographic processes
(\(\Sigma E_{a_{\text{ij}}}^{S^{\sigma}}\)) at the between-populations
hierarchical level. Colors represent different taxonomic orders with
Primates occupying the right-hand side. Silhouettes: a)Brachyteles hyphoxantus , b) Gorilla beringhei , c)Cercopithecus mitis , d) Urocitellus columbianus, e)Mustela erminea , f) Erythrocebus patas , g) Lepus
americanus , h) Rattus fuscipes , i) Ovis aries , j)Homo sapiens , k) Macropus eugenii , and l) Felis
catus . The vertical axis delineates the values of the probability
density function, indicating the frequency of populations at each value
of \({\Sigma E}_{a_{\text{ij}}}^{S^{\sigma}}\). The placement of data
points (species/populations) along the horizontal axis corresponds to
their calculated values of \({\Sigma E}_{a_{\text{ij}}}^{S^{\sigma}}\)and is arranged linearly, while the placement along the y-axis is random
for improved visual comprehension.
Figure 3: First- and second-order effects on population growth
rate, λ1 (corresponding to elasticities and
self-second derivatives of population growth rate, respectively) for 16
mammal species. The 16 plots represent populations where the MPMs built
by ages were available in the COMADRE Animal Matrix Database. The
yellow-blue colour scale represents elasticity values for each of the
demographic processes in the MPM, where yellow cells represent high and
blue cells low elasticity of deterministic population growth rate to
changes in demographic processes. No colour means elasticity=0. The
black dots represent negative self-second derivatives ofλ1 - corresponding to concave selection - and the
white dots represent positive self-second derivatives ofλ1 - ditto convex selection. The dot sizes are
scaled by the absolute value of self-second derivatives, where the
smaller the dot, the closer a self-second derivative is to 0, indicating
weak or no nonlinearity. Thus, large dots indicate strong nonlinear
selection forces, either concave (black) or convex (white). Since the
derivatives of population growth rate are confounded by eigen-structure
(Kroon et al. 2000), the scaling of the elasticity values and
second-derivative values is species specific - i.e. , each plot
has its own scale. Species-specific scales can be found in Supplementary
material (Table S2).
BOX 1:
The demographic buffering hypothesis : Stemming from
Tuljapurkar’s approximation (Tuljapurkar 1982), Pfister (1998) showed
that the penalisation term representing the variance-covariance
structure, tends to be reduced when elasticities of demographic
processes and their coefficients of variation covary negatively.However, the term demographic buffering was only coined later (sensu Morris & Doak 2004). The demographic buffering hypothesis
is also referred to as “adaptive buffering” (sensu Le Coeur et
al. 2022), suggesting that selection acts to minimize the
negative impacts of environmental variation by reducing the temporal
variance of key demographic processes (e.g. , survival,
development, reproduction) that have the highest sensitivity/elasticity
to population growth rate, a fitness proxy (Gaillard & Yoccoz 2003;
Pfister 1998).
Demographic buffering is a broader concept than the demographic
buffering hypothesis; it refers to a population’s capacity to withstand
environmental variability by keeping essential demographic processes
stable over time (Gascoigne et al. 2024a, b; Hilde et al. 2020; Morris & Doak 2004; Pfister 1998). It is worth noting that this
term does not explicit allude to the evolutionary mechanisms that
include selection, which are predicted by the demographic buffering
hypothesis (Le Coeur et al. 2022).
Demographic lability : A population’s ability to accommodate
fluctuations in demographic processes in response to temporal variations
in environmental conditions (Le Coeur et al. 2022; Jäkäläniemiet al. 2013; Koons et al. 2009). The relationship between
the labile demographic process and the environment can be convex,
concave, or linear. A labile vital rate in a variable environment will
have an average value that is greater than, less than, or equal to the
vital rate estimated in the mean environment, depending on the shape of
the relationship. Similar as for the demographic buffering hypothesis,
the demographic lability hypothesis relies on selection for
demographic process to track environmental fluctuations in a way that
increases the long-term fitness (λs ). This
process occurs when the increase in demographic process mean — due to
convexity — overcomes the detrimental effect of temporal variance in
annual population growth rates (Le Coeur et al. 2022).
Sensitivity : Represented by a first-order partial derivative of
population growth rate with respect to each demographic process (Caswell
1978, 2001; Ebert 1999), sensitivity measures the absolute change in
fitness that a change in a demographic process would cause.
Elasticity : Proportional sensitivity. A measure of proportional
change in fitness caused by a proportional change in demographic
process. Elasticities can be of different types (Grant et al. 2007; Haridas et al. 2009; Haridas & Tuljapurkar 2005, 2007; de
Kroon et al. 1986; Kroon et al. 2000; Van Tienderen 2000;
Tuljapurkar et al. 2003) and with respect to both, the stochastic
and the deterministic population growth rates.
Tuljapurkar’s approximation : To overcome dealing with complex
probability distributions that describe demographic fluctuations through
time, the approximation captures the essence of the effect of temporal
variability, at least for small amounts of variability (i.e. small noise). It states that the logarithm of the long-term stochastic
population growth rate equals the geometric mean growth rate plus a
penalty term containing the demographic process variance-covariance
structure (Tuljapurkar 1982).