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).