λ1 with respect to each demographic process) reveals the type of selection acting on temporal variability of demographic patterns.
A strong first-order effect of variation on fitness implies in a linear relationship between a demographic process and fitness. When linearity is assumed, the self-second derivatives are near zero, which means selection changes the mean of demographic processes, but not their variance (Shyu & Caswell 2014). Nonzero self-second derivatives indicate nonlinear relationships between fitness and a demographic process, revealing additional aspects of selection on the variances and covariances of demographic processes (Brodie et al. 1995; Carslake et al. 2008; Shyu & Caswell 2014). Furthermore, the second-order derivatives measure how sensitive the population growth rate is to temporal autocorrelation of demographic processes.
We argue that the joint interpretation of first- and second-order effects of variation on λ1 provides the needed platform to address our prediction of demographically buffered populations displaying concave selection pressures. To address our hypothesis, we:
  1. Place populations along a continuum defined by\({\Sigma E}_{a_{\text{ij}}}^{S^{\sigma}}\) values.
  2. Identify the demographic processes with highest elasticities for each population.
  3. Associate the same demographic processes identified in (2) with negative self-second derivatives, indicating concave selection.
We showcase these steps on an imaginary wolf population (Fig. 1B). In this wolf population, individuals remaining in the fourth stage (MPM element a4,4 ) have the most impact overλ1 , with the highest elasticity value (Fig. 1B, yellow square). However, Figure 1C reveals a weak second-order effect of element a4,4 on λ1 , thus implying a weak selection pressure to reduce a4,4 temporal variance. A combination of a strong first-order and near zero second-order effects on fitness coincides with a strong linear influence of a change in the mean of a4,4 onλ1 . However, in this example, there is no evidence of concave selection on a4,4 , as we expected based on the positioning of wolf population on the left (unbuffered) side of the variance continuum (Fig. 1A).
We found evidence of concave selection in the fertility of individuals in the second and third stages of the hypothetical wolf species (Fig. 3C, MPM elements a1,2 and a1,3 , respectively, large black dots). Both fertility elements in this wolf population reveal low elasticities (Fig. 3B), but highly negative self-second derivatives. Such a pattern coincides with strong concave selection acting to reduce temporal variance in wolves’ second- and third- stage fertilities. These patterns also reveal that temporal autocorrelation in second- and third- stage fertilities affect population fitness. Nonetheless, the absence of concave selection in the fertility of individuals in the fourth stage (Fig. 3C, MPM element a1,4, small black dot) might suggests a pattern consistent with senescence.
Although not our primary goal, we briefly introduce steps to evidence demographic lability. Compelling lability evidence requires sufficient data across environments [over time or space; but see Perret et al. (2024)] to construct reaction norms depicting demographic responses to environmental changes (Drake 2005; Koons et al. 2009; Morriset al. 2008). Non-linear relationships between demographic processes and the environment must be established based on the reaction norms. Demographic processes where an increase in the mean value has a stronger positive impact on population growth rate than the detrimental effect of increased variance need to be identified. The latter condition is only met when the process-environment reaction norms are convex (Drake 2005, Koons et al . 2009, Morris et al . 2008). However, Barraquand & Yoccoz (2013) show that even with log-concave reaction norms, environmental variability can positively affect population growth under certain conditions, such as constant survival or density-dependent growth. Importantly, species may not be purely buffered or labile some processes may be buffered, others labile, and others insensitive to environmental variability (e.g. , Doaket al . 2005). Deciphering these patterns is a primary research interest in the field.