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:
- Place populations along a continuum defined by\({\Sigma E}_{a_{\text{ij}}}^{S^{\sigma}}\) values.
- Identify the demographic processes with highest elasticities for each
population.
- 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.