Demographic processes, their first- and second-order effects onλs, and types of selection on temporal variance
The following method delves into within-population level by calculating
the partial derivatives of λ1 (obtained by
averaging sequential MPMs across the study duration) concerning each
separate matrix element aij of the MPMA (Fig. 1B). This step reveals a first-order effect of
demographic process variation on fitness — the elasticities ofλ1 to changes in demographic processes. We then
evaluate a second-order effect using self-second derivatives ofλ1 for each aij (Fig. 1C)
(Caswell 1996; Shyu & Caswell 2014).
First- and second-order effects of the variation in demographic
processes on fitness are evidence of average selection pressures
over time (Carslake et al. 2008; Caswell 2001; Kajin et
al. 2023; Shyu & Caswell 2014; Tuljapurkar et al. 2023). While
elasticities can be considered a proxy for selection gradients (Lande
1982), a second-order effect (measured as a self-second derivative of