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