Second-order elasticities for ecology and evolution: Unraveling nonlinear fitness responses to perturbations

In ecology and evolutionary biology, understanding the relationship between vital rates (e.g., survival, development, reproduction) and population growth is essential to elucidate how life history strategies are shaped by natural selection. Yet most established demographic methods rely on linear assumptions that rank the importance of vital rates but are insensitive to whether the returns of the population growth rate are diminishing or accelerating. Here, we introduce the second-order elasticities of mean individual fitness in the population, the S-elasticity. The S-elasticity is a proportional, element-wise measure that can quantify the curvature of mean individual fitness with respect to single matrix population model (MPM) elements (self S-elasticity) and to pairs of elements (mixed S-elasticity). S-elasticities (1) classify concavity versus convexity of local fitness surfaces, (2) provide a relative-curvature diagnostic by scaling second-order to first-order effects, (3) sum to zero across elements-facilitating comparisons across populations and species-and (4) connect fitness curvature to biologically feasible trade-offs via the elasticity ratio K. We provide a simple inequality based on the S-elasticities and the K-ratio that detects local fitness peaks along constraint directions corresponding to each pair of matrix elements. We illustrate the framework with an average 4 × 4 MPM for Didelphis aurita (Marsupialia, Didelphidae) and compare our results to classical second-order derivatives of population growth rate applied to the same dataset. First-order elasticities identify survival of weaned and subadult survival as dominant linear levers of fitness; their negative self S-elasticities reveal diminishing returns for increases and amplified losses for decreases relative to linear expectations. Mixed S-elasticities expose a costly trade-off between survival of weaned and aged females' fertility: The curvature-only surface is concave near the baseline and the trade-off direction is a local peak, whereas several other element pairs lie along flatter, more substitutable directions. Together, S-elasticities provide a compact diagnostic for when linear rankings suffice, when nonlinearity dominates, and where feasible substitutions are most constrained. Because they are proportional, additive (sum-to-zero), and rooted in life cycle pathways, S-elasticities complement classic elasticity and make second-order demographic information directly comparable across systems-yielding actionable guidance for evolutionary inference and for prioritizing management actions under realistic trade-offs.