Second derivatives of population growth rate: Calculation and applications

The rate of increase of a population depends on the age- or stage-specific vital rates (survival, reproduction, etc.). Sensitivity analysis explores this dependence by calculating the fir st derivatives of the rate of increase with respect to each of the vital rates, and has become a standard part of demographic analysis. This paper presents a formula for the second derivatives, for the case where the rate of increase is measured by the dominant eigenvalue lambda of a population projection matrix. Because the first derivatives of lambda are written in terms of the eigenvectors, the second derivatives of lambda can be written in terms of the first derivatives of the eigenvectors. I present examples of the second derivatives of lambda in age- and size-classified life cycles, and discuss several potential applications of the results: (1) the perturbation analysis of elasticity patterns with applications to conservation biology, (2) the sensitivity analysis of stochastic growth rates, (3) the inclusion of second-order terms in the decomposition analysis of life table response experiments, and (4) the analysis of directional, stabilizing, and correlational selection on life history traits.