Elasticities in variable environments: Properties and implications

Elasticities in stochastic matrix models are used to understand both population and evolutionary dynamics. We examine three such elasticities: stochastic elasticity E-ij(S) with respect to the (i, j) matrix element, the elasticity with respect to the mean mu(ij) of the matrix element, and the elasticity E-ij(S mu) with respect to the variability sigma(ij) of the matrix element. We show that the stochastic elasticity E-S does not accurately describe the effect of variability; one should use E-s sigma and E-S mu. We establish two general properties of these elasticities: a sum rule that connects them and a limit on the sum of the E-S sigma. We discuss the implications of these properties for the analysis of buffering and selection on the average rates versus the variability of rates.