New developments in matrix perturbation analysis: Transient, nonlinear, and stochastic

Perturbation analysis (sensitivity and  elasticity) has become a major component of demography, especially in the study of linear models. For linear models, the main focus has been on the effect of parameter changes on the asymptotic population growth rate and its periodic and stochastic counterparts. I will present a selection of new results that extend perturbation analysis to (1) effects on short-term transient dynamics, (2) effects on equilibria, cycles, and transient dynamics of density-dependent matrix models, (3) effects on stable age and stage distributions, (4) effects on measures of generation time, and a wide range of other quantities. The keys to this development are the application of matrix calculus and the trick of deriving dynamic expressions for the derivatives of quantities, expressions that are evaluated along with the quantities themselves. The new results presented here provide both sensitivity and elasticity, with respect to both matrix entries and lower-level vital rates, for both age- and stage-classified matrix models.  As an added bonus, they are easy to compute.