## Introduction

Sustained changes to the probabilities of survival, growth and reproduction of members of threatened, pest or exploited populations, caused by pollutants, climate or biotic interactions, will alter future population dynamics, but rarely in a linear fashion. Population management decisions require identification of the life-history stages that should be targeted in order to achieve, mostly easily or with least expense, a desired population response. Predicting the effects of pollution, genetic modification and climate change requires the ability to link the results of small-scale life-history experiments to population-level responses. The development of analytical techniques that link individual life cycles to population dynamics will therefore find application in the risk assessment of genetically modified organisms (Kareiva, Parker & Pascual 1996; Bullock 1999), conservation management (Burgman, Ferson & Akcakaya 1993; Silvertown, Franco & Menges 1996; Fisher, Hoyle & Blomberg 2000; Kaye *et al*. 2001; Norris & McCulloch 2003), ecotoxicology (Caswell 2000), harvesting (Marboutin *et al*. 2003) and pest control (e.g. Woolhouse & Harmsen 1991; Jarry, Khaladi & Gouteux 1996).

Population projection matrices (PPMs) summarize the transition, per unit time, of members of a population between ages (Leslie 1945) or stages (Lefkovitch 1965) of their life cycle (Caswell 2001). Eigenvalues of the PPM predict future dynamics of the population. In particular, the dominant eigenvalue, λ_{max}, predicts the asymptotic rate of increase of the population. A fundamental problem is how to predict the effects of perturbations to life histories on this rate of increase? Differentiating the change in λ_{max} with respect to change in matrix entries *a*_{i,j} yields sensitivity analysis (Caswell 1978) (or elasticity when standardized by the magnitude of transition rates (de Kroon *et al*. 1986)). Sensitivities are most commonly used to describe the importance of life-history transitions to population behaviour, but via extrapolation can also predict approximately (Mills, Doak & Wisdom 1999; de Kroon, van Groenendael & Ehrlen 2000) the effects of small perturbations. Precise predictions involve calculating the eigenvalues of simulated, perturbed matrices (Mills *et al*. 1999).

An analytical alternative to simulation methods involves the *transfer function* of a perturbed matrix (Pritchard & Townley 1989; Hinrichsen & Kelb 1993; Rebarber & Townley 1995). The transfer function is the building block of *robust control* theory and is readily applied to PPMs. It promotes generic and systematic understanding of matrix properties and response to perturbation, using analytical tools that can deal with multiple and structured perturbations to single or multiple transition rates or their vital rate components. The direct and indirect costs (whether financial or ecological) of poor population management decisions can be great, therefore we believe there is a need for precise and analytical techniques as long as they are not significantly more complicated than sensitivity and elasticity approximations. The transfer function introduces an analytical framework that will help tackle these problems.