### Summary

- Top of page
- Summary
- Introduction
- Life cycle perturbations and the corresponding transfer function
- Graphical analysis of the transfer function: a recipe
- Utility of the transfer function: an illustrative example
- Structured perturbations: trade-offs and vital rates
- Discussion
- Synthesis and application
- Acknowledgements
- Supplementary material
- References
- Supporting Information

- 1An important task in applied population ecology is to understand how changes to individual life-history parameters, such as survival, growth and fecundity, affect population dynamics. Parameter changes, or perturbations, may be caused by deliberate attempts to manage populations (e.g. in pest control, harvesting or conservation) or they may be side-effects of pollution, genetic modification and climate change.
- 2For organisms with complicated life cycles, links between individual life histories and population dynamics are made using population projection matrix (PPM) modelling.
- 3Changes to individual, or groups of, life-history transition rates within a PPM have a nonlinear impact on the resulting eigenvalues. Conventional sensitivity analysis calculates the derivative of the perturbation-eigenvalue curve to provide tangential linear extrapolation. Until now, only the simulation of perturbed PPMs has captured nonlinear perturbation effects.
- 4Here we describe the
*transfer function*of a matrix perturbation. The transfer function captures analytically the true relationship between perturbation magnitude and PPM eigenvalues. This analytical link extends easily to multi-transition and multiple perturbations, promotes an understanding of matrix properties, and provides a simple method to predict the perturbation required to achieve a desired population rate of increase. - 5We use the transfer function approach to analyse a PPM for the desert tortoise
*Gopherus agasizzii*Cooper, in the context of conservation management decisions. - 6
*Synthesis and applications.*The transfer function offers a novel and powerful framework for the analysis of population projection matrices (PPMs), giving precise predictive power and analytical understanding of population-level responses to life-history perturbations, for example in the design of conservation, pest control and population harvesting strategies, prediction of population effects of pollution in ecotoxicology, and in ecological risk assessment. A useful focus is to set a target for the desired rate of increase (or decline) of a population, and use the transfer function to determine how best to achieve this rate.