1. The dominant eigenvalue of the population projection matrix provides the asymptotic growth rate of a population. Perturbation analysis examines how changes in vital rates and transitions affect this growth rate. The standard approach to evaluating the effect of a perturbation uses sensitivities and elasticities to provide a linear approximation.
2. A transfer function approach provides the exact relationship between growth rate and perturbation matrix. An alternative approach derives the exact solution directly by calculating the matrix characteristic equation in terms of the perturbation parameters and the asymptotic growth rate. This may be calculated numerically or by using symbolic algebra, and here we focus on the symbolic algebra approach.
3. The direct approach provides integrated sensitivities and plots of the exact relationship. The same method may be used for any perturbation structure, however complicated, including perturbations to vital rates that determine the elements of the population projection matrix.
4. The simplicity of the direct approach is illustrated through two examples, the killer whale and the lizard orchid.
5. Synthesis and applications. In this paper we describe three different methods for exact perturbation analysis. It is shown that each has its own merits, and the associated online computer code will encourage wider use of this analysis in the future.