## Introduction

This paper provides a new technique, based on symbolic algebra, for the exact perturbation analysis of matrix population models and compares and contrasts the approach with two other exact procedures. In a matrix population model, the life-history transitions of an individual, which may be based on either age or stage classification, are arranged in a population projection matrix **A** = (*a*_{ij}). If **n**(*t*) is the population state vector at time *t*, then the model takes the form

where *t* takes discrete time steps (typically annual). Key characteristics of the model can be obtained by analysing the nature of the population projection matrix. Three important components of the analysis are the asymptotic population growth rate (given by the dominant eigenvalue λ_{max} of **A**), the stable-stage/age distribution (given by the right eigenvector **w** associated with λ_{max}) and the reproductive value vector (given by the left eigenvector **v** associated with λ_{max}; Caswell 2001).

Classical asymptotic perturbation analysis examines the response of λ_{max} to changes in the entries of **A**. The simplest approach to this problem involves computing the dominant eigenvalue numerically for a series of perturbed matrices to provide a graphical analysis of the relationship between asymptotic growth rate and perturbation. Additional numerical analysis is necessary for the inverse problem of finding the growth rate for a particular perturbation. Illustrative R programs for the numerical approach applied to the examples in this paper are provided in the online Appendices S1–S3. An advantage of using R is that the package is freely available. Although computations depend on the size of the projection matrix, programs are unlikely to take long to run, even for ‘large’ matrices. However, numerical analysis does not provide the algebraic detail of either of the other two exact methods considered in this paper, and it lacks the analytic elegance of those methods. The standard approach has been to use linear approximations (incorporating sensitivity and elasticity; Caswell 2001). Relatively recent methods based on transfer function analysis provide a different and elegant way of obtaining the exact relationship between perturbation and asymptotic growth rate (Hodgson & Townley 2004; Hodgson, Townley & McCarthy 2006). This paper presents a new simple way to obtain an exact sensitivity analysis, which is easily implemented, and compares the alternative methods.

### Review of existing methods and techniques

The standard approach to perturbation analysis has been based on sensitivity and elasticity analysis. It assumes a linear relationship for the effect of the perturbation on the asymptotic growth rate of the system, conveniently quantifying the effect through the calculation of single measures. A sensitivity measures the effect of an absolute perturbation of a particular model parameter, whilst an elasticity considers the effect of a relative perturbation. Sensitivity is defined as the rate of change in the asymptotic growth rate with respect to a parameter of interest. The sensitivity matrix is defined as

and it is easily calculated using standard matrix algebra, specifically

where is the complex conjugate of **v**, and 〈**w**,**v**〉 is the scalar product of the eigenvectors **w** and **v** (Caswell 1978). To deal with proportional changes to the parameters, the elasticity matrix is defined as

This can be easily calculated from the sensitivity matrix as

where ○ denotes the Hadamard (elementwise) product. Both sensitivities and elasticities provide measures of the perturbation effect on λ_{max} through considering infinitesimal changes. Taylor series expansions are described in Section 3.3.7 of Bolker (2008). By applying a first-order Taylor series expansion, linear approximations estimate the impact of larger perturbations. It is also possible to consider second derivatives, which reflect the sensitivity of the sensitivities, allowing for a second-order Taylor series approximation (Caswell 1996). The methodology has also been extended to multiple parameter perturbations by allowing simple trade-off relations between parameters through the development of integrated sensitivities and integrated elasticities (van Tienderen 1995; van Tienderen 2000). Sensitivities and elasticities may be subject to mathematical constraints, and this is investigated by Carslake, Townley & Hodgson (2009).

A method proposed in Hodgson & Townley (2004) and further developed in Hodgson *et al.* (2006) uses a transfer function. Note that a transfer function is so called as it provides a mathematical relationship between the input and output of some system and is typically used in signal processing and control engineering. For target population growth rates, if there is only a single perturbation to just one matrix entry, the transfer function is given by

where **b** and **c** are row vectors that determine the entry of **A** that is to be perturbed. For any target growth rate λ_{max}, it is simple to show that the magnitude of the perturbation corresponding to that growth rate is given by , provided such a perturbation exists, of course. The methodology is extended in Hodgson *et al.* (2006), allowing for multiple perturbations to matrix elements. The transfer function *G*(*z*) can be written as a ratio of two polynomial expressions in *z*, and the Matlab function *ss2tf*, which is available in the Matlab (MathWorks, Natick, Massachusetts, USA) Control Theory toolbox, evaluates these expressions; an alternative approach is to use symbolic algebra. Note that this function is only called once in any perturbation analysis. The expression, , easily provides the value of the perturbation that results in a specified growth rate λ_{max}, provided the perturbation exists, and a simple Matlab program provided in association with Hodgson & Townley (2004) provides a way of plotting *δ* for a fixed range of λ_{max}, and inversion then gives a graph of λ_{max} as a function of *δ*. This can be followed for all the real eigenvalues of a matrix, not just the dominant eigenvalue. This technique provides the exact relationship between perturbation and growth rate, removing the need for linear approximations, which can be inappropriate, as shown by Hodgson & Townley (2004). A particularly appealing feature of the work of Hodgson *et al.* (2006) is the mathematics of how this basic simple approach extends naturally to cases of more than a single perturbation.