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.
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 = (aij). 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 λmax of A), the stable-stage/age distribution (given by the right eigenvector w associated with λmax) and the reproductive value vector (given by the left eigenvector v associated with λmax; Caswell 2001).
Classical asymptotic perturbation analysis examines the response of λmax 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 λmax 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 λmax, 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 λmax, 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 λmax, and inversion then gives a graph of λmax 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.
Methods: The characteristic equation approach to perturbation analysis: using symbolic algebra
An alternative method for exact perturbation analysis uses symbolic algebra, through software such as Maple, Mathematica or the symbolic toolbox of Matlab. In contrast to the transfer function approach, which finds the required perturbation for a target asymptotic growth rate, the impact of a perturbation is considered by deriving the functional relationship between perturbation and any eigenvalue λ symbolically. This offers a wide range of tools that can be used to analyse the impact of perturbation on the growth rate.
The characteristic equation approach obtains eigenvalues of a perturbed matrix model, where the perturbation is controlled by one or more parameters. This gives a function that links the perturbation parameters and λ, from which it is possible to obtain both summary measures and graphical plots. The functional dependence between the entries of the population projection matrix and an eigenvalue λ is given implicitly by the characteristic equation (Caswell 2001)
Through the use of symbolic algebra, it is feasible to evaluate the characteristic equation for any matrix model. If there are several perturbation values, denoted by δ, then the characteristic equation involves both the elements of δ and λ, thereby providing an implicit relationship between the size of perturbations and λ. The equation can be expanded using a symbolic algebra package, yielding an implicit expression that is a polynomial function of the elements of δ and λ. The attraction of this approach is that it applies to any perturbation structure, however complicated, as we shall see in Results: Applying the characteristic equation method.
Perturbation analysis tools
The implicit expression of the characteristic equation can be used to construct a graphical representation of the exact behaviour of the perturbed asymptotic growth rate by using an implicit plotting procedure, such as implicitplot within the symbolic algebra package Maple. An implicit plotting procedure plots the perturbed characteristic equation as a surface defined by the perturbation parameters and then estimates the contour that is equal to zero, providing a two-dimensional representation of the relationship in the single perturbation parameter case. The characteristic equation describes the behaviour of all eigenvalues, and so implicitplot provides perturbation analysis of all real eigenvalues. The dominant eigenvalue of a population projection matrix is always real and positive when the population projection matrix is primitive and non-negative, which is typically true (Caswell 2001; Hodgson et al. 2006). In such cases, the plot of the characteristic equation provides a graphical summary of the relationship between λmax and the perturbation parameters.
The plot of the characteristic equation may also provide information about the damping ratio, which is defined as the ratio of the magnitudes of the largest and second largest eigenvalues, and provides a measure of the rate of convergence to the stable population structure. This quantity can be used to determine whether the use of an asymptotic analysis is appropriate for short-term perturbation analyses where the system is known not to be in equilibrium. If the second eigenvalue remains real across the range of perturbation, the implicit plot directly provides graphical information on the damping ratio. In the case where a sub-dominant eigenvalue is complex, a numerical procedure is required to calculate and plot the magnitude of all eigenvalues. Both of these cases are considered in the examples in Results: Applying the characteristic equation method.
Deriving a functional relationship between λmax and the perturbation parameters also allows the exact calculation of both the perturbation required to achieve a specified asymptotic growth rate and the value of λmax for any perturbation. To find the perturbation required for a specified asymptotic growth rate, the value of λmax is substituted into the implicit expression, and the equation is then solved in terms of the perturbation parameters. In the simplest case, where a single parameter defines the perturbation structure, the solution is simply the magnitude of the perturbation parameter. When two or more perturbation parameters are used, an infinite set of solutions is provided. The asymptotic growth rate can be found from a particular set of perturbations by substituting the appropriate values into the characteristic equation and solving for λmax.
The implicit form of the characteristic equation can also be used to derive sensitivities and elasticities, providing a link to the classical approach. Note that if, for example, two variables x and y are connected by means of a mathematical expression, but it is not possible to express y as an explicit function of x, it can still be possible to obtain the derivative dy/dx by differentiating the entire expression with respect to x, and this is called implicit differentiation. The sensitivity is obtained as the first derivative of Eqn 1 evaluated at the appropriate parameter value, and so implicitly differentiating the characteristic equation with respect to the perturbation parameters enables the computation of individual sensitivities.
An advantage of our approach to sensitivity analysis is the ease with which integrated sensitivities can be computed. By defining the perturbation structure to reflect the required trade-off relations, it is possible to compute the characteristic equation in terms of λmax and δ, and evaluate the partial derivatives of λmax with respect to the elements of δ at δ = 0 to give the required integrated sensitivity. Through scaling the sensitivity appropriately to take into account the ratio of the size of the parameter value to λmax, elasticities and integrated elasticities can also be computed.
Results: Applying the characteristic equation method
The killer whale, Orcinus orca
The life cycle of the killer whale comprises four stage classifications: yearlings, juveniles, mature adults and post-reproductive adults, and it is only the second and third stages that contribute to the reproductive component of the life cycle. Using the estimated transition rates from Brault & Caswell (1993), the population projection matrix takes the form
having a dominant eigenvalue of λmax = 1·0263. In the matrix representation, the first column corresponds to the transitions from yearlings, the second column to the transitions from juveniles and so forth. The existence of post-reproductive adults implies that the matrix AKil is reducible and non-ergodic (Stott et al., 2010). However, λmax gives the asymptotic growth rate of the population provided that the initial population does not consist entirely of post-reproductive individuals.
The simplest type of perturbation is a single-parameter, single-entry perturbation, and so we consider a modification of δ to the (3,3) entry, corresponding to stasis for reproductive adults. The perturbation can be represented in the form of the matrix
Because the probability of survival for mature adults is 0·9986, we consider only negative values of the perturbation parameter δ. To find the functional relationship between the perturbation parameter and the asymptotic growth rate, the determinant of the matrix (AKil + P1 − Λ) is calculated, where Λ is the diagonal matrix with λ in each diagonal entry.
The characteristic Eqn 1 is a polynomial function of δ and λ and can be expressed as
which is evidently linear in δ.
This equation can be used directly to generate a range of perturbation analysis summaries, and a step-by-step summary of the Maple code required to generate the perturbation analysis summaries is given in the online Appendix S4. The graph of the exact relationship between perturbation and asymptotic growth rate is given separately in Fig. 1, which displays the behaviour of the four real eigenvalues of the matrix for different levels of perturbation. In this application, the classical linear approximation approach appears to be appropriate. Figure 1 can also be used to investigate the impact of perturbation on the damping ratio in this instance, as both of the largest eigenvalues are real. The magnitudes of these eigenvalues are comparable in this example, giving a high ratio across the whole range of perturbation, and indicating that use of the asymptotic growth rate is not valid for short-term population projections.
There are several other uses of the characteristic equation in this example. For the given perturbation structure, the sensitivity of λmax to the magnitude of the perturbation can be computed by differentiating the equation implicitly with respect to δ and evaluating the subsequent equation at δ = 0 and λmax = 1·0263. The Maple commands required for this calculation are shown in online Appendix S4. The value of δ that yields a stable population size (i.e. λmax = 1) can be found by substituting λmax = 1 into the characteristic equation and solving for δ, giving δ = −0·0516.
The lizard orchid, Himantoglossum hircinum
The lizard orchid Himantoglossum hircinum is most abundant in southern Europe, where the species thrives in the Mediterranean conditions, with the development of populations in northern parts of Europe limited by both low summer and low winter temperatures (Carey & Farrell 2002). Of particular interest is the growth of new populations in north-east Germany and southern parts of the United Kingdom, and their spread may be attributed to climate change (Pfeifer, Heinrich & Jetschke 2006; Pfeifer et al. 2006). Matrix population models can be constructed using historical data that track individual plant histories from 1987 to 2008. Research on the target orchid population by P. Carey is ongoing. Plants are classified by size, giving a six-state model that includes flowering and dormancy states. Estimates of transition rates are obtained from maximum-likelihood methods, and the reproductive components are estimated using a least-squares approach. This yields the overall population projection matrix
In the matrix ALiz, the first four columns represent the size classes 1–4, the fifth column is for flowering plants and the final column for dormant plants, and the transitions are again described as going from columns into rows. For reproduction, it is the (1,5) element that is the appropriate term. The dominant eigenvalue is λmax = 1·0198, corresponding to a slowly increasing population.
The elasticity values can be used to identify influential components of the population projection matrix. They range from 0·0001 to 0·0971, the largest value corresponding to stasis for flowering plants, which is intuitively sensible. Two examples are constructed to illustrate the characteristic equation method: one using a single-perturbation parameter and one using two perturbation parameters.
A single-parameter perturbation
Motivated by the location of the largest elasticity mentioned earlier, this example considers the effect of changing the proportion of flowering plants that remain flowering; this is balanced by equal and opposite changes to the proportion of flowering plants that recur as non-flowering Stage 3 or Stage 4 plants. In matrix representation, the perturbation structure takes the form
Computing the characteristic equation of (ALiz + P2) gives an implicit expression relating δ and λ which is shown in Fig. 2. In this case, the roots of the characteristic equation do not all remain real across the range of perturbation, and we also plot |λ| when this does not occur, following numerical evaluation of all eigenvalues. By considering the absolute values of the sub-dominant eigenvalue of the matrix, the graph demonstrates that the damping ratio remains relatively low and constant. This indicates that an asymptotic analysis of the growth rate is applicable for short-term population projections. Once again, the linear approximation appears appropriate.
The previous example is extended by introducing a new parameter ε that controls the proportion of Stage 2 plants that remain in Stage 2 the following year when compared to those that become Stage 3, giving the perturbation matrix
The location of the ε-perturbations corresponds to entries with high elasticity. The characteristic equation of (ALiz + P3) is a function of the two perturbation parameters, δ and ε, and λ. Figure 3 is a contour plot, showing how λmax changes with different combinations of δ and ε. In this instance, nonlinearity is present.
The characteristic equation can also be used to identify the locus of parameter values that provide a stable population size. This is achieved by substituting λmax = 1 into the characteristic equation, yielding the expression below
Applying the characteristic equation method to the lizard orchid example illustrates the ease by which different perturbation structures can be investigated. In particular, the algebraic form of the characteristic equation may be used to derive integrated sensitivities, to control the nature of the perturbation structure and to identify stability loci for multi-parameter perturbations.
A range of methods of perturbation analysis have been discussed – the classical approximate approach using sensitivities and elasticities, the direct numerical approach, the transfer function method and a symbolic algebra alternative to obtain exact analysis. Sensitivities and elasticities are easy to calculate for matrix models and provide single summary values that quantify the influence of each parameter. This is useful for identifying which matrix elements are most important to the dynamics of the population.
The transfer function approach applies robust control theory to determine exact perturbation effects on asymptotic growth rate. As outlined in Hodgson & Townley (2004) and Hodgson et al. (2006), techniques such as pseudospectra, spectral value sets and stability radii can also be developed to provide analyses of transient effects. We have only described how the transfer function method works for a single perturbation to a single matrix element. In contrast to the method of this paper, the procedure becomes more complex as the complexity of the perturbation increases.
The characteristic equation approach provides a link between the classical sensitivity approach and transfer function analysis. The method tackles the problem from the same angle as the classical method by considering the characteristic equation of the population projection matrix. However, since the equation is expanded in terms of the perturbation parameters and any eigenvalue λ, it is possible to obtain an exact representation of the relationship. Plotting this equation gives an exact graphical representation of the behaviour of the dominant and real sub-dominant eigenvalues, providing information on the behaviour of both the asymptotic growth rate and the damping ratio of the system, when the second largest eigenvalue is real. When a sub-dominant eigenvalue is not always real, additional numerical procedures are required.
The main strength of the approach is the ease by which all types of perturbation structure can be investigated in exactly the same way – the perturbation parameters are simply entered into the population projection matrix. The approach can be used in exactly the same way when the matrix entries are functions of model parameters and interest lies in the effect of these parameters on the population growth rate. Such cases arise naturally in population ecology, see for example Morris & Doak (2004). The entries of perturbed population matrices could then well be non-linear functions of the value(s) of the perturbations, and this is a situation that is outside the structures considered by Hodgson et al. (2006). It is also possible to derive the sensitivity and elasticity equations by implicit differentiation, again highlighting the link with the classical approach. This is particularly useful when computing integrated sensitivities and elasticities of more complicated perturbation structures, because the result required is given by a single implicit differentiation of the exact expression. Other advantages of having the implicit form of the characteristic equation include manipulation of perturbation structures to minimise or maximise sensitivities at particular perturbation levels, and the ability to extract loci of parameter values to achieve stability. Sensitivities also measure selection gradients, and the identification of situations where the sensitivity to a trade-off parameter is zero can identify evolutionary stable strategies. The characteristic equation method gives a simple, intuitive approach to exact perturbation analysis, enabling the investigation of non-linear behaviour between perturbation and asymptotic growth rate. An attraction of the symbolic approach is that it furnishes explicit algebraic formulae. This provides an opportunity to examine the entire model parameter space simultaneously for sensitivity, as in response–surface design methodology, using the analysis of variance for example (Fang, Li & Sudjianto 2006).
The authors thank Stuart Townley and David Koons for useful comments on comparisons of methods of perturbation analysis. Three referees and an Associate Editor produced a most helpful review of the paper.