D.J. Hodgson, School of Biological and Chemical Sciences, University of Exeter, Exeter EX4 4PS, UK (e-mail email@example.com).
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.
6Synthesis 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.
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, λmax, 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 λmax with respect to change in matrix entries ai,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.
Life cycle perturbations and the corresponding transfer function
Perturbation to a PPM can be written as A + P, where A is the PPM of interest and P is a structured matrix of perturbation magnitudes. By recognizing different forms of the perturbation matrix, a suite of robust control techniques, algorithms and concepts becomes readily available to population ecology. For the sake of this introduction to these tools, we restrict our attention to a single structured perturbation of a PPM:
( eqn 1)
Here b and c are column and row vectors, respectively, that define the transitions within A to be perturbed and p is a scalar that defines the magnitude of perturbations. Basic perturbation analysis wishes to determine the dominant eigenvalue of A + bpc. Simple algebraic manipulation (see Appendix S1 in Supplementary Material) leads to the following result: if λpert is an eigenvalue of A + bpc then
In equation 3, I is the identity matrix of the same dimension as the PPM.
In other words, for any given PPM and perturbation structure, we can calculate the nonlinear relationship between λpert, the dominant eigenvalue of the perturbed PPM, and p, the magnitude of perturbation. The function G(z) plays a fundamental role in systems theory and control engineering. It is the transfer function for the triple (A, b, c). Note that instead of calculating λpert as defined by p, we vary λpert and determine, via 1/G(λpert), the perturbation p required to achieve this eigenvalue.
Graphical analysis of the transfer function: a recipe
The transfer function is typically nonlinear and is therefore best analysed graphically. We illustrate the utility of transfer function analysis using the PPM of the desert tortoise Gopherus agassizii Cooper (Doak, Kareiva & Kleptetka 1994). For this PPM the tortoise life cycle is divided into eight stages: stages six, seven and eight all represent reproductively mature individuals. We restrict attention to the matrix representing medium–high tortoise fecundities (Caswell 2001) (Fig. 1). A typical population management problem would be to identify the life-stage transitions or vital rates that will most ‘improve’ the target population's future dynamics. In the case of the desert tortoise, a declining population, conservation managers aim to cause population increase. To demonstrate the application of our approach we first consider perturbation of the rate of stasis of individuals within stage class 6, i.e. a6,6. In this case we use b = [0 0 0 0 0 1 0 0]T, where T represents the transpose of a vector or matrix, c = [0 0 0 0 0 1 0 0], and calculate
( eqn 4)
From the equation pG(z) = 1, with G(z) given by equation 4, we see that λpert is an eigenvalue of the perturbed matrix if
( eqn 5)
The function 1/G(z) is rational and a complete plot of 1/G(z) against z provides a complicated graph, with the eigenvalues of the unperturbed matrix represented by z-axis values where 1/G(z) = 0. We evaluate 1/G(z) for all z but record only admissible values of 1/G(z) (Fig. 2a). We then swap the axes of our restricted plot to produce a graph of the eigenvalue of the perturbed matrix, z, against 1/G(z) which equals the level of perturbation, p (Fig. 2b). This step aids interpretation: for any given admissible perturbation, what is the resulting asymptotic population growth rate? Here, ‘admissible’ depends on the application, but objectively the perturbations p should not reduce any entry ai,j to less than zero, and in the case of probabilities (i.e. of stasis within a stage class, or growth) should not increase the sum of transitions from category i into other categories above 1. Admissible changes to rates of reproduction are less easy to place upper bounds on.
This simple recipe is easily applied to each non-zero entry in the desert tortoise PPM (Fig. 3). Using mathematical syntax, to perturb a single transition rate, ai,j, requires
( eqn 6)
Here e(m) is a column vector of zeroes except for a 1 at entry m. This is the case when perturbation to each transition rate is independent of other transitions: for example, when a change in the probability of stasis within a stage class does not affect the probability of growth into the next stage class. This will not always be true, but robustness can also easily incorporate life-history trade-offs (see below). The algorithm used to calculate these robustness graphs is presented as MATLAB code in Appendix S2. This algorithm is easily adapted to specific needs and other PPMs.
Utility of the transfer function: an illustrative example
As a second example of the utility of the transfer function, we consider the problem of identifying the point of zero net population growth. Pest control aims to cause λpert < 1, conservation λpert > 1, sustainable exploitation λpert ≥ 1, and risk assessment to demonstrate λpert < 1. What level of perturbation (pcrit) is required to achieve λpert = 1? The transfer function provides an immediate answer: from equation 2,
( eqn 7)
We can use knowledge of pcrit to rank population management strategies. We use the tortoise PPM as an example (Table 1). Using sensitivities, we extrapolate the tangent of the perturbation-eigenvalue curve to predict the perturbation required for each non-zero transition in the PPM to reach λpert = 1. Alternatively, the precise perturbations required can be calculated instantly by equation 7. Note that precise calculations using simulations would have required unnecessary iterations. Note also that sensitivity and transfer function analysis produce different rankings for population management strategies (Table 1). Sensitivity analysis consistently underestimates the required increase in rates of growth between size classes, and overestimates the required increase in rates of stasis. This is easily explained by the fact that sensitivity does not capture the respective accelerating and decelerating responses of λpert to these transition rates: see Fig. 3. Transfer function analysis identifies nine transition rates (not including fecundities) that could be targets for conservation management, compared to sensitivity's identification of five.
Table 1. Summary of the stasis and growth transitions in the desert tortoise life cycle that can be independently perturbed to achieve positive asymptotic population growth. Sensitivity predictions are based on tangential extrapolations. Transfer function predictions are precise, capturing the nonlinear relationship between changes to the transition rates and the resulting dominant eigenvalue of the PPM. The absolute magnitude of perturbation predicted to achieve λpert = 1 is denoted pcrit. Transitions for which the required perturbation is unfeasible (either because the perturbation would raise the transition rate itself above 1, or would raise the sum of stasis and growth rates from that size class to above 1) are highlighted
Transfer function analysis
Structured perturbations: trade-offs and vital rates
The transfer function may be applied to more complicated problems than just considering the effect of single transition rate perturbations. Population dynamics are prone to (i) structured impacts on systems, and (ii) trade-offs or correlated responses across matrix transitions (van Tienderen 1995). Projection analysis often requires prediction of perturbation effects on (iii) components of transition rates (i.e. vital rates such as mortality and growth), or perturbation effects on (iv) seasonal matrices (Caswell 2001). For all these special cases of structured perturbations, transfer function analysis needs only structure the perturbation matrix P appropriately. For trade-off analysis, information is required on which transitions covary; for vital rate or seasonal matrix perturbations, life cycle graph analysis allows the correct structuring of P.
Here we interpret two simple, hypothetical, trade-off examples that assess the effect of perturbation of multiple transition rates on population growth rate. In each case, only the ‘perturbation structure’ vectors b and c are altered. These new vectors can then be used in the transfer function analysis algorithm.
(1) Consider a simple structured perturbation, recognizing that perturbation of the rate of stasis (individuals remaining in life-stage category i can trade-off directly against the rate of growth (individuals moving into the next life-stage, i + 1). In this case,
b = e(i) − e(i+1), c = (e(j))T, p ∈ ℜ.
This is the case when, for example, increased stasis does not affect mortality, but prevents individuals progressing to the next life-stage. As an example, if an increase in the stasis of size 3 tortoises (a3,3) directly reduces the rate of growth to size 4 (a4,3) by the same amount (slope of the phenotypic regression = −1), then:
b = [0 0 1 − 1 0 0 0 0 ]T and c = [0 0 1 0 0 0 0 0] .
(Fig. 4c)(2) Consider a partial trade-off example, where the slope of the relationship between the rate of growth and the rate of stasis [(dai+1,i)/(dai,j)] is –β. In other words, a unit change in rate of stasis is compensated (1 –β) by a change in mortality, and β by a change in progression to the next size class. We write this b = e(i) − βe(i+1), c = (e(j))T, p ∈ ℜ. As an example, suppose the rate of stasis of size 3 tortoises (a3,3) is increased (by p). Due to phenotypic trade-off, 70% of this increase is compensated by a decrease in mortality (i.e. mortality rate decreases by 0·7p), whilst 30% is compensated by a decrease in growth rate (i.e. a4,3 decreases by 0·3p). Then:
b = [0 0 1 − 0·3 0 0 0 0]T and c = [0 0 1 0 0 0 0 0] (Fig. 4b).
Visual assessment of these graphs demonstrates that the nature and magnitude of covariation or trade-offs between traits can radically alter not just the direction of population response to perturbation but also the lack of precision of extrapolations based on sensitivity analysis. The curvature of the perturbation-eigenvalue curve changes with the strength of the trade-off between a3,3 and a4,3: with weakly negative trade-off relationships, increasing a3,3 slows the asymptotic rate of decrease (increased λpert), but with strong negative trade-offs, increasing a3,3 harms the population due to the correlated decrease in a4,3.
The effects of perturbation on the dominant eigenvalue of the resulting perturbed PPM differ dramatically between the two special cases described above (Fig. 4b,c), and the single-rate perturbation described in Figs 2, 3, 4(a). Other examples (D. J. Hodgson & S. Townley, unpublished data) have revealed unexpected nonlinearities that would severely affect decisions based on sensitivity approximations. Furthermore, a useful feature of the transfer function approach is that, having computed the data corresponding to each perturbed individual transition rate of interest (e.g. Figure 3), trade-off effects can be assessed via simple manipulation and combination of this data. Instead of re-computing equation 4 (or in the case of matrix simulations, starting again with new simulated perturbed matrices), one needs only combine the perturbation-eigenvalue relationships according to the newly structured perturbation vectors b and c.
Transfer function analysis, and companion techniques of robust control such as pseudospectra (Trefethen 1991), spectral value sets (Hinrichsen & Kelb 1993) and stability radii (Hinrichsen & Pritchard 1990), are firmly established in many other quantitative disciplines for dealing with perturbations and uncertainty in dynamical systems. We are currently developing these companion techniques for translation to ecological problems: they will aid in the analysis of highly complicated perturbation structures (stability radii provide information on the range of perturbations that will provide a desired set of eigenvalues), and the graphical assessment of how prone PPM predictions are to transient effects (pseudospectra and spectral value sets gauge how well-conditioned is a matrix model under conditions of unstructured and structured parameter uncertainty, respectively).
The precision offered by transfer function analysis can also be captured by eigenvalue calculation of simulated perturbed PPMs (Morris & Doak 2002). To debate the relative computational expense of transfer function vs. eigenvalue simulation methods seems trivial given the computing power available to most users of PPMs, but we note that the relative efficiency of transfer function analysis increases with increasing dimension of the matrix, with relevance to the use of large PPMs for metapopulations, and numerical analysts acknowledge the existence of matrices for which eigenvalue computations are unreliable. The main strength of transfer function analysis, however, is that it provides not only an analytical solution, but also a generic framework for understanding perturbation effects, which will be more revealing of qualitative features than prevailing methods.
Population ecology can profit from the application of these tools to projection matrices. However, robust control must be refined so as to address specific problems. For example:
1Stochastic effects. Biological systems are prone to stochastic variation (Grant & Benton 1996; Engen & Saether 1998) that may obscure, prevent or alter asymptotic dynamics. Stochastic robust control techniques deal simultaneously with stochastic uncertainty and perturbation parameters to predict the range of possible future system behaviour.
2Non-linear transitions. Many biological systems are nonlinear. This is epitomised by density-dependent population growth (Grant & Benton 2000; Freckleton et al. 2003). Non-linear robust control must also be translated to ecology to deal with nonlinear projection dynamics.
Given the apparent ubiquity of stochastic environmental variation and density-dependent influences on natural populations, it perhaps seems strange to develop precise tools for the prediction of deterministic, exponential population dynamics. However, this is just the first step in the translation of the analytical tools of robust control to population ecology. The models that applied ecologists use should be simple enough to be easily understandable, and precise enough that expensive mistakes are avoided. We are not aware of explicit manipulative experiments that compare the relative value of different methods for population dynamic predictions based on PPMs. We hope that recognition of the transfer function approach will motivate such experiments.
Synthesis and application
Modern problems in population ecology, such as invasive genes and genotypes, habitat fragmentation, overexploitation and climate change, could impose life-cycle transition perturbations of sufficient magnitude to warrant precise, predictive perturbation analyses of system dynamics. Transfer function analysis offers an analytical approach to making the best predictions possible, and should stand alongside sensitivity and matrix simulation as a powerful tool for predictive empirical modelling in population ecology. The transfer function approach retains the simplicity of sensitivity and elasticity analyses, but emphasizes that the effect of perturbation on population responses depends not just on the specific life-history transition rate(s) being perturbed, but also on the magnitude of perturbation. Proponents of sensitivity analysis often claim that life-history perturbations are rarely large enough to warrant calculation of more than the approximate population response determined by sensitivity. However, we propose that modern pressures on natural populations of any organism may significantly alter life-history transitions, especially when we consider harvesting strategies and genetically modified traits such as pathogen resistance. Transfer function analysis is particularly useful when there is a ‘target’ rate of population increase to be achieved. For example, in pest control, what life-history stages should be removed to best cause population decline? In conservation, what vital rates should be increased (or decreased) to promote population increase? In harvesting, can we maximize the removal of certain life-history stages while maintaining a fixed population size (no net decline)? These are important questions that deserve precise analytical tools to help solve them.
The authors thank Dan Doak for contribution of data to this study, and James Cresswell, Martin Hoyle and anonymous referees for helpful comments on earlier versions of the manuscript.