Erratum et addendum: transient amplification and attenuation in stage-structured population dynamics


  • Stuart Townley,

    1. Mathematics Research Institute, School of Engineering, Computing and Mathematics, University of Exeter, Exeter, Devon, EX4 4QF, UK; and
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  • David J. Hodgson

    Corresponding author
    1. Centre for Ecology and Conservation, School of Biosciences, University of Exeter, Cornwall Campus, Tremough, Penryn, Cornwall, TR10 9EZ, UK
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*Correspondence author: E-mail:


  • 1Not all members of natural populations contribute equally to population growth or decline. Populations that are disturbed away from stable stage structure will amplify (i.e. get bigger than expected) and/or attenuate (i.e. get smaller than expected) in the short term.
  • 2We provide mathematical bounds for the magnitude of this amplification and attenuation, both in terms of absolute population change and population change relative to the long-term rate of population increase.
  • 3Our results correct an important error in an earlier analysis of transient population amplification, and provide new transient bounds for the analysis of population attenuation.
  • 4Synthesis and applications. Bounds on transient amplification and attenuation help population managers to gauge ‘worst case’ and ‘best case’ scenarios for the response of stage-structured populations to disturbance and management strategies. Such bounds help to create an envelope of possible future population scenarios around the mean, long-term predictions made by eigenvalues and eigenvectors of projection matrix models. Transient amplification, caused by stage structures biased towards reactive life stages, may be exploited by conservation managers wishing to boost population densities in the short term and may be avoided in pest species by stage-specific control strategies. Similarly, transient attenuation should be avoided by conservation managers and exploited by pest managers.


Traditionally, stage-structured models of population dynamics have been used to predict the long-term dynamics of populations with a stable stage structure. However, when natural populations are subject to environmental change and disturbances, their structure and dynamics may never settle to behaviour predicted by long-term projections of empirical models. Instead, populations can show transient amplification, attenuation or oscillations (Fox & Gurevitch 2000; Koons et al. 2005), with exogenous perturbations or disturbances resetting the process (Neubert & Caswell 1997; Neubert, Klanjscek & Caswell et al. 2004). In a previous study (Townley et al. 2007), we compared a set of mathematical bounds, using numerical linear algebra for population projection matrices (PPMs) (Caswell 2001; Hodgson & Townley 2004; Hodgson, Townley & McCarthy 2006) that help to describe how large populations may get in the short term (transient amplification), following a disturbance to the population's stage structure.


We regret an important typographical error in that paper (Townley et al. 2007, p. 1246, column 1), which claimed that the maximum population magnitude at any time in the future (relative to an initial magnitude of 1) was equal to the maximum population magnitude after one time-step (reactivity), raised to the power of time. In fact, reactivity raised to the power of time provides an UPPER bound for longer-term transient amplification. Using the induced-norm of the projection matrix to describe reactivity, our statement should have read:

image(eqn 1)

This correction helps to emphasize the fact that bounds on transient dynamics are only exact in special cases. For example, there exists a population stage structure which amplifies by a multiplicative factor of inline imageAinline image in one time-step, but this need not be the same stage structure that achieves maximum amplification (ρt, sensu Neubert & Caswell 1997) in any future projection time-step. Instead, transient bounds should be considered as tools which help to narrow the envelope of possible future population dynamic trajectories.

Addendum: indices of transient attenuation

Having fixed this mistake, we also take this opportunity to extend our list of transient bounds to describe transient population attenuation. For each index of population amplification, there is a partner index which bounds how small populations could get in the short term (assuming the projection matrix model is correct and well-parameterized). To emphasize these partnerships, we describe indices of amplification with over-bars, and indices of attenuation with under-bars (Table 1). We believe the most relevant measure of transient population magnitude is total population size/density, that is, the sum of the components of the population stage structure x(t), which is calculated using the 1-norm. For some attenuation indices, we use the minimum column sum of a matrix, which we call minCS in the absence of a relevant, established norm. Finally, we note that absolute transient amplification loses meaning in growing populations, as does absolute transient attenuation in declining populations. For this reason, our application of these bounds to a real population projection matrix (Fig. 1) studies transient amplification and attenuation relative to the short-term dynamics of a population initiated at stable stage structure. To achieve this, we divide the PPM by its dominant eigenvalue:  = A/λ, and apply transient bounds to Â. The new Kreiss bound for attenuation (based on Kreiss 1962) gains all the benefits of parametric dependence described in Townley et al. (2007): it is possible to study the impacts of demographic change, caused by management strategies, on this index.

Table 1.  A summary of metrics that can be used as indices or measurements of transient system magnitude in population projection models in discrete time. New symbols (following Caswell 2001; Townley et al. 2007): v is the dominant left eigenvector of A, also called the reproductive value vector, and vmax and vmin are its largest and smallest entries, respectively. w is the dominant right eigenvector. In the 1-norm, is a vector of zeroes except for a 1 in the same row as the largest value in v; v is its attenuation counterpart, a vector of zeroes except for a 1 in the same row as the smallest value in v. minCS represents the minimum column sum of a matrix. In all cases, inline imageThumbnail image of
Figure 1.

The transient envelope, and bounds on transient dynamics, for the population projection matrix of medium-high fecundity desert tortoises (Doak, Kareiva & Kleptetka 1994). Population size (y-axis) is plotted on a log scale to spread out bounds on attenuation visually, and to emphasize geometric relationships. The projection matrix was divided by its dominant eigenvalue so that all projected dynamics are relative to the expected population size (thick, solid, horizontal line) achieved by a population of tortoises initiated at stable stage structure, described by the dominant right eigenvector w. Thick solid curves describe the maximum and minimum population sizes achieved by any disturbance of the stage structure at time zero. Dotted lines represent one time-step amplification or attenuation, while dashed lines describe outer transient bounds: together these bounds contain all possible transient population sizes. Dot-dash lines represent the actual maximum amplification and attenuation for any projected time-step. Thin solid lines describe the upper and lower Kreiss bounds, which for this example coincide with asymptotic amplification and attenuation respectively. In other examples, K need not equal ρ.


Population dynamics of wild animals and plants are notoriously difficult to predict, even when those populations are not strongly influenced by density-dependent processes. External drivers of variation, such as unpredictable climates, natural disturbances, fluctuations in resource availability and attack by natural enemies, interact with internal drivers that include demographic stochasticity, age- and stage-structuring, and short-term (transient) responses to disturbance. When humans intervene with natural population dynamics, for example, by culling pests or releasing cohorts of individuals (whether accidentally or on purpose), a crucial task is to make useful predictions of the effect of each intervention.

There are two established methods for predictive population ecology, both of which rely on the creation of a predictive model. Given this model, and knowledge of the impacts of intervention, one could simulate population dynamics into the future, many times, for each set of initial conditions, and create a histogram-based envelope of possible outcomes. The alternative is an analytical approach which is less expensive of computations and offers deeper understanding of the model and its response to interventions. Population ecologists have only recently begun to use tools from numerical linear algebra that go beyond describing long-term average effects of disturbance and perturbations. Neubert & Caswell (1997) introduced the concept of reactivity and amplification (measures of short-term population size) to rival the more traditional use of damping ratios (measures of settling-times after disturbances) which were themselves based on eigenvalues and therefore asymptotic measures of transient effects. Recently, analytical techniques have been translated into ecology that allow the prediction of transient responses to fixed, known disturbances (Fox & Gurevitch 2000; Koons et al. 2005; Caswell 2007). Our contribution is to provide analytical tightening of the bounds which describe impacts on future population dynamics of the full range of possible disturbances and initial conditions faced by any population.

Why might these bounds be useful for applied ecologists? We identify two important applications. First, ecologists are now well aware that predictions of system dynamics require measures of uncertainty around the mean, equilibrium or asymptotic dynamics. In a statistical setting, these will often be confidence intervals. Even in the absence of stochasticity, stage-structured population models deserve their own envelopes that describe the possible impacts of transients. Given a prediction of, for example, extinction time for a population of a long-lived organism, these transient envelopes can now provide worst-case and best-case extinction times that surround the asymptotic prediction. Similarly, the rate of spread of an invasive weed will have an asymptotic value predicted by an eigenvalue, but its short-term dynamics could lie anywhere within the transient bounds and these deviations could influence dramatically the future density of the weed even after the transients have disappeared. There exist few published examples where the predictions of population dynamics have been tested in the field. In one classic system (Bierzychudek 1999), PPM eigenvalues failed to provide even a qualitative prediction of plant population growth; in such cases, attaching transient bounds and considering the impacts of disturbances on population stage structures could reveal the mechanisms driving observed dynamics.

The second application of our indices of transient bounds is that they are amenable to analytical perturbation analysis (Hodgson & Townley 2004). In other words, we can study the impact on the size of these bounds of any changes to the demographic transition rates contained in the PPM. Usually, such analyses are performed using sensitivities and elasticities, which are derivatives of the functional relationship between the index (whether it is an eigenvalue or a growth bound) and the magnitude of the demographic parameter. We prefer perturbation analyses that capture these nonlinear relationships without the need for simulations. One motivation for using the Kreiss bound is that it is amenable to such analyses (Townley et al. 2007). Hence, for example, one could ask how changes in rates of seed dormancy will influence the transient attenuation of weeds; or how conservation efforts that target juvenile vs. adult turtles (Crowder et al. 1994; Heppell, Crowder & Crouse 1996) will affect the potential for transient amplification of these populations. We believe there will be important applications of these transient bounds for captive-rearing strategies (in which short-term boosts in population density are desired) and for pest control (in which transient amplification should be avoided but attenuation could be very useful (Hastings, Hall & Taylor 2006)). Population managers could ask how changes in stage-specific demographic rates, caused by intervention strategies, affect not just the long-term predictions of population dynamics but also the potential for short-term growth and decline.


This research was supported by the Natural Environment Research Council, Leverhulme Trust and the European Social Fund.