Predicting transient amplification in perturbed ecological systems

Authors

  • STUART TOWNLEY,

    1. Mathematics Research Institute, School of Engineering, Computer Science and Mathematics, University of Exeter, Exeter EX4 4QF, UK; and
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  • DAVID CARSLAKE,

    1. Centre for Ecology and Conservation, School of Biosciences, University of Exeter, Cornwall Campus, Tremough, Penryn TR10 9EZ, UK
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  • OWEN KELLIE-SMITH,

    1. Mathematics Research Institute, School of Engineering, Computer Science and Mathematics, University of Exeter, Exeter EX4 4QF, UK; and
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  • DOMINIC MCCARTHY,

    1. Mathematics Research Institute, School of Engineering, Computer Science and Mathematics, University of Exeter, Exeter EX4 4QF, UK; and
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  • DAVID HODGSON

    1. Centre for Ecology and Conservation, School of Biosciences, University of Exeter, Cornwall Campus, Tremough, Penryn TR10 9EZ, UK
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Stuart Townley, Mathematics Research Institute, School of Engineering, Computer Science and Mathematics, University of Exeter, Exeter EX4 4QF, UK (fax + 44 1392 26-4067; e-mail s.b.townley@exeter.ac.uk).

Summary

  • 1Ecological systems are prone to disturbances and perturbations. For stage-structured populations, communities and ecosystems, measurements of system magnitude in the short term will depend on how biased the stage structure is following a disturbance.
  • 2We promote the use of the Kreiss bound, a lower bound predictor of transient system magnitude that links transient amplification to system perturbations. The Kreiss bound is a simple and powerful alternative to other indices of transient dynamics, in particular reactivity and the amplification envelope.
  • 3We apply the Kreiss bound to a discrete-time model of an endangered species and a continuous-time rainforest model.
  • 4We promote the analysis of transient amplification relative to both initial conditions and asymptotic dynamics.
  • 5Transient amplification of ecological systems, following exogenous disturbances, has been implicated in the success of invasive species, persistence of extinction debts and species coexistence.
  • 6Synthesis and applications. The Kreiss bound allows simple assessment of transient amplification in ecological systems and the response of potential amplification to changes in system parameters. Hence it is an important tool for comparative analyses of ecological systems and should provide powerful predictions of optimal population management strategies.

Introduction

Analyses of population and community dynamic models have tended to focus on asymptotic (long-term) model behaviour (Hastings 2004). Recognizing that natural dynamics tend to fluctuate through time, a historical debate (Kendall et al. 1999; Turchin 1999) has focused on the relative contribution of endogenous effectors of complicated dynamics (May 1974) and exogenous, stochastic causes of population (Bjornstad & Grenfell 2001) or community (Ives et al. 2003; Boyce, Haridas & Lee 2006) fluctuations. Recently, transients have gained theoretical and empirical attention as a third mechanism of complicated dynamics (Hastings 2001, 2004). Transient dynamics are now implicated in promoting species invasions and epidemics (Drake 2005), food web complexity (Chen & Cohen 2001; Caswell & Neubert 2005; Noonburg & Abrams 2005), the amplitude of population cycles (Costantino et al. 1998) and species co-existence (Nelson, McCauley & Wrona 2005). A better understanding of transient dynamics should allow their exploitation in pest management, harvesting and conservation strategies.

When natural systems are subject to environmental change and stochastic disturbances, their structure and dynamics may never settle to behaviour predicted by model asymptotics. Instead, populations and communities can show transient amplification, decrease or oscillation, with exogenous perturbations or disturbances resetting the process (Neubert & Caswell 1997; Neubert, Klanjscek & Caswell 2004). Transients can dominate the dynamics of systems that are described by ‘non-normal’ projection matrices. Non-normality is now well-established as a mechanism for strange system behaviour that defies explanation using eigenvalues (Trefethen et al. 1993; Trefethen & Embree 2005). Ecological projection matrix models are often extremely non-normal.

This paper introduces the Kreiss bound, an index of transient amplification of ecological dynamics based on stage-structured matrix models (Kreiss 1962), and develops a simple formula showing how this index depends on model parameters. Throughout, the simplicity and utility of the Kreiss bound is compared with two prevailing methods, reactivity and the amplification envelope (Neubert & Caswell 1997), however, we also refer to a wider array of potentially useful indices of transient dynamics (Table 1).

Table 1.  A summary of metrics that can be used as indices or measurements of transient system magnitude in ecological projection models in discrete or continuous time, computed using the 1-norm. D, discrete time models; C, continuous time. New symbols (following Caswell 2001): v is the dominant left eigenvector of A, also called the reproductive value vector. w is the dominant right eigenvector. In the 1-norm, v̂a is a vector of zeroes except for a one in the same row as the largest value in v
IndicatorWhat is it?UseParametricPrecisionComments
λ(A)Max eigenvalue of A
image
YesPoorUseful only if A is normal
Transient bound Pt
image
Upper-bounds future system magnitudeYesGood
image
Asymptotic amplification ρ(A)
image
image
YesGoodExcellent for t >> 1
Asymptotic amplifier
image
Often achieves ρtYesGoodAsymptotic amplification, t→∞
D, reactivity
image
image
YesGood for t << ∞
image
C, reactivity µ(A)
image
image
YesGood for t << ∞eµt useful when combined with Pt
Amplification envelope ρt
image
image
NoExcellentComputational, no meaningful parameterization
Maximum amplification ρmaxmaxt0ρt
image
NoExcellentComputational, no meaningful parameterization
Kreiss bound inline image
image
Lower bounds ρmax (D: θ= 1; C: θ= 0)YesGoodCompromise

Projection matrix models

The tools we describe apply to linearized, stage-structured projection models, including (i) population projection matrices (PPM), introduced to ecology by Leslie (1945) and Lefkovitch (1965) and recently developed further by Caswell (2001) and Morris & Doak (2002); (ii) compartment models of ecosystems (McGinnis et al. 1969); (iii) food web models (Pimm & Lawton 1977); and (iv) sets of difference/differential equations linearized to explore dynamics near equilibria (Gurney & Nisbet 1998).

The basic model structure is:

image( eqn 1)

Here x(t) is an s-dimensional vector representing the stage-structured system profile (e.g. the density of individuals in each stage class of a population). A is an s-by-s matrix containing rates of transition between stage classes: in discrete and continuous time, these rates are geometric and exponential, respectively. We can therefore write predictions of future stage structure as:

image(eqn 2)

Asymptotic analyses of projection models rewrite them using the eigenmode expansions:

image( eqn 3)

where the αi are scalars resulting from an eigenvector expansion of the initial population profile. Hence s eigenvalues, λi, and s eigenvectors, vi, describe the asymptotic behaviour of the model. For primitive non-negative projection matrices in discrete time, the Perron–Frobenius theorem states that the eigenvalue of largest magnitude (λmax) is real and positive, and represents the asymptotic, geometric rate of increase of the system. In continuous time, the eigenvalue with maximal real part is real and represents the asymptotic exponential rate of increase. From any non-negative, real, initial condition, system dynamics will eventually settle to this asymptotic rate with an asymptotically stable stage structure defined by the dominant eigenvector. Subdominant eigenvectors define the asymptotic rate at which system dynamics settle to the asymptote, but the scalar multipliers αi can have a huge impact on the actual transient dynamics. See Caswell (2001) and Horn & Johnson (1999) for further mathematical details.

Prevailing techniques

The analysis of transient dynamics aims to characterize what happens before a system settles to the asymptote after a disturbance (a change to the stage structure, x(0)) or perturbation (a change to the projection matrix, A), particularly how big or small the system becomes in the short term. Indices of return times, i.e. how long a system takes to settle to asymptotic dynamics after a disturbance, include the damping ratio, λ1/|λ2|, of a PPM (Caswell 2001) and the resilience (Holling 1973), –Re(λ1), of food web models (Pimm & Lawton 1977; Loreau 1994; Chen & Cohen 2001). Both of these indices are asymptotic, measuring the rate of return to stability as t→∞. Transient system dynamics that differ from asymptotic dynamics cannot be detected by eigenvalue analysis; instead, we need indices of transient system magnitude. Relevant measures of system magnitude at any time t include the 1-norm, 2-norm or ∞-norm of x(t) (we use the standard notation ‖x1,2 or ), and/or the respective induced-norm of At or eAt (Horn & Johnson 1999). The 2-norm describes the length of a vector in Euclidean space, and is the norm most amenable to mathematical manipulation (Neubert & Caswell 1997). However, it is not easy to ascribe a biological meaning to the 2-norm. The 1-norm is the sum of a non-negative vector (or the maximal column sum of a projection matrix), with a clear interpretation as the total number or density of ‘individuals’ across all stage classes. The ∞-norm is the largest entry in a vector (or the maximal row sum of a projection matrix), which may be useful, for example, when specific ages or stages of organisms are to be harvested from a population.

For the case of continuous time models, Neubert & Caswell (1997) provided two indices of transient system amplification, calculated using the 2-norm. Reactivity (following the notion of ‘initial growth rate’ from Lozinskiï & Dahlquist cited in Hinrichsen & Pritchard 2005), denoted by µ(A), is the maximum possible instantaneous rate of amplification. It is defined by:

image(eqn 4)

The discrete-time reactivity is simply ‖A‖ (using whatever norm is most relevant). However, maximal transient amplification may not be realized in the first time step. The maximum amplification, ρmax, is the largest value of the amplification envelope, ρt, where ρt = ‖eAt‖ in continuous-time and ρt = ‖At‖ for discrete-time models. We call the time of maximal amplification t*. Reactivity is parametric and therefore readily amenable to sensitivity analysis (Neubert & Caswell 1997) but maximum amplification requires exploration of ρt for all values of t and is not so readily parameterized. Reactivity does give immediate bounds for ρt; in discrete time ‖At‖ = ‖At, in continuous time ‖eAt‖ = eµ(A)t. There are many other indicators that attempt to capture or estimate ρt. We summarize these, and their usefulness, in Table 1. Two useful and novel parametric bounds for ρt are identified: the amplification envelope must lie below Pt and above the Kreiss bound, K.

We promote the use of the Kreiss bound (K) as a predictor of the possible amplification of a disturbed natural system represented by a linearized model. This is a time-independent lower bound for transient amplification (Kreiss 1962). It avoids the ‘instantaneous’ or ‘first time-step’ weakness of reactivity. Also, it is relatively insensitive to peaks and troughs in transient dynamics (. 1). Finally, it is calculated using the resolvent of A (Horn & Johnson 1999), hence its dependence on model parameters can be calculated via the Sherman–Woodbury–Morrison equation (Henderson & Searle 1981). This analytical description of parametric dependence allows powerful perturbation analyses, even for perturbation effects that are distributed (and possibly correlated) across the projection matrix. We show that the Kreiss bound mimics the behaviour of maximum amplification in response to system perturbations, but is easier to calculate.

Figure 1.

Transient dynamics of disturbed and perturbed populations of desert tortoise, relative to initial density (a) and to asymptotic population decline (b). Simulated trajectories of tortoise populations were initiated at a total density of 1 but with stage structures corresponding to a stable-stage structure (lower dotted lines) or with all individuals in a single stage (dashed lines). The amplification envelope, caused by an initial bias of stage-8 tortoises, is highlighted as the upper dotted line. Superimposed are three indices of transient amplification: the maximum point of the amplification envelope (ρmax), reactivity (reac) and the Kreiss bound (K*). (c) and (d) show the relationship between r and K(r) for Kreiss bound analyses relative to initial conditions and asymptotics, respectively.

We develop these concepts in the context of a discrete-time PPM for the desert tortoise Gompherus agassizii (Doak, Kareiva & Kleptetka 1994) and a continuous-time ecosystem model (McGinnis et al. 1969; Neubert & Caswell 1997). These two examples serve to highlight the utility of the ‘resolvent’ approach, the versatility of the Kreiss bound, and the relevance of different norms for the measurement of the magnitude of ecological systems.

Transient dynamics of the desert tortoise PPM

The population projection matrix for a population of desert tortoises with medium-high fecundity (Doak, Kareiva & Kleptetka 1994) is:

image

Simulation analyses use equation 2 to assess the range of transient dynamics in disturbed populations before they settle to asymptotic dynamics (Fig. 1a). These simulations highlight three phases typical of non-normal population dynamics. There is initial growth or reactivity, followed later by a transient amplification, followed by eventual settling to asymptotic growth or, in this case, decline. Estimates of ρt attempt to capture these three phases.

The modelled tortoise population declines asymptotically (λ1 = 0·958). Simulated populations are initiated with a total density of 1, and are disturbed so that all members of the initial population are aggregated in a single stage class c, i.e. x(0) is a vector of zeroes except for a 1 in row c. We compare these disturbed dynamics to the stable dynamics expected of a population initiated by a stage structure proportional to the dominant eigenvector of A. For this example, we predict the total density of tortoises in the population using the 1-norm. Reactivity is ‖Atort1 = 3·43. In less mathematical terms, the system shows maximal instantaneous growth when all individuals at t= 0 are in the stage class that produces and/or retains the most new individuals. This is stage class 8, which has the largest column sum of Atort. The maximum transient amplification of this population is achieved by a tortoise population initiated by a cohort of stage-8 tortoises. The amplification envelope is maximized at t* = 4, with an amplification inline image

It is natural to compare transient dynamics with predictions based on eigenvalues, in other words how ‘big’ the system is at time t relative to its magnitude had it started out with a stable stage structure. To do this we divide A by λ1. Reactivity relative to asymptotics is ‖Atort11 = 3·58 (Fig. 1b). The maximum transient amplification of this population, relative to asymptotics, is achieved by a cohort of stage-8 tortoises: the standardized amplification envelope is maximized at t* = 5, with maximum amplification inline image

calculating the kreiss bound

Norms of the powers of A grow geometrically and describe the transient magnitude of the system. We wish to compare the norm of At to a baseline geometric growth rate described by θt. The Kreiss bound provides a lower bound for this transient magnitude. To calculate the Kreiss bound, choose r > θ. If:

image(eqn 5)

where I is the identity matrix of same dimension as A. The choice of θ will affect the interpretation of the Kreiss bound. In discrete time, choosing θ= 1 provides a lower bound for transient population increase relative to initial conditions (i.e. ‖x(0)‖, which is normalized to equal 1 in our simulations) (Fig. 1a). Choosing θ=λ1 = 0·958 provides a lower bound on the possible future amplification, standardized relative to asymptotic dynamics (Fig. 1b). In continuous time, we use θ= 0 to study transient amplification relative to ‖x(0)‖, and θ= Re{λ1} (the real part of the dominant eigenvalue) for amplification standardized relative to asymptotics. Standardizing by the dominant eigenvalue is useful when studying systems that increase asymptotically.

The magnitude of K changes with r (Fig. 1b,d). To find the best lower bound, we use:

image(eqn 6)

The matrix (rI – A)−1 is the resolvent of A, hereafter called R. The value of r that maximizes K is called r*, and R* is the resolvent using r*. R(r) describes R as a function of r. In biological terms, the parameter r* represents the growth rate, exceeding θ, that is relatively easiest to achieve by perturbing the projection matrix.

dependence of k* on perturbations to a

Our general approach to matrix perturbations (Hodgson & Townley 2004) is to write the perturbed matrix as:

image(eqn 7)

where D and E are matrices that define the structure of the perturbation (i.e. which entries in A are to be perturbed) and Δ is a matrix that defines the magnitude(s) of the perturbation (i.e. how much each entry is perturbed). The resolvent of the perturbed matrix is Rpert= (rI − A −DΔΕ)−1, so an unwieldy formula for the Kreiss bound of a perturbed matrix is:

image(eqn 8)

To describe Kpert in terms of the unperturbed projection matrix, and also highlight the dependence on Δ, we use the Sherman–Morrison–Woodbury formula (Henderson & Searle 1981). This states that for compatible matrices T, U and V:

image(eqn 9)

Choosing T= (rI – A), U=D, and V=–ΔE, we obtain:

image( eqn 10)

approximations based on unperturbed a

Predicting the effects of perturbation to A on K* still requires maximization over a range of r, just as calculations of ρmax for a perturbed A must be maximized over t. Approximations can be made by using r* and t* of the unperturbed matrix A:

image(eqn 11)
image(eqn 12)

simple matrix perturbations

We consider a special case of equation 7 in which a single entry in A, ai,j, is perturbed by a scalar δ. In this case, D and E are vectors of zeroes except for ones in positions i and j, respectively. Now the induced 1-norm of a matrix can be computed via pre-multiplication by a row vector of ones (C) and post-multiplication by a column vector (B) containing zeroes except for a one in row m where column m of (rI – A)−1 has the greatest absolute sum. Hence the 1-norm Kreiss bound for a simple perturbation is:

image(eqn 13)

Because δ is scalar:

image( eqn 14)

When δ is small, B will not change and (IERD)−1I, so that:

image(eqn 15)

Thus we can write the Kreiss bound of a perturbed matrix in terms of the magnitude of perturbation and the resolvent of the unperturbed matrix. Note also that for small δ, r* and R* of A will remain valid, but this can be checked by returning to the previously calculated R(r) to find r* and R* of Apert, hence inline image.

A useful corollary is that similar calculations apply when using the ∞-norm, only with C a row vector containing zeroes except for a one in row n where row n of (rI – A)−1 has the greatest absolute sum, and B a column vector of ones.

sensitivity of k to changes in a

Although equation 15 describes the non-linear relationship between K* and δ, a useful index of this relationship is the derivative of K* with respect to δ (i.e. sensitivity of K* to change in ai,j):

image( eqn 16)

kreiss bound for the desert tortoise model in discrete time

We calculated R for a range of r > 1 (Fig. 1c,d), then identified K* using equation 6. Values of r and their corresponding R(r) were stored for further analysis. For the desert tortoise PPM, the Kreiss bound relative to initial conditions is inline image = 3·83 (occurs at r* = 1·20), which lies between the reactivity and maximum amplification indices (Fig. 1a). Measured relative to asymptotic decline, inline image = 5·21 (at r* = 1·004), which again lies between reactivity and maximal amplification (Fig. 1b). This raises the question why calculate an index that describes neither immediate nor maximal transients? The answer has four parts: first, like ρmax, K* is insensitive to the transient fluctuations of ρt seen in many matrix projections; secondly, K* tracks the behaviour of ρmax in response to matrix perturbations; thirdly, we can use the stored information on R(r) to characterize precisely the dependence of K* on matrix perturbations; and fourthly, Kreiss’ theory guarantees that transient amplification will equal or exceed K* for some initial stage structure.

Indices of transient amplification respond to perturbations of A in a (sometimes dramatically) non-linear fashion. In Fig. 2a,b, we show the response of K* to perturbations in the rate of progression from stage 2 to stage 3 tortoises (a3,2; a transition rate to which λ1 is highly sensitive). Figure 2a shows the response of K* to the life cycle perturbation relative to initial conditions, while Fig. 2b shows transients relative to asymptotic decline. Figure 2c,d describes the response of K* to perturbations of the rate of stasis of stage-8 tortoises (a8,8; the transition rate which, when locally perturbed, causes the biggest change in ρmax), relative to initial conditions and asymptotics, respectively. In all cases we performed simple reactivity calculations and the more complicated maximizations of ρmax, across the same perturbed-parameter space, for comparison. The range of perturbations was chosen so that the biggest negative change reduced the relevant transition rate ai,j to zero, and the biggest positive change resulted in λ1 = 0·99 (i.e. close to the point of population stability). K* tracks very closely the response of ρmax to changes in A. This is generally true, based on calculations for other transition rates and our tests on many other PPM. Reactivity (at least in the 1-norm) is influenced little by perturbations unless they either change a transition rate in the column of A with maximal sum (causing a linear response to perturbation) or cause a new column sum to exceed the maximum column sum of the unperturbed matrix (causing an abrupt change from no response to a linear response) (e.g. Fig. 2c,d at δ=–0·583). When using θ=λ1, these effects are compounded by the effect of perturbation on λ1: the slow decline in reactivity with increasing δ (Fig. 2b) is caused by a concomitant, gradual increase in λ1 in response to perturbations (hence θ changes with magnitude of perturbation in equation 6).

Figure 2.

The response of the Kreiss bound (solid line), reactivity (dotted line) and maximum amplification (dashed line) to perturbations of transition rates in the desert tortoise PPM. Transient responses to perturbations were assessed for transition rates a3,2 (a, b) and a8,8 (c, d), relative to initial conditions (a, c) and asymptotic decline (b, d).

The response of the Kreiss bound to matrix perturbations provides a focus for some interesting possibilities in the design of conservation management strategies. The asymptotic rate of increase is most sensitive to small changes in transition rate a8,7. However, transfer function analysis (Hodgson & Townley 2004) shows that asymptotic population growth is best achieved by increasing a7,7. In some situations, transient amplification may be a useful short-term target while plans are made for strategies that will improve long-term rates of increase. The non-linear response of K to perturbations (Fig. 2b–d) suggests that a large increase in a8,8 may be a useful short-term management option.

Food web model in continuous time

To demonstrate the versatility of the Kreiss bound, we extend its use to the continuous-time compartmentalized rainforest model of McGinnis et al. (1969), modified by Neubert & Caswell (1997). In this model, matter flows through different ecological compartments with measurable rates: compartments (stage classes) 1–9 represent leaves, stems, litter, soil, roots, fruits, detritivores, herbivores and carnivores, respectively.

image

In continuous time, reactivity µ(A) is the maximal instantaneous rate of system change in response to disturbance of x(0). The amplification envelope, ρt, is the maximal system magnitude at time t > 0, i.e. ‖eAt‖, and inline image This raises two numerical issues. First, ρt must be maximized over a continuum of t. Secondly, the exponential formulae for ρt and ρmax are prone to numerical errors. The Kreiss bound tends to be maximized over a much smaller range of r, and, being rational in parameters, does not suffer any potential problems caused by exponentiation.

A third issue that affects both amplification envelope and Kreiss bound calculations is which norm provides the best description of system magnitude. Depending on model formulation, the 1-norm could describe the total number or density of biological individuals. However, members of different species or trophic levels may not be considered equivalent. The ∞-norm could describe the density of the numerically dominant system component. Certain combinations of 1- and 2-norms may be useful as indices of diversity. Choice of norm will be application-specific: here we give examples using 1-, 2- and ∞-norms (Fig. 3b–d). We graph transient dynamics (Fig. 3a) and indices of transient amplification (Fig. 3b–d), relative to initial conditions.

Figure 3.

Transient dynamics of a rainforest compartment model. The rainforest model was projected through time (a), based on a stable-stage structure (lower dashed line) and the initial biased structure that provided maximal transient amplification (upper dashed line). Superimposed are two indices of transient amplification: the Kreiss bound (K*) and the maximal amplification (ρmax). (b–d) show the response of K* (solid lines) and ρmax (dashed lines) to perturbations of single transition rates in the projection matrix: a3,8, a4,7 and a6,6, respectively. In each case, transient amplification was measured relative to initial conditions in the 1-, 2- and ∞-norm.

Working first in the 2-norm (following Neubert & Caswell 1997), the reactivity of Aforest= 65·44, indicating a highly reactive system despite asymptotic decline, but this rate of amplification is extremely short-term. The Kreiss bound of the rainforest model, relative to initial conditions, is:

image

which is close to ρmax,θ = 0 = 2·81. K was maximized at r* = 0·06. In contrast, ρ was maximized at t* = 6·84. There is a striking fit between the responses of ρmax and K* to perturbations of Aforest. We chose to perturb a3,8, a4,7 and a6,6, to display a range of perturbation responses in each norm. Given the close fit between K* and ρmax (Fig. 3b–d), our main message from this model is that K* calculations required only equation 6 and an initial computation of R(r), while ρmax required intensive exponentiations and maximizations over t for all perturbation magnitudes. K* is therefore both simple and useful as a lower bound predictor of transient system amplification.

The Kreiss bound relative to asymptotic decline is:

image

which is exactly inline image. This equality is caused by the largest possible amplification, relative to asymptotics, occurring at t=∞. This is precisely equivalent to
inline image according to the final value theorem for Laplace
transforms (Schiff 1999). In this example, the largest amplification is actually an asymptotic amplification that is precisely captured by the Kreiss bound. We have not graphed the dependence of K* relative to asymptotics on perturbations. Indeed asymptotic decline is so slow as to make the graphs nearly identical to Fig. 3b–d. Furthermore, K* and ρmax are completely equivalent across all magnitudes of life-cycle perturbations.

Discussion

Ecologists now recognize the importance of transient dynamics in stage-structured biological systems (Hastings 2001, 2004). Asymptotic analyses describe long-term stability, while predictions of short-term dynamics often require intensive, projection-based simulations and/or analytical manipulations of the eigenmode expansion (Fox & Gurevitch 2000; Yearsley 2004), or the use of matrix calculus (Caswell 2007). We now require useful and simple indices of transient dynamics that can be used to predict and compare the response of ecological systems to disturbance and perturbation. While reactivity and the amplification envelope are precise indices of instantaneous and medium-term transients (Neubert & Caswell 1997), the former is usually too immediate and the latter is not readily amenable to simple perturbation analyses. Instead we promote the utility of the Kreiss bound, now favoured by applied mathematicians in the fields of engineering, fluid dynamics and climatology (Trefethen & Embree 2005), especially when used alongside several other parametric bounds for transient amplification. Simple numerical linear algebra allows the assessment of possible future system amplification in response to exogenous disturbances, and is amenable to predicting the effects of system perturbation.

We have used the desert tortoise and forest food web examples for purposes of illustration. Uses of the Kreiss bound in real ecological applications will be system- and problem-specific. However, we suggest that a better understanding of transient amplification could be exploited in captive-rearing programmes (Balmford, Mace & Leader-Williams 1997), the design of optimal-harvesting strategies (Acevedo & Waller 2000) and the risk assessment of invasive species (Bullock 1999). The use of vector and matrix norms promotes a better understanding of how to measure the magnitude of natural systems. In many systems, total number or density may be the goal, in others we may seek to maximize biodiversity or perhaps the density of a particular system component. The ease of standardization of the Kreiss bound, allowing us to assess transient amplification relative to initial or asymptotic conditions, promotes consideration of the type of amplification that is to be studied.

Exploration of amplification relative to asymptotic behaviour has revealed a novel feature of linearized system dynamics, which we call asymptotic amplification and denote by ρ: given an appropriately biased, but small stage-structured, perturbation away from equilibrium, the magnitude of the perturbed population profile will asymptotically exceed the magnitude of a population initiated at stable-stage structure. In general, ρ = ‖v‖/vTw, where w is a normalized stable age structure and v is the reproductive value. In reality this relies on the assumption of model linearity but it could provide an important index of long-term effects of transients.

Given the recent vast increase in computing speed and power, critics often query the need for parametric indices of system dynamics, particularly in low-dimensional systems (few published PPM contain more than 10 stage classes). We dispute this for four reasons. First, even low dimensional systems can be dynamically unpredictable to the untrained eye. Secondly, simulation analyses of ecological perturbations and disturbances can never lend the same deep level of understanding as analytical relationships. Thirdly, ecological projection matrices can be of very high dimension: examples include metapopulation projection models (Hunter & Caswell 2005), discretized integral projection models (Ellner & Rees 2006) and population genetic projection models (Kruger & Lindstrom 2001). Fourthly, parametric links between ecological dynamics, perturbations and disturbances should extrapolate to any future developments in empirical population dynamics.

A common criticism of asymptotic (and even medium-term) analyses of linearized models is that they ignore non-linear dynamics (Benton & Grant 1999). A prime example in ecology is the influence of density dependence, even in declining populations. However, the analysis of models linearized near equilibria is still standard practice (Hastings 2004) and is often the best we can do in the absence of detailed information on the intensity and demographic details of density dependence. Adaptations of algebraic indices of transients for non-linear situations are underdeveloped, but for now we point out that transient amplification, predicted from linearized models, could exaggerate density-dependent effects: a problem that could not be predicted from asymptotic analyses. We intend to develop uses for Kreiss bound and transfer function analysis for non-linear processes in ecology. Given that different norms predict different severities of transients, we believe that a better understanding of links between density dependence and system structure will guide better informed choices of which norm to use in the study of transient and asymptotic system magnitudes. Predictions of system-level responses to perturbations can be used to assess the susceptibility of system behaviour (i.e. both transient and asymptotic dynamics) to non-linear effects, errors in parameterization and/or model construction (Trefethen & Embree 2005). There is also the potential to develop ‘structured’ Kreiss bounds that use simple matrix algebra to define the subset of biologically feasible disturbances to x(t). Furthermore, the tools offered by transfer function and Kreiss bound algebra easily extend to pseudospectrum analysis, which is becoming established as superior to eigenvalue analyses in fields as diverse as fluid dynamics, oceanography and climate modelling (reviewed in Trefethen & Embree 2005).

Acknowledgements

The authors thank Paulette Bierzychudek, Hal Caswell, Tim Benton, Richard Rebarber and three anonymous referees for constructive criticism of earlier versions of this work. Thanks also to Stephen Pring, Jonathan Armond and Chris Lloyd who contributed to an undergraduate workshop on Kreiss bound mathematics. This research was supported by the Natural Environment Research Council, Leverhulme Trust and the European Social Fund.

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