Boom or bust? A comparative analysis of transient population dynamics in plants


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1. Population dynamics often defy predictions based on empirical models, and explanations for noisy dynamics have ranged from deterministic chaos to environmental stochasticity. Transient (short-term) dynamics following disturbance or perturbation have recently gained empirical attention from researchers as further possible effectors of complicated dynamics.

2. Previously published methods of transient analysis have tended to require knowledge of initial population structure. However, this has been overcome by the recent development of the parametric Kreiss bound (which describes how large a population must become before reaching its maximum possible transient amplification following a disturbance) and the extension of this and other transient indices to simultaneously describe both amplified and attenuated transient dynamics.

3. We apply the Kreiss bound and other transient indices to a data base of matrix models from 108 plant species, in an attempt to detect ecological and mathematical patterns in the transient dynamical properties of plant populations.

4. We describe how life history influences the transient dynamics of plant populations: species at opposite ends of the scale of ecological succession have the highest potential for transient amplification and attenuation, whereas species with intermediate life history complexity have the lowest potential.

5. We find ecological relationships between transients and asymptotic dynamics: faster-growing populations tend to have greater potential magnitudes of transient amplification and attenuation, which could suggest that short- and long-term dynamics are similarly influenced by demographic parameters or vital rates.

6. We describe a strong dependence of transient amplification and attenuation on matrix dimension: perhaps signifying a potentially worrying artefact of basic model parameterization.

7.Synthesis. Transient indices describe how big or how small plant populations can get, en route to long-term stable rates of increase or decline. The patterns we found in the potential for transient dynamics, across many species of plants, suggest a combination of ecological and modelling strategy influences. This better understanding of transients should guide the formulation of management and conservation strategies for all plant populations that suffer disturbances away from stable equilibria.


The dynamics of natural systems are highly complex and as such often prove difficult to predict. Yet, it is in the interest of ecologists to be able to predict the natural dynamics of ecological systems. Populations have received perhaps the most attention in the ecological modelling literature, with population projection matrices (PPMs) emerging as a prevailing modelling tool. This is hardly surprising, considering their wide application in pest control (Smith & Trout 1994; Shea & Kelly 1998; Hastings, Hall & Taylor 2006; Dudas, Dower & Anholt 2007), harvesting (Cropper & DiResta 1999; Souza & Martins 2006) and conservation (Price & Kelly 1994; Esparza-Olguín, Valverde & Vilchis-Anaya 2002; Linares et al. 2007), and although there is an increasing interest in habitat- and ecosystem-level practice (Harding et al. 2001), the population still remains a popular target for management. Often population dynamics defy simple model predictions, and current explanations for fluctuations in dynamics of populations range from deterministic chaos (Hastings et al. 1993) to environmental stochasticity (Dennis & Costantino 1988) and various combinations of these (Dennis et al. 1997, 2001; Bjørnstad & Grenfell 2001).

Transient dynamics have recently emerged as a further explanation for unpredictable population behaviour (Hastings 2001). Typically, ecological systems are not at equilibrium, and disturbances are integral to ecosystem function (Wallington, Hobbs & Moore 2005). As such, populations may be subject to frequent exogenous disturbances to stage structure (Koons et al. 2005; Townley et al. 2007; Townley & Hodgson 2008), such as weather events, disease, human impacts, invasive species or migration. Following a disturbance, short-term transient dynamics often prove to be radically different from long-term trajectories (Caswell 2007). Thus, in reality, population dynamics are likely to be dominated by transient responses to exogenous disturbances (Townley et al. 2007) – deterministic reactions to largely stochastic events. Additional complications arise because studies of plant demography usually last only a few years (Jongejans et al. 2010), and short-term goals are often the norm in management strategies (Hastings 2004).

With only a relatively recent interest in transients and their quantification, PPM models have commonly undergone asymptotic analyses (but see Hastings 2004; Koons et al. 2005; Townley et al. 2007; Caswell 2007; Townley & Hodgson 2008; Maron, Horvitz & Williams 2010) – eigendata of matrices, used to describe such long-term trends, have proven analytically tractable and conceptually simple (Caswell 2001). Studies have considered the asymptotic responses of plant populations to disturbances such as fire (Hoffmann 1999), grazing (O’Connor 1993) and hurricanes (Batista, Platt & Macchiavelli 1998). In addition to this, there are established means of analysing the contributions of demographic parameters to long-term growth rates in the forms of sensitivity and elasticity analyses (Caswell 2001) and, more recently, transfer function analysis (Hodgson & Townley 2004). Many studies have utilized such analyses, with results often advocated for use in informing policy and management of species (e.g. Fiedler 1987; Menges 1990; Nault & Gagnon 1993). Larger comparative analyses have found general patterns according to life history (Silvertown et al. 1993; Franco & Silvertown 2004; Burns et al. 2010). But, if transients do dominate ecological systems, then policy, management and conservation decisions based on asymptotic analyses may be severely limited, risking the waste of resources on suboptimal population management strategies. Indeed, in plants, there is evidence that long-term (asymptotic) trends in population models can provide very poor predictions of the true long-term behaviour of populations over time (Bierzychudek 1999).

Transients and their sensitivities, however, have proven to be difficult to quantify. Published methods of transient analysis have often either lacked applicability to every PPM model (Neubert & Caswell 1997) have needed large amounts of demographic information (Fox & Gurevitch 2000) or have required intensive computation (Yearsley 2004; Caswell 2007). The largest drawback of published methods of transient analysis is that none have truly managed to divorce the need for knowledge of initial population structure – something that asymptotic analysis benefits hugely from. Population ecologists have hence been ill-equipped to utilize transient analysis to inform management and conservation strategies. Thus, although now gaining greater attention in the literature (e.g. Drake 2005; Koons et al. 2005), transients have yet to gain wide popularity in practice.

However, Townley & Hodgson (2008) recently developed a computationally simple new set of indices that builds in part on the work of Neubert & Caswell (1997), aimed at providing a means of calculating best- and worst-case scenarios following disturbance. These included the Kreiss bound, a flexible parametric index that may be calculated independently of initial population structure. They also extended transient measurements to include both population amplification (transient increase in population size or density) and attenuation (transient decrease in population size or density) following disturbance events. It is these indices that we employ in this study. We have chosen not to analyse other established indices of transient population size or density, such as population momentum and inertia (the long-term population amplification following perturbations to demographic rates; see Koons, Holmes & Grand 2007). These differ from our indices primarily in that they measure responses to specified disturbances or perturbations – here, we focus instead on indices of maximal transient amplification and attenuation as measures of the overall transient dynamical properties of populations.

This article reports the first large-scale analysis of global patterns in transient dynamical properties of populations. We scrutinize a data base of models for 108 plant species with the aim of detecting deterministic biological and mathematical signals in patterns of transient dynamics. With relevance to plant ecology, we describe how life history influences the transient dynamical properties of populations. Our results also indicate relationships between magnitudes of transient departure from long-term growth and long-term growth itself, perhaps suggesting that short- and long-term dynamics are similarly influenced by the vital rates (e.g. survival, growth, fecundity) of members of populations. In an attempt to further inform the building of more predictive models and reliable analyses, we describe dependence of transients on matrix dimension and explore the influence of matrix structure on transients.

Materials and methods

Data base

Population projection matrices were collected from the literature, with the current data base of plant PPMs at the time of analysis numbering 500 matrices for 113 plant species of 11 different Subclasses and 52 Families. The taxonomy for each species was found using Tudge (2000) for the taxonomic level of Order and above and Systema Naturae 2000 (Brands 1989–2005) for taxonomic levels below Order.

One matrix was chosen for each species for use in analyses. This ‘species reference matrix’ was, where possible, an average of a number of subpopulation matrices of spatial and/or temporal replication. Where it was not biologically justifiable to take an average, the selected species reference matrix was chosen as the matrix for the (sub)population subject to conditions that most closely resembled the species’ ecological ‘norm’ (e.g. before, rather than after, hurricane disturbance). Matrices with redundant seed stages (the ‘seeds’ problem, highlighted in Caswell 2001, pp. 60–62) were corrected in the appropriate manner.

Each matrix was tested for reducibility using the argument that a square matrix A is irreducible if, and only if, (I + A)s−1 is positive (i.e. every element of the new matrix is greater than zero), where I is the identity matrix of same dimension as A, and s represents the dimension (number of columns or rows) in the matrix (Caswell 2001). Reducible matrices were not used in analyses because they often represent biologically implausible life cycles and defy asymptotic analysis. One hundred and eight species reference matrices remained after reducible matrices were removed (see Appendix S1 in Supporting Information for a full list).

Indices of transient dynamics

The indices of transient dynamics used were measures described by Townley & Hodgson (2008). The method of calculation uses theoretical stage-biased disturbances to initial population structure (i.e. with all individuals in a single stage class and a density of 1), thus modelling extreme transient dynamics of populations in response to such disturbances. We studied transient amplification and attenuation of each population relative to the long-term growth or decline predicted by the dominant eigenvalue of its PPM. This avoided the problem of there not being any upper bounds on the transient growth of an asymptotically growing population, or any lower bounds on the transient attenuation of an asymptotically declining population. Hence, all our transient indices describe how much bigger or smaller a population could get, relative to how big it would be if it started out at stable stage structure. This is equivalent to measuring the dynamics of a geometrically weighted population λtN(t), where N(t) is the population size or density at time t. In practice, we did this by measuring transient indices, not of the PPM A itself, but of the ‘standardized’ PPMinline image(equal to A divided by its dominant eigenvalue λ). This standardized λ of all inline image to be 1, allowing direct comparison of both amplified and attenuated dynamics across all populations (Fig. 1). This particular method also has the advantage of removing the effect of the dominant eigenvalue on the indices that we measure, allowing us to infer relationships between those indices and the asymptotic growth rate of the population as predicted by the PPM A.

Figure 1.

 Graphical representation of the indices of transient dynamics used in analyses. Each solid line represents a single stage-biased projection (of the standardized matrix) initiated with a total density of one (i.e. each projection starts with all individuals clustered in a single stage class: a population initiated at stable stage distribution would follow a flat line at y = 1). Therefore, in this example from a six-stage PPM, there are six projections. The vertical axis is on a logarithmic scale to illustrate the multiplicative nature of the measurements. Note that maximum amplification inline image and reactivity inline image may or may not result from the same initial stage structure, which is also true for first-timestep attenuation inline image and maximum attenuation inline image and inline image and inline image. Example PPM used for projection is for the palm Iriartea deltoidea (Pinard 1993).

Six indices were calculated for each PPM, with three indices describing amplified dynamics and three analogous indices describing attenuated dynamics. Figure 1 illustrates the indices graphically with respect to the matrix projections. Table 1 provides formulae for calculation and biological interpretations of each index. In brief, our amplification indices are reactivity (the largest possible population density achieved in one timestep after disturbance), maximum amplification (the largest possible population density achieved in any timestep after disturbance) and the upper Kreiss bound (an analytical lower bound for maximum amplification). These are partnered by indices of attenuation, which are first-timestep attenuation, maximum attenuation and the lower Kreiss bound, respectively. Appendix S2 contains codes for calculation of the indices in R. Table 2 is a correlation matrix of the indices, which shows that by nature, these indices are interrelated. To account for this non-independence, we conducted a principal components analysis (PCA) on the six indices.

Table 1.   The transient indices used in analyses, accompanied by formulae for calculation, and biological descriptions. After Townley & Hodgson (2008). See also Fig. 1. minCS, minimum column sum of the matrix. Note that r used in Kreiss bound calculations is different from r used to describe intrinsic population growth rate
DescriptionAmplification indexAttenuation indexBiological meaning (amplification/attenuation)Strengths/weaknessesAdditional information
Immediate (first-timestep) transient indexReactivity
inline image
First-timestep attenuation
inline image
The largest/smallest possible density (relative to asymptotic dynamics) that may be reached by the population in the first projection intervalSimple; amenable to perturbation analysis/does not always capture largest transientinline imageinline image
Kreiss boundUpper kreiss bound
inline image
Lower kreiss bound
inline image
The density a population must amplify/attenuate to (relative to asymptotic dynamics) before reaching its maximum/minimum overall size; therefore inner bounds on transient amplification/attenuationAmenable to perturbation analysis; captures largest transient more effectively/inner rather than outer bound on transient magnitudeinline image
Maximum transient indexMaximum amplification
inline image
Maximum attenuation
inline image
The largest/smallest density (relative to asymptotic dynamics) that may be reached by the population overallCaptures outer bound, maximum transient/not amenable to perturbation analysisMay or may not result from the same stage-bias as inline image
Table 2.   Spearman’s rank correlation matrix for the log10-transformed transient indices used in analyses. Values reported are Spearman’s ρ values. All P < 0.001
 inline imageinline imageinline imageinline imageinline image
inline image0.99    
inline image0.970.96   
inline image−0.66−0.65−0.65  
inline image−0.48−0.49−0.460.92 
inline image−0.44−0.41−0.560.700.55

Principal components analysis

Principal components analysis distils variation in a large number of correlated (i.e. non-independent) variables into just one or two uncorrelated measurements, called principal components. In our case, this means that the variation held in the six indices of transient dynamics can potentially be described instead by just one or two measurements, thus simplifying the analytical process and reducing the probability of type I statistical errors by accounting for the high correlates between indices.

The PCA was conducted on log10-transformed indices, as this describes more accurately the multiplicative nature of these transient dynamics. Hence, amplified transients become positive, whereas attenuated transients become negative. Detailed results for the PCA are presented in Appendix S2. These results suggest that more than 90% of the variance can be accounted for by principal components 1 and 2. Each index has an associated ‘loading’, which is the coefficient associated with that index in the principal component’s linear function. In our case, for principal component 1 (PC1), amplified transients have a positive loading and attenuated transients have a negative loading. For principal component 2 (PC2), both amplified and attenuated transients have a positive loading. This means that PC1 describes the overall tendency of the population to produce transients of large magnitude (i.e. a large PC1 corresponds to a population with both amplified and attenuated transients of large magnitude, and a small PC1 corresponds to a population with both amplified and attenuated dynamics of small magnitude). PC2 describes the tendency of the population to be biased towards either relatively larger amplified dynamics or relatively larger attenuated dynamics (i.e. a large PC2 corresponds to a population with amplified dynamics that are large relative to its attenuated dynamics, and a small PC2 corresponds to a population with attenuated dynamics that are large relative to its amplified dynamics).

Statistical models

To find ecological and model parameterization patterns in the observed variation in transient dynamics, we regressed PC1 and PC2 against four different explanatory variables using general linear modelling. The life history of a plant is a categorical variable equivalent to those described in Franco & Silvertown (2004): monocarpic (semelparous) plants from open and/or disturbed habitats, perennial (iteroparous) herbs from open and/or disturbed habitats, perennial herbs from forest habitats, shrubs and trees. The long-term (asymptotic) intrinsic population growth rate r is equal to ln (λ), where λ is the dominant eigenvalue of PPM A. The dimension of the matrix is equal to the number of matrix columns or rows, and is equivalent to the number of stage-classes in the life cycle model. Last, the matrix type is distinguished by the position of non-zero data in the matrix and represents a qualitative description of the complexity of the demographic model. Our descriptions are consistent with Carslake, Townley & Hodgson (2009). Leslie+ matrices are traditional Leslie matrices (Leslie 1945) incorporating growth and fecundity but also allowing for stasis of the last stage class, therefore containing positive data in the first row, the first subdiagonal and the last entry. These are usually age-structured models, in which ‘old’ individuals are grouped into a single stage class whose members die at a fixed rate per projection interval. Progression matrices are like Leslie+ matrices, but additionally incorporate stasis of all stage classes (individuals can remain within a stage-class between projection intervals; e.g. where stages are determined by the size of the individual) and therefore contain data in the first row, the first subdiagonal and the diagonal. Growth matrices are like progression matrices, but allow for skipped stage classes and therefore may include additional data anywhere in the lower triangle of the matrix. Lefkovitch matrices (after Lefkovitch 1965) represent a generalized, stage-structured life cycle which can contain data anywhere in the matrix, including regression through the life cycle: individuals can, e.g. get smaller, undergo fission or produce smaller vegetative offspring.

Statistical models considered the effects of all explanatory variables simultaneously on a single principal component, although interactions between explanatory variables were not considered. We used general linear models with Gaussian error structure, identity link function and stepwise model simplification from the maximal model, in order of least significance, using F-test analysis of deviance model comparisons and a significance threshold of P = 0.05, until a minimal adequate model was achieved (i.e. one where only significant effects remain). One clear outlier was removed in all analyses that involved r as an explanatory variable (Digitalis purpurea, λ = 11.8; = 2.5). All minimal adequate models were checked for normality of standardized residuals and homoscedasticity.

All analyses were conducted in a TIPS framework, where every species is treated as an independent data point (Silvertown & Dodd 1996; the name of the analysis refers to using the data from the ‘tips’ of the phylogenetic tree). However, previous statistical analyses that were conducted on the raw transient indices employed both TIPS and phylogenetically independent contrasts (PICs), where the degree of relatedness between species is controlled for (Felsenstein 1985; Freckleton 2000). Each included matrix dimension as a covariate (cf. Salguero-Gómez & Casper 2010). The results from PIC analyses corroborated those of TIPS analyses, although they are not presented here. Phylogeny was obtained using Phylomatic, using taxonomic groups recognized by the Angiosperm Phylogeny Group (2003).

All mathematical and statistical modelling was performed using R version 2.8.0 (R Development Core Team 2008).


Effects of life history

Plants with different life histories exhibit different potential magnitudes of transient amplification and attenuation. Analyses showed there to be a significant relationship between life history and PC1 (F4,100 = 3.6, P = 0.009; Fig. 2), however, there was no significant relationship between life history and PC2 (F4,100 = 0.5, P = 0.72). Monocarpic plants and trees are probably to exhibit both amplified and attenuated transients of greater magnitude than perennial, iteroparous herbs from open habitats and shrubs, which in turn are probably to exhibit amplified and attenuated transients of greater magnitude than perennial, iteroparous herbs from forest habitats. There is no bias towards either relatively larger amplification or relatively larger attenuation. Statistically, life histories can be grouped as indicated in Fig. 2 with perennials (both open and forest) and shrubs having similar transients, being lower in magnitude than those of monocarps. Trees can be grouped with either monocarps or perennials and shrubs, having transients of a lower magnitude than monocarps but of a higher magnitude than perennials and shrubs.

Figure 2.

 On average, plants at the extremities of life history complexity (i.e. monocarps and trees) tend to have transients of greater magnitude than those of mid-range complexity (i.e. perennial herbs and shrubs). Lowercase letters signify groupings of life history by statistical significance. M = monocarpic plants from open and/or disturbed habitats, O = perennial herbs from open and/or disturbed habitats, F = perennial herbs from forest habitats, S = shrubs, T = trees. Means and SD are plotted for an average matrix dimension and average r where relevant, using the minimum adequate model.

Relationships between transients and asymptotic growth

Populations that grow faster in the long term show both amplified and attenuated transients of a greater magnitude than those that are slower-growing or declining (significant positive relationship between r and PC1; F1,100 = 4.2, P = 0.042; Fig. 3), with no bias towards either relatively larger amplification or relatively larger attenuation (no significant relationship between r and PC2; F1,99 = 0.04, P = 0.83).

Figure 3.

 The potential magnitudes of transient dynamics (both amplified and attenuated measures) have a positive association with r, the measure of long-term population growth or decline, although there is a lot of scatter in the relationship, as is evident from the graph. The fitted line is plotted for an average matrix dimension using the minimal adequate model (but excluding life history because of the difficulty of calculating an ‘average’ value for this parameter).

Influence of matrix structure

Matrices with a greater dimension are probably to exhibit both amplified and attenuated transients of a greater magnitude than matrices with a smaller dimension (significant positive relationship between matrix dimension and PC1; F1,100 = 36.2, P < 0.001; Fig. 4), with no bias towards either relatively larger amplification or relatively larger attenuation (no significant effect of dimension on PC2; F1,104 = 0.3, P = 0.60).

Figure 4.

 The dimension of the matrix model (equal to the number of stages chosen in the life cycle model) shows a positive relationship with the potential magnitudes of transient dynamics (both amplified and attenuated transient measures). Fitted curves are plotted for average r where relevant, using the minimum adequate model (but excluding life history due to the difficulty of calculating an ‘average’ value for this parameter).

Matrix type was not influential in determining the transient dynamical properties of the PPM models: it had no significant effect on PC1 (F1,99 = 0.52, P = 0.47) or PC2 (F1,105 = 2.1, P = 0.15).


Relationships between transients and life history

Analyses showed there to be a U-shaped relationship between life-history strategy, defined along a continuum of ecological succession, and the magnitude of both amplified and attenuated transient dynamics. Monocarpic plants (a grouping of annuals and monocarpic perennials) and trees, representing opposite ends of a scale of ecological succession, generally have amplified and attenuated transients of greater magnitude than shrubs and iteroparous perennials. This is an intriguing observation – the expected relationship might be one of monotonic decline with increasing life history complexity, at least for amplified dynamics: several studies have shown that annuals and species of early succession have greater reproductive allocation than perennials and species of later succession (Gleeson & Tilman 1990; Aarssen & Taylor 1992; Silvertown & Dodd 1996; Fenner 2000).

Selective environmental pressures acting on life-history trade-offs may account for our findings. Early-successional species inhabiting open habitats will benefit from a quick pre-emption of abundant resources. For trees, gaps in the forest are ephemeral and may be filled by a few individuals at most. In both situations, under models of ‘lottery’ recruitment, where successful establishment of juveniles is random and indiscriminate (Chesson & Warner 1981; Chesson 1991; Turnbull, Crawley & Rees 2000), producing many viable offspring may give the best chance of successful colonization (which increases the potential for transient amplification). Due to subsequent space constraints and/or competition for resources, juvenile mortality would be high (which increases the potential for transient attenuation). This would account for the boom-and-bust transient properties exhibited by these life histories. Perennials in forest habitats, in contrast, experience a relatively constant environment, where colonization opportunities are more predictable. Therefore, whilst still experiencing space and resource constraints once established, they might be less exposed to the pressure of ‘lottery’ colonization. Thus, their best strategy may be to weigh the costs of (unnecessary) reproduction against the benefits of survival and vice versa, resulting in decreased fecundity and increased juvenile survival (and therefore more stable transient properties as a result). There is ample evidence for trade-offs such as these in plants (Mitchell-Olds 1996a,b; Ehrlén & van Groenendael 1998 and citations therein). It is worth bearing timescale in mind when considering this: tree populations are slow-growing, and their ‘transient state’ will therefore probably last far longer than that of an annual. Transient dynamics can be just as extreme for both, but over entirely different timescales.

The discrepancy between our findings and results for reproductive allocation theory (Silvertown & Dodd 1996) perhaps reflect the differences between studying individual demographic traits and demographics (the size, structure and dynamics of populations) per se. The number of offspring produced is a product not only of reproductive allocation, but of many vital rates. For example, trees may allocate comparatively less resources to reproduction than annuals, but given increased pollination success, greater number of seeds produced or greater seed set, a tree may produce as many individual offspring in a single year, and hence have the capacity to exhibit the same magnitude of transient amplification. Indeed, seed size (and therefore number) is not necessarily proportional to individual size across species, and shows great variation across taxa and life histories (Moles et al. 2005). In addition, measures of reproductive allocation do not account for adult survival: increased adult survival in trees will probably contribute to greater population amplification.

Further analysis of how life history and transient dynamics are related is needed. Sensitivity, elasticity and/or transfer function analyses of transient indices would inform on whether, as for asymptotic elasticity (Silvertown et al. 1993; Silvertown, Franco & Menges 1996; Franco & Silvertown 2004), specific life histories show individual patterns in their responses to perturbation. Such results would equip population managers with the information on the ability of populations of certain species to amplify and/or attenuate.

Relationships between transients and long-term dynamics

Populations with rapid asymptotic growth (greater r) are likely to harbour the potential for greater magnitudes of both transient amplification and attenuation.

For transient amplification, this relationship is relatively intuitive: vital rates such as survival and fecundity that contribute to long-term population growth also contribute greatly to transient amplification. However, the relationship with transient attenuation is less intuitive, since attenuation is associated with low sub-adult survival, and low or zero fecundity. Again, given trade-offs in allocation of resources to survival and reproduction, a plant that has high adult survival (hence increasing asymptotic growth rates and the potential for transient amplification) may invest less in sub-adult survival (hence simultaneously increasing the potential for transient attenuation). Our discovery of positive relationships between long-term rates of growth and the magnitude of short-term population amplification and attenuation may suggest that all of these are similarly influenced by vital rates of members of the population, which in turn will be determined by evolutionary history or environmental resources and stressors. We explore this possibility by relating our results to reported patterns in both asymptotic and transient sensitivities and elasticities.

The popularity of asymptotic analyses is accompanied by an abundance of literature on sensitivities of long-term population dynamics to vital rates of plants. Comparative studies utilizing empirical models have explored patterns in elasticity of asymptotic growth to both compound transition rate elasticities (Silvertown et al. 1993; Silvertown, Franco & Menges 1996; Crone 2001) and vital rate elasticities (Franco & Silvertown 2004; Burns et al. 2010). Simulation studies have also explored relationships between elasticities of matrix elements (Carslake, Townley & Hodgson 2009). In the vast majority of these cases, it has been reported that elasticity of asymptotic growth to survival is greater than elasticity to fecundity. As outlined earlier, we might intuitively expect transient amplification to respond to adult survival (as well as fecundity), but attenuation to respond to sub-adult survival.

Despite the wealth of studies of sensitivities of asymptotic growth rates to vital rates, there is a dearth of similar studies for the sensitivity of transient dynamics to recruitment, survival and growth in plants. Perhaps, the only investigation into patterns of sensitivity in transient dynamics was made by Koons et al. (2005). They considered three bird and three mammal species at various points on the fast–slow life-history continuum. They found that for populations close to stable stage structure, transient sensitivities showed a greater dependence on sub-adult survival than on fecundity (which is similar to results for asymptotic elasticity in mammals and birds – see Heppell, Caswell & Crowder 2000 and Sæther & Bakke 2000). Conversely, for populations disturbed away from stable stage structure, transient sensitivities showed a greater dependence on fecundity and adult survival than on sub-adult survival. Our indices utilize initial structures that represent significant departures from stable structure, and as such they would be more likely to exhibit this second pattern. This agrees with expectations as outlined above for amplified dynamics, but not with expectations for attenuated dynamics. It is noteworthy that Koons et al.’s results are not directly comparable with those described for asymptotic dynamics as one is a measure of sensitivity and the other is a measure of elasticity (for formal definitions see Caswell 2001). However, it appears that, for animals at least, transient sensitivities do not conform to the neat, global patterns of asymptotics and are far more attuned to variation in initial conditions.

It is difficult to infer how vital rates may determine the transient dynamics of plant populations using studies of asymptotic dynamics in plant populations or transient dynamics in animal populations. However, patterns of transient sensitivity in plants could be complex, and amplified and attenuated dynamics might show differing responses to variation in vital rates. A study explicitly concerning transient sensitivity, elasticity and/or transfer function analyses in plants needs to be performed to infer these relationships. For now, we are left with the intriguing observation that plant populations predicted to grow faster in the long term tend to exhibit the potential for larger magnitudes of transient amplification and attenuation than (asymptotically) slower-growing or declining populations.

Effects of matrix structure on transient indices

Both amplified and attenuated dynamics increase in magnitude with increasing matrix dimension.

One explanation for this is that it is an artefact of model design. A larger matrix (i.e. one with a larger number of stages in the life cycle model) will house comparatively more demographic transition rates and hence will capture peak rates of fecundity and mortality more efficiently. A small (low-dimension) matrix used to describe the same life cycle will effectively average out variation in fecundity and mortality rates, so that peak rates of a small matrix will always be of lesser magnitude than those of a large matrix for the same population. This ‘averaging effect’ is a parsimonious explanation for an increase in transient magnitude with increasing matrix dimension, since maximum (and minimum) column sums of a larger matrix are likely to be larger (and smaller) than for an averaged, smaller matrix. This is an important consideration, because it highlights that populations may amplify or attenuate beyond bounds predicted by simplified models.

However, if the number of stages chosen is a true reflection of the life cycle, then this should not be the case – modelled transient dynamics would be indicative of the true transient dynamical properties of the population. An organism with greater stage specificity (i.e. greater heterogeneity in demographic rates across the life cycle) will require a life cycle model with more stages (and hence a larger projection matrix) to capture that heterogeneity. In this case, our finding that magnitudes of transient dynamics increase with increasing matrix dimension would indicate that organisms with greater stage specificity exhibit more extreme transient dynamics.

As a comparative study, ours does not control for stage specificity and as such we cannot reliably distinguish between these two explanations. However, a recent study by Tenhumberg, Tyre & Rebarber (2009) provides some support for the former. They collected demographic data from the pea aphid, Acyrthosiphon pisum, and used it to create matrices of differing dimension. They then compared model predictions from these matrices to empirical dynamics of laboratory populations. They found that larger matrices modelled greater transient growth rates and captured observed transient growth rates more effectively than smaller matrices. This further supports our hypothesis that magnitudes of transient dynamics increase with increasing matrix dimension.

Approaches to parameterizing matrix models are often based upon data availability (Vandermeer 1978; Moloney 1986) or are taxon-specific (e.g. Noon & Sauer 1992), whilst reviews of methods for modelling complex factors such as environmental stochasticity (Fieberg & Ellner 2001) or spatial heterogeneity (Day & Possingham 1995) reveal a diversity of approaches. Matrix dimension is known to have a significant effect on the elasticities of matrix elements (Enright, Franco & Silvertown 1995). However, its effect on population size, density or growth rate has not often been studied (but see de Matos & Silva Matos 1998; Lamar & McGraw 2005; Ramula & Lehtilä 2005; Tenhumberg, Tyre & Rebarber 2009) and is less often controlled for in demographic analyses. Our results indicate that it could have consequences for these measures and hence for management strategies based upon them. Perhaps there is a need for more stringent rules in matrix parameterization to assure that models are predictive and comparative. Enright, Franco & Silvertown 1995 suggested that matrix dimension should either be a function of longevity or equal for different models that require comparison (see also Salguero-Gómez & Casper 2010 for homogenizing matrix dimension across models). Our results indicate that either solution could have an effect on transient dynamics depending on the population under study. We also note that the discretization of infinite-dimensional integral projection matrices (see Ellner & Rees 2006), for numerical analysis, should consider the impacts of dimensionality not just on asymptotic predictions, but also on the transient properties of the discretized model.

Matrix type did not show a significant relationship with transient magnitude and so appears to have no effect on the transient dynamics of the model. This may be because real-world populations do not conform rigidly to the definite boundaries of classification used to describe empirical models. For example, inclusion of just one extra stasis parameter in a Leslie+ matrix changes it into a growth matrix. However, this single parameter is unlikely to have huge effects on the model output. Whilst it is a good idea to incorporate possible model effects in analyses, perhaps a more robust definition of matrix ‘type’ is required.


The indices of transient dynamics that we employ here should prove to be useful in many scenarios. In the fields of in situ conservation and pest control, measures such as reactivity and first-timestep attenuation would allow demographers to calculate ‘best-case’ and ‘worst-case’ scenarios, so establishing the ability of disturbance to cause population booms and crashes in the very near term, and enabling attempts to design ‘quick fixes’ to buffer against population disturbances. In the contrasting fields of ex situ population management (including species reintroduction) and harvesting, indices such as maximum amplification and attenuation may enable demographers to utilize population disturbance to their advantage and enable population managers to maximize population size and viability without compromising costs or harvest rates. In fundamental research, these indices (and in particular, the Kreiss bounds) are useful in comparative analyses as measures of the overall transient dynamical properties of a population or species. Primarily, however, all of these indices would be of use to anyone who does not have knowledge of initial population structure. Where information on population structure is available, other methods may prove more pertinent. For example, the measures of transient growth employed in Maron, Horvitz & Williams (2010) may be more useful to demographers who have information on current population structure and are interested in population growth rate rather than size (e.g. to identify whether a population is currently undergoing short-term decline or increase). Population momentum and inertia (Koons, Holmes & Grand 2007) may be more useful to demographers who are interested in population size or density but have information on current population structure. In research, measuring transient growth rates or population inertia and/or momentum would be more useful when comparing within, rather than between, species or populations exposed to differing conditions, e.g. as a measure of fitness.

Our results have shown that there are strong correlations between the Kreiss bounds and immediate and maximum measures of population amplification and attenuation. The relationships of these indices with one another, and with other indices of transient dynamics such as population momentum and inertia, warrant further exploration. In addition, we have uncovered several patterns in the transient dynamical properties of populations. First, life history and transient dynamical properties of populations are related. Second, transient and long-term dynamics are positively correlated. To shed more light on both of these findings, a study that looks at the sensitivity of transients to changes in transition rates would be ideal. Last, transient dynamics are sensitive to matrix dimension with consequences for management strategies based upon such models. It is debateable whether this is an artefact of the model or because of larger matrices modelling organisms with greater stage specificity and life cycle complexity. In either case, matrix dimension ought to be controlled for in any transient analysis of population dynamics. Transient analysis is likely to see increased popularity in the near future – these results and further studies of transient population dynamics should aid population managers and conservationists working with wild plant populations.


We thank the organizers and participants in the Ecological Society of America’s session on ‘Projection Matrix Models: Investigating General Patterns in Plant Demography’ for feedback on this research. The work was funded by NERC and European Social Fund grants to D.J.H., and a Leverhulme visiting fellowship to S.T.