### Introduction

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgements
- References

Understanding the drivers of population dynamics is one of the fundamental themes in ecology. Theoretical ecology has a long history of using models to help elucidate the processes that influence populations (Kingsland 1995; Turchin 2001). Population models are inevitably simplifications of real systems, but in them we hope to capture the most important elements of a population's biology. Traditionally, ecologists have focused on the eventual (asymptotic or equilibrium) behavior of a model system in order to understand the current state of a population (May 1973). This approach assumes that the conditions implied by the model have been in place long enough for the dynamics in the current population to reflect the equilibrium conditions of the model. Over the past few decades, ecologists have recognized the importance of short-term, transient dynamics in understanding population dynamics ( Hastings 2001 2004). Due to environmental variability, populations may be constantly experiencing short-term responses to changing conditions, creating long-term dynamics that are not supported under equilibrium theory (Pickett 1985). This theoretical shift in how ecologists approach population dynamics, from emphasizing the role of asymptotic versus transient dynamics, has important implications in conservation and management (Hastings 2004; Benton *et al*. 2006). Managers are often interested in the most efficient way to increase (or decrease) the size of a population and analysis of transient dynamics can help assess how best to achieve this result (Fox & Gurevitch 2000; Koons *et al*. 2006; Ezard *et al*. 2010). Consequently, several studies have called for an increased focus on transient responses in common applications (Townley *et al*. 2007; Ezard *et al*. 2010).

One difficultly in transitioning to transient analyses is determining what measure of transient response is most appropriate. Many new methods have been developed for describing the transient behavior of population models (reviewed in Stott *et al*. 2011), and these methods differ in subtle but important ways. Transient responses that lead to larger population densities than expected based on asymptotic behavior (i.e. amplification of the population) are treated separately from responses that lead to smaller population densities (i.e. attenuation). Furthermore, these responses depend on both time-scale and initial conditions in the population. Asymptotic methods focus on the long-term, stable behavior of a model, essentially assuming that a population is initiated in its equilibrium stable state. In contrast, many transient methods focus on the initial condition that produces the most extreme behavior compared to asymptotic dynamics (e.g. Caswell 2007; Haridas & Tuljapurkar 2007, Townley *et al*. 2007). While some transient methods can be modified to produce analygous, case-specific measures of transient response, the more general forms are used to set upper and lower bounds on potential transient behavior (Stott *et al*. 2011). These bounds represent the ‘transient potential’ of a model based on a particular, extreme initial condition (Stott *et al*. 2011; Williams *et al*. 2011). In practical applications of transient dynamics, the crucial issue is not necessarily how large responses could be at an extreme initial condition, but whether large responses are common, given the set of conditions that a population might experience. Understanding the implications of transient dynamics requires a renewed focus not just on a model's theoretical potential for either amplified or attenuated transient responses, but also on the set of initial conditions that populations commonly experience (Williams *et al*. 2011).

To this end, I used transient analyses of a commonly applied model in population management, demographic matrix models, to examine how frequently populations experienced responses that were comparable to theoretical measures of transient potential. Discrete time, stage- or age-based matrix models are one of the most commonly used models in management (Menges 2000; Morris & Doak 2002; Reed *et al*. 2002). The models are based on matrices containing vital rates for distinct groups of individuals in the population and can be analysed to yield measures of dynamics for the population as a whole. These models have been the basis for a developing framework for analysing the transient potential of population models, based on descriptions of transient bounds (Stott *et al*. 2011). I relied on stochastic simulations for five plant species, for which long-term modeling results were available, to get a baseline of behavior around the equilibrium stable state (i.e. the stable stage distribution of the matrix model) assumed in traditional, asymptotic analyses. These simulations provide an empirically derived set of initial conditions with which to examine transient behavior, in particular focusing on two questions:

- How similar are initial conditions required by measures of transient potential to conditions produced based on variation in natural populations (i.e. how realistic are the initial conditions)?
- Does theoretical transient potential reflect the transient responses that simulated populations experience?

It is important to note that although this approach relies on a stochastic model to generate a set of expected initial conditions, thus far, measures of transient potential focus on the response of a single deterministic matrix (i.e. a population mean matrix). Especially for demographic matrix models, transient bounds have been used as an indicator for how important transient responses might be in the population dynamics of a species (e.g. Stott *et al*. 2010; Williams *et al*. 2011). By improving our understanding of how well measures of transient potential reflect actual population dynamics (based on observed variability in demographic rates within populations), we can make more informed inferences on how and when more complicated analyses will be necessary in applications of demographic matrix models.

### Results

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgements
- References

Distances from SSD required for producing theoretical measures of amplification and attenuation were far larger than distances observed in the simulated populations. Across populations, the average population mean distance from SSD was 0.08±0.02 (mean ± SE; Fig. 1). Differences between population means contributed < 10% of the total variation in simulated distances (Intraclass correlation coefficient, ICC = 0.07); however, populations differed in the amount of variation in distance from stable. Among individual populations, one population of *Haplopappus radiatus* produced the largest variation in distances (estimated variance in observed distances: ), while *Aspasia principissa* had the smallest variance (). In terms of maximum distance expected over a 20 year time period (i.e. the 5% and 95% quantiles of the distribution of distances), this variation lead to differences between SSD and observed stage structure that ranged from −0.54 to 1.24 from populations of *H. radiatus* and −0.07 to 0.08 for *A. principissa*. On average across the populations, these empirical maximum distances from SSD ranged from 0.758±0.119 (mean ± SE) for amplification and −0.396±0.028 for attenuation.

In comparison, the average distance from SSD required to match theoretical transient bounds were 1.48±0.27 for amplification and −0.67±0.04 for attenuation. Treating these sets of distances in each population as theoretical versus empirical bounds defining a range of possible stage structures, the empirical range of variation on average encompassed only 35.8% ± 7.0 of the theoretical range across populations (Fig. 2). Distances required for the theoretical bounds were between 1.27 and 21.62 times greater than the mean empirical distances for amplification (median = 4.20 times larger; Fig. 3a) or between 0.23 and 4.58 times greater than the maximum distances reached in the simulations (median = 1.16; Fig. 3b). For attenuation, differences between empirical distances and those required to produce theoretical measures were more similar, ranging from 2.00 to 23.50 times further from SSD for theoretical mean distances or 0.15 to 5.84 for maximum distances.

To define empirically-derived bounds on transient behavior that would be analogous to measures of transient potential, the quantity of interest is the response of the mean matrix to the range of stage structures that were produced for each population. Based on the 5% and 95% quantiles of distances observed in the simulations, representing 20-year maximum and minimum distances from SSD, the empirically-derived reactivities () ranged from 1.06 for *A. principissa* to 2.55 for *H. radiatus*, with a mean of 1.68±0.16 across populations. In comparison, the transient potential for reactivity (), which represents the upper bound of transient responses that are theoretically possible, ranged from 1.21 for *A. principissa* to 7.12 for one population of *Astragalus tyghensis* (mean across populations: 3.21±0.34). For the theoretical bounds, most populations reached their maximum amplification () in the first time step (i.e. ); however, with the empirical bounds, the largest transient responses took slightly longer to develop (average increase in years). Still, these increases in amplification over time were not large enough to approach the theoretical bounds (Population means: for empirical bounds versus for theoretical bounds).

Theoretical bounds for transient responses were predictive of the amount of variation in distance from SSD in the simulations (Fig. 5). Due to colinearity issues among the measures of transient potential, I used a principal component analysis based on the log-transformed measures to define axes with greatest variation. In this analysis, the first two principal components described 97.6% of the variation among the four measures of transient potential (Table 3). The first principle component had a significant positive effect on log-transformed ; Fig. 5). This PCA axis involved positive associations with the two measures of amplification and negative associations with the measures of attenuation. The measures of attenuation had stronger effects than measures of amplification, and first step measures (reactivity and first step attenuation) were more influential than the prolonged responses. Together, the measures of transient potential explained 46.3% of the variation in among the populations.

Table 3. Principal components analysis on four indices of transient dynamics | Axis 1 | Axis 2 | Axis 3 | Axis 4 |
---|

Loadings |

| −0.618 | −0.465 | 0.626 | −0.098 |

| −0.507 | −0.379 | −0.770 | 0.078 |

| 0.455 | −0.576 | −0.084 | −0.674 |

| 0.392 | −0.555 | 0.089 | 0.728 |

Standard deviation | 0.685 | 0.488 | 0.130 | 0.026 |

Cumulative proportion of variance | 0.648 | 0.976 | 0.999 | 1.000 |

### Discussion

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgements
- References

These analyses indicated that under ordinary levels of environmental stochasticity measures of transient response overestimate the difference between asymptotic and realized population dynamics. In keeping with this result, many examples used to motivate analyses of transient dynamics have focused on extreme situations, such as colonizing populations with only seeds (Caswell and Werner 1978; McMahon and Metcalf 2008; Maron *et al*. 2010) or populations with only reproductive adults (Tenhumberg *et al*. 2009). However, transient methods are increasingly discussed as more realistic and thus more appropriate for many ecological situations (Bierzychudek 1999; Koons *et al*. 2005; Fefferman and Reed 2006; Koons *et al*. 2006, , 2007; Townley *et al*. 2007; Tenhumberg *et al*. 2009; Ezard *et al*. 2010; Maron *et al*. 2010; Stott *et al*. 2010). Several studies have pointed out that observed stage structures tend to differ from SSD (Bierzychudek 1999; Ramula and Lehtilä 2005; Williams *et al*. 2011); however, these differences may not lead to large effects on transient dynamics (Fig. 4; see also Williams *et al*. 2011). Similarly, simulated transient responses to known disturbance have also produced results that were qualitatively similar to asymptotic analyses (e.g. Satterthwaite *et al*. 2002). Together, these results and this growing body of literature indicate that while analysis of transient bounds improve our understanding of extreme or theoretical scenarios, they may not reflect dynamics in many typical populations.

In addition to forecasting different population scenarios, transient measures have increasingly been used in comparative life history analyses. For example, longer lived organisms are expected to show larger transient effects in terms of transient potential (Stott *et al*. 2010), dampening ratios (Koons *et al*. 2006), and distances from SSD (Williams *et al*. 2011). Similarly, models with more stage classes or greater complexity may lead to greater transient responses (Ramula and Lehtilä 2005; Tenhumberg *et al*. 2009; Stott *et al*. 2010). These results broadly support the results of these studies in the sense that measures of transient potential were predictive of the variation in simulation distances and correlated with realized responses. However, in general, the correlations found both between longevity and measures of transient response and between matrix size and transient response in previous studies were not strong. Taken together with the difference between theoretical and realized responses found in this study, these relationships are likely to be weaker under ordinary environmental variation.

The results in this study are based on a very small subset of species for which matrix models are applied and may not be indicative of the transient behavior across other species. The perennial plant studies included in this analysis were selected based on study length, where a fully parameterized matrix could be drawn with equal probability in each year. This excluded some species with known disturbance cycles (e.g. Quintana-Ascencio *et al*. 2003; Menges and Quintana-Ascencio 2004; Menges *et al*. 2006; Menges, 2008), where matrix selection depends on the previous year's state. Although these species are likely to experience potentially large transient responses, the analyses presented here are based on disturbance and response to a consistent set of conditions represented by the mean matrix (Stott *et al*. 2011), which would not be comparable for species with cyclic dynamics. It is possible that populations that lend themselves to long-term study tend to be more stable, leading to a conservative estimate of the amount of disturbance observed during the study period. Furthermore, one might expect that species that are more mobile or in environments with high temporal variation might have greater potential to get pushed further from their SSD, and thus may get closer to their theoretical transient response (e.g. Ujvari *et al*. 2010). However, the finding that most of the populations included here remained centered on their SSD, with variation well within the theoretical bounds for amplification and attenuation, challenges the view that populations should tend to be away from their SSD and instead points continuing to develop an understanding of SSD as an equilibrium steady state.

Another consideration in interpreting these results is whether variation in demographic rates alone can describe the actual variation in population stage structures observed in the field. For a subset of populations in this study for which data were available, the range of distances from standing stage structures (i.e. the observed proportion of individuals in each stage in the population) was very similar to the range of simulated distances, although there were also exceptions to this pattern. For example, during the study period for the *H. radiatus*, the populations experienced a large seedling pulse that briefly moved the populations out of the simulated range of variation. Similarly, in a analysis of observed stage structures, Williams *et al*. (2011) found that although around 80% of the standing stage structures were within Such events might occur if populations experience conditions that are either outside of the range of demographic variability or that are not well described by the modeling framework.

The lack of evidence for dramatic transient responses in these long-term datasets does not imply that transient dynamics are inconsequential to the species' population dynamics. In these simulations, transient responses contributed as much variation to population growth rates as variability in vital rates. Variation in deterministic growth rates from the annual transition matrices for a population growth rate contributed, on average, 52% to the combined variation based on year to year differences in vital rates and non-stable age structures, suggesting that even in absence of external disturbances transient responses have an influential role in determining population-level variability. Several methods, such as stochastic elasticities (Tuljapurkar *et al*. 2003) and stochastic life table response experiments (Caswell 2010; Davison *et al*. 2010), incorporate transient responses as a contributing factor to stochastic dynamics, may be a more practical way of incorporating transient effects in understanding how underlying factors affect population dynamics. Understanding the interacting effects of stochasticity and transient behavior remains a challenge in understanding the importance of transient dynamics to population behavior.

In sum, simulated populations remained relatively close to SSDs and produced much smaller transient responses than reflected by indices of transient potential, suggesting that for some species, asymptotic measures may provide a reasonable description of short-term population responses, assuming other model assumptions are met. Transient bounds on amplification and attenuation, as measures of transient potential, provide an indication of species that may respond more strongly to disturbances; but managers should focus on the particular effects of a disturbance on stage or ages distributions in a population in order to accurately assess the role of transient dynamics in a specific system.