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# Beyond sensitivity: nonlinear perturbation analysis of transient dynamics

Article first published online: 11 JUN 2012

DOI: 10.1111/j.2041-210X.2012.00199.x

© 2012 The Authors. Methods in Ecology and Evolution © 2012 British Ecological Society

Additional Information

#### How to Cite

Stott, I., Hodgson, D. J. and Townley, S. (2012), Beyond sensitivity: nonlinear perturbation analysis of transient dynamics. Methods in Ecology and Evolution, 3: 673–684. doi: 10.1111/j.2041-210X.2012.00199.x

#### Publication History

- Issue published online: 30 JUL 2012
- Article first published online: 11 JUN 2012
- Received 6 January 2012; accepted 26 February 2012 Handling Editor: Robert Freckleton

### Keywords:

- inertia;
- perturbation analysis;
- population management;
- population projection matrix;
- sensitivity;
- transfer function;
- transient dynamics

### Summary

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgement
- References
- Supporting Information

**1.** Perturbation analyses of population models are integral to population management: such analyses evaluate how changes in vital rates of members of the population translate to changes in population dynamics. Sensitivity and elasticity analyses of long-term (asymptotic) growth are popular, but limited: they ignore short-term (transient) dynamics and provide a linear approximation to nonlinear perturbation curves.

**2.** Population inertia measures how much larger or smaller a non-stable population becomes compared with an equivalent stable population, as a result of transient dynamics. We present formulae for the transfer function of population inertia, which describes nonlinear perturbation curves of transient population dynamics. The method comfortably fits into wider frameworks for analytical study of transient dynamics, and for perturbation analyses that use the transfer function approach.

**3.** We use case studies to illustrate how the transfer function of population inertia may be used in population management. These show that strategies based solely on asymptotic perturbation analyses can cause undesirable transient dynamics and/or fail to exploit desirable transient dynamics. This highlights the importance of considering both transient and asymptotic population dynamics in population management.

**4.** Our case studies also show a tendency towards marked nonlinearity in transient perturbation curves. We extend our method to measure sensitivity of population inertia and show that it often fails to capture dynamics resulting from perturbations typical of management scenarios.

### Introduction

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgement
- References
- Supporting Information

A thorough understanding of how external factors shape population density and growth is essential to population management. Management goals will often explicitly incorporate elements of density and growth, for example, to promote growth and persistence of populations of conservation concern (e.g. Esparza-Olguín, Valverde & Vilchis-Anaya 2002), reduce density and spread of pests and invasive species (e.g. De Walt 2006), or sustain optimal yields from exploited populations (e.g. Pinard 1993). Commonly, it is necessary to explore how potential management actions may impact population density and growth, to evaluate the most ecologically and economically effective means of achieving management goals (e.g. Baxter *et al.* 2006). Alternatively, an assessment of how potential environmental change may impact population density and growth may be required, to explore the possible detrimental impacts of uncontrolled events that are antagonistic to management goals (e.g. Mumby, Vitolo & Stephenson 2011). It is important in either case to know which vital rates or stages of the life cycle should be targeted during management, to ensure optimum outcomes for minimum effort and cost.

In practise, these assessments are commonly carried out using perturbation analyses of empirical population models. Such analyses evaluate how changes in the vital rates of a population (survival, growth, regression, fecundity, or underlying parameters that influence these) affect measures of population density and growth. Frequently, analytical methods have focussed on linear approximations of the effect of small perturbations on long-term growth. Termed asymptotic sensitivity and elasticity analyses (Caswell 2001), these are typically mathematically and computationally simple and easily interpreted.

However, asymptotic sensitivity and elasticity analyses have two major drawbacks. First, long-term growth is often a very poor predictor of real population dynamics, as short-term ‘transient’ dynamics are often very different to long-term trends (Hastings 2001; Townley *et al.* 2007). There is an increasing awareness of the need to incorporate analyses of transient dynamics into ecological and evolutionary studies (Ezard *et al.* 2010; Stott *et al.* 2010a), and many methods have been developed in the last decade to quantify the sensitivity of transient dynamics to perturbation. Most of these have focussed on population projection matrix (PPM) models, which are arguably the most widely used empirical models in population ecology. Methods vary in their approach and may focus on time-invariant, density-independent models (Fox & Gurevitch 2000; Yearsley 2004), stochastic models (Tuljapurkar, Horvitz & Pascarella 2003), density-dependent models (Tavener *et al.* 2011) or combinations thereof (Caswell 2007). A second drawback of asymptotic sensitivity and elasticity analyses is that the linear approximations they offer may be very poor predictors of population dynamic response to large perturbations. The actual relationship between a perturbation to the life cycle and resultant population dynamics is often substantially nonlinear (Townley & Hodgson 2008), and there has been interest in modelling nonlinearity in perturbation analyses of long-term growth (Hodgson & Townley 2004; Hodgson, Townley & McCarthy 2006; Miller *et al.* 2011). These methods have been limited to non-stochastic, density-independent PPM models, which is more a by-product of the tractability of analytical nonlinear perturbation analysis than an ignorance of the need for such analyses in stochastic and density-dependent models. That said, given the relatively data-demanding nature of even basic PPM models, it is likely that non-stochastic, density-independent models dominate the literature, especially in conservation.

However, to date, there do not exist any analytical methods that can assess nonlinear perturbation of transient dynamics. In this paper, we address this issue to conceive a formula that calculates the transfer function of population inertia for linear, time-invariant PPM models. This builds on existing methods that use population inertia as an index of transient dynamics and that use transfer function methods for nonlinear perturbation analysis of population models.

Transient dynamics of ‘non-stable’ populations (with non-stable demographic distributions) result in a phenomenon termed population inertia. A ‘stable’ population (with stable demographic distribution) is predicted to grow or decline at a fixed geometric rate. Population inertia occurs because non-stable populations exhibit very different patterns of transient growth and/or decline before eventually settling to this stable geometric rate. These transient dynamics mean that a non-stable population achieves long-term population densities at a fixed ratio either above or below that of an equivalent stable population. Population inertia (Koons, Holmes & Grand 2007) is the value of this ratio. Therefore, a population with inertia greater than 1 becomes and remains larger than had it started out with a stable demographic distribution, whilst a population with inertia smaller than 1 becomes and remains smaller. The magnitude of inertia indicates how much larger or smaller, relatively, the population becomes. Somewhat paradoxically, inertia is a measure of the asymptotic effects of transient dynamics that cause departures from asymptotic growth in the short term. Hence, it measures the persistency of transients, yet it has been shown to correlate very strongly with near-term transient indices such as reactivity and transient amplification (Stott, Townley & Hodgson 2011). As an index of transient dynamics, population inertia has a number of strengths: it is standardised measure and thus can be easily compared among different models, and as we show here, it is amenable to analytical perturbation analysis.

Transfer function analyses help to elucidate relationships between system inputs, outputs and feedback mechanisms. Their application to PPM models in population ecology began with the modelling of nonlinear perturbations of asymptotic growth (Hodgson & Townley 2004; Hodgson, Townley & McCarthy 2006). A particular strength of transfer function methods is that they are able to model complex perturbation structures involving multiple simultaneous perturbations (for example, multiple management strategies or life cycle trade-offs). However, they require no extra data to calculate than traditional sensitivity and elasticity analyses.

Combining population inertia and the transfer function approach yields a method for nonlinear perturbation analysis of transient dynamics, which we show to be flexible in its ability to model realistic management regimes, and comparable both within and among models. We illustrate these methods using worked examples that demonstrate the importance of considering transient perturbation analysis alongside asymptotic perturbation analyses for population management. We derive measures of the linear sensitivity of population inertia, to compare these with their corresponding transfer functions. Examples show that transient dynamic response to perturbation is often extremely nonlinear, rendering sensitivity a poor approximation to nonlinear perturbation analysis of transient dynamics.

### Materials and methods

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgement
- References
- Supporting Information

#### Transfer Function Framework

The transfer function analysis (TFA) framework is a powerful method for analysing linear, time-invariant mathematical systems (Ogata 2010). TFA describes relationships between system inputs, system outputs and the feedback mechanisms within the system. In the case of a PPM model, the ‘system’ is the PPM model itself, and a ‘linear, time-invariant’ model is one that is non-stochastic and density independent. An ‘input’ is the current demographic structure, the ‘outputs’ of interest are measures of population density or growth, and the ‘feedbacks’ being studied are the vital rates of the life cycle, that is, the elements of the PPM or underlying parameters that influence them. Thus, transfer functions may be used to determine the precise nonlinear relationship between a perturbation to vital rates of the life cycle and resulting population dynamics, which can then be used to inform management regimes or anticipate population response to environmental change.

When presenting transfer function formulae, we use a common notation throughout where matrices are presented upper-case bold (e.g. **A**), vectors are presented lower-case bold (e.g. **a**), numbers and scalars are presented in normal font (e.g. a) and functions are rendered in italics (e.g. *a*(λ)). The transfer function of long-term, or ‘asymptotic’, population growth has already been described (Hodgson & Townley 2004). For the purposes of population management, it may be desirable to know what the long-term growth of a perturbed PPM **A**_{δ} is, where:

- (eqn 1)

In this equation, the original PPM **A** is perturbed by magnitude δ using a ‘perturbation structure’ determined by column vectors **d** and **e**, which when multiplied together as in equation 1 (with **e**^{T} representing the transpose of **e** from a column to a row vector) form a matrix that determines which vital rates are perturbed. (It is noted that **d** and **e** are analogous to **b** and **c**, respectively, of Hodgson & Townley 2004, although we have decided to change notation here to remain in line with the accepted norms of the control systems literature). **A** has some emergent properties relating to asymptotic population dynamics: the dominant eigenvalue λ_{max} represents the long-term (asymptotic) growth of the population, the dominant right eigenvector **w** represents the ratio of stages in the stable demographic distribution and the dominant left eigenvector **v** represents the reproductive value of each stage. **A**_{δ} in turn has its own, different, asymptotic growth rate, stable stage structure and reproductive value that are determined by the structure and magnitude of perturbation.

If δ is used to represent a continuous set of perturbation values, then there is a corresponding set of numbers λ that represents all of the associated λ_{max}(**A**_{δ}). The exact relationship between δ and λ can be described using:

- (eqn 2)

where **I** is the identity matrix of same dimension as **A** (Hodgson & Townley 2004; please see Appendix S1 in Supporting Information for all proofs). This equation is the transfer function for asymptotic growth. It is possible therefore to specify a range of perturbation magnitudes (δ_{min} to δ_{max}), calculate λ_{max}(**A**+ δ_{min}**de**^{T}) and λ_{max}(**A**+ δ_{max}**de**^{T}) to find the maximum and minimum of the associated range of λ, and use equation 2 to easily evaluate the precise relationship between λ and δ. As **d**, **e** and the range of δ are user-specified, the only required known parameter is the PPM. Hence, transfer function analysis of asymptotic growth may be executed with the same basic information as sensitivity and elasticity analysis. Another strength of TFA is that it may be used to perturb more than one vital rate at a time, as illustrated later (and whilst we concentrate here on single-parameter, rank-one perturbations, TFA may also be extended to model multiple perturbations simultaneously: we refer to Hodgson & Townley 2004 and Hodgson, Townley & McCarthy 2006 for more information on specifying these complex perturbation structures).

#### The Transfer Function of Population Inertia

An extension to equation 2 allows evaluation of the transfer function of population inertia. Population inertia (P_{∞}) is calculated using:

- (eqn 3)

Here, **v** is the dominant left eigenvector of **A** and **w** is the dominant right eigenvector of **A** (as described earlier), is the vector representing the demographic distribution of the population (scaled to sum to 1) and ||**w||**_{1} denotes the one-norm, or column sum, of **w** (Koons, Holmes & Grand 2007; Stott, Townley & Hodgson 2011). Replacing with **v**_{max} or **v**_{min}, the standard basis vectors that maximise and minimise population inertia, will give the upper bound () or lower bound () on population inertia, respectively (Stott, Townley & Hodgson 2011), and this is true for all subsequent formulae. Working with this basic equation, it can be shown that population inertia of the perturbed matrix **A**_{δ} is equivalent to:

- (eqn 4)

where **I** is the identity matrix of same dimension as **A** and **c** is a column vector of ones of equal dimension to **A** (see Appendix S1 for proofs). Hence, given the same parameters as before (**d**, **e** and a range of δ), plus one extra parameter (, **v**_{max} or **v**_{min}), it is possible to first use equation 2 to evaluate the relationship between λ and δ as described earlier and then use equation 4 to evaluate the relationship between population inertia and λ. The result is three sets of numbers: a set of δ values (the perturbation magnitude), a set of λ values (the corresponding asymptotic growth) and a set of population inertia values (the corresponding transient population density). It is then possible to plot the precise relationships between population inertia and δ, between population inertia and λ, and between λ and δ.

#### Sensitivity of Population Inertia

Transfer functions are extremely useful for informing population management, but sensitivity measurements may remain useful for comparative statistical analyses among models, populations or species (e.g. Franco & Silvertown 2004; McMahon & Metcalf 2008). In such analyses, the scalar value of sensitivity is more amenable than a complex, nonlinear transfer function.

Using the transfer function method, it is fairly simple to obtain an estimate for the sensitivity of population inertia through differentiation of the transfer functions themselves. Equation 4 may be represented as:

- (eqn 5)

where

- (eqn 6)

The derivatives of the functions *f*_{1}(λ), *f*_{2}(λ) and *f*_{3}(λ) separately are:

- (eqn 7)

Then, the product rule and the quotient rule are used to differentiate equation 5 with respect to λ:

- (eqn 8)

Here, using the limit as λ approaches λ_{max} is a necessary step, as matrices used in the equation become singular (i.e. their inverse cannot be computed), when λ = λ_{max}.

Equation 2 can be directly differentiated with respect to δ to give the sensitivity of λ to δ:

- (eqn 9)

Again, the limit as λ approaches λ_{max} must be used to avoid singularities. Third, the chain rule is used to find the sensitivity of population inertia to δ:

- (eqn 10)

(see Appendix S1 for proofs). All code used in analyses is freely available as part of the R package ‘popdemo’ (R Development Core Team 2011; Stott, Hodgson & Townley 2012).

### Results

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgement
- References
- Supporting Information

In this section, we provide two worked examples using the analyses outlined earlier. Both are empirical PPM models parameterised from field data, and previous studies of both populations have led to very clear management recommendations based on asymptotic perturbation analyses. We explore the transfer function for population inertia of each, using parameter inputs based on their recommended management strategies, and discuss how the results from these analyses might affect the outcome of proposed management. We highlight how the nonlinear predictions of transfer functions compare with the linear predictions of sensitivity analysis. For each model, we evaluate the effect of management on the predicted inertia of the current population structure, which provides the most accurate prediction of perturbed dynamics. We also explore the effect of management on the upper and lower bounds on inertia. Management is often unlikely to operate near the inertia boundaries, although there are some notable scenarios when this may be the case: for example, in population invasions and reintroductions, individuals are likely to be concentrated in a single stage class. For the case studies presented here, the bounds on inertia provide a good indication of the effect of perturbation on the range of transient dynamics, which is useful knowledge if there is uncertainty in the estimation of current population structure or if there is the potential for disturbance to cause that structure to change.

#### Cactus (*Neobuxbaumia macrocephala*)

*Neobuxbaumia macrocephala* is a rare species of columnar cactus, endemic to the Tehuacan valley, central Mexico. Esparza-Olguín, Valverde & Vilchis-Anaya (2002) parameterised two size-based PPM models for the species using data collected in the field from 1997 to 1999. Based on elasticity analysis and simulated perturbation analysis of long-term growth, they recommended prolonging survival of the largest individuals as the best management strategy to increase population growth of the species.

Here, we work with the 1997–1998 matrix (the same matrix used in simulation analyses in Esparza-Olguín, Valverde & Vilchis-Anaya 2002), which we call **A**_{cactus} (Table 1). We perturb stasis of the largest size class (element [10,10]), in accordance with the recommended management regime: therefore, the perturbation structure is determined by the vectors **d**_{cactus} and **e**_{cactus} (Table 1). The population structure is known (Table 1; Fig. 1a), and we use this to calculate the transfer function of case-specific population inertia. However, there is uncertainty in the number of seedlings in the population: none were detected, but this is likely a result of the fact that they are small and hard to find. Therefore, evaluation of the transfer functions of the outer bounds on inertia provides an indication of the range of values of transient dynamics that may result from any population structure. Given that stasis of the largest individuals is 0·869, we perturb over a realistic range of −0·1 ≤ δ ≤ 0·1, to give a maximum perturbed stasis value of 0·969.

Stage | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
---|---|---|---|---|---|---|---|---|---|---|---|

A_{cactus} | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 4·130 | 11·456 | 10·238 | 31·370 |

2 | 0·013 | 0·435 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

3 | 0 | 0·217 | 0·677 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

4 | 0 | 0 | 0·097 | 0·865 | 0 | 0 | 0 | 0 | 0 | 0 | |

5 | 0 | 0 | 0 | 0·108 | 0·875 | 0 | 0 | 0 | 0 | 0 | |

6 | 0 | 0 | 0 | 0 | 0·025 | 0·900 | 0 | 0 | 0 | 0 | |

7 | 0 | 0 | 0 | 0 | 0 | 0·050 | 0·850 | 0·077 | 0 | 0 | |

8 | 0 | 0 | 0 | 0 | 0 | 0 | 0·150 | 0·615 | 0 | 0 | |

9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0·308 | 0·700 | 0·091 | |

10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0·300 | 0·869 | |

0 | 0·1116 | 0·1505 | 0·1797 | 0·1942 | 0·0971 | 0·0971 | 0·0631 | 0·0485 | 0·0582 | ||

d_{cactus} | Long term | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |

Transient | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

e_{cactus} | Long term | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |

Transient | 0 | 0 | 0 | 0 | 0 | 0 | 0·13 | 0·36 | 0·32 | 1 |

The population currently has a relatively large inertia and is expected to increase to a density around 400% of that predicted by λ_{max} (Fig. 1b). Given that the goal is to increase population density and growth, this is good news: although the population is predicted to decline in the long term, current population structure dictates that it will in fact increase rapidly in the short term and should not dip below current density within the next 70 years (Fig. 1b). The recommended management regime is beneficial in the long term; however, it has a negative impact on population inertia: any increase in stasis of the largest stage class will cause a drop in the value of population inertia (Fig. 1d). The magnitude of this decline is not captured by sensitivity, which predicts a far less pronounced decline in inertia. Thus, a significant benefit of management is not seen for several decades, because of the dampening of transient dynamics. Evaluation of the outer bounds on population inertia indicates further detriment of managing for adult survival. Fluctuations in the upper bound on inertia (Fig. 1c) and a decrease in the lower bound (Fig. 1e) cause an increase in the range of potential transient density. Sensitivity analyses do not accurately describe these changes over large perturbation range – in particular, the upper bound shows highly nonlinear dynamics that are far from those predicted by sensitivity.

Managing for transient dynamics can yield a better result. Transfer functions of inertia across the life cycle (Appendix S2) indicate that fecundity has the most positive impact on transient dynamics. Using a perturbation structure that acts on fecundity across all reproductive life stages (Table 1) increases population inertia (Fig. 2d). A doubling of fecundity significantly increases short-term population size (Fig. 2b), giving much more rapid results than managing for increased adult survival, despite the fact that this management scheme has a negligible effect on long-term growth. Additionally, this management practise increases both upper (Fig. 2c) and lower (Fig. 2e) bounds on inertia: given uncertainty in the population structure, this is favourable as it shifts the potential range of transient dynamics upwards on the scale of transient density. Therefore, it may be favourable in this case to manage for increased fecundity rather than increased adult survival, as the immediate returns are better. Alternatively, it may be desirable to implement two different management strategies: first, to manage for transient dynamics to boost population density and then to manage for asymptotic dynamics to maintain population persistence. However, this will of course inevitably depend on the relative costs of each management strategy.

#### Koala (*Phascolarctos cinereus*)

The Koala, *Phascolarctos cinereus*, is an arboreal marsupial native to Australia. It is widespread within the country, although the status of populations varies from region to region: in some areas, it is overabundant, whilst in others, populations are in persistent decline. Baxter *et al.* (2006) explore potential management strategies for an overabundant population on Snake Island, with a goal of reducing population density. They parameterise a stage-based PPM with stages defined by the tooth wear of koalas, which is an indication of age. The study uses asymptotic elasticity analysis that explicitly incorporates costs of management, therefore identifying those strategies that are both ecologically and economically viable. Based on these analyses, they recommend investing in reduction of both fecundity and survival, in an approximate 0·3:1 ratio. Subdermal contraceptives are considered as a mechanism to decrease fecundity, whilst translocation of adults to more sparsely populated areas is recommended to decrease survival by proxy, without the need for culls.

It is possible to evaluate the precise impact of this management strategy on the transient dynamics of the population using transfer function analysis. We perturb the koala PPM **A**_{koala} using a structure determined by vectors **d**_{koala} and **e**_{koala} (Table 2). Recommended management targets all adult life stages, but modelling this would require a multi-parameter, multi-rank perturbation structure. As an analytical compromise, we have chosen to focus on the vital rates of stage 2 individuals, as they have both largest fecundity and highest survival. The perturbation structure acts on several vital rates simultaneously: fecundity of stage 2 individuals (element [1,2]) is perturbed by factor 0·3, whilst stasis and growth of stage 2 individuals (elements [2,2] and [3,2], the life cycle transitions that incorporate survival) are both perturbed by factor 0·5 each. This gives the 0·3:1 investment ratio recommended by the study. We evaluate transfer functions based on the known structure of the population (McLean 2003; Table 2), although again we evaluate the transfer functions of bounds on inertia to get an idea of the range of uncertainty surrounding this estimate. We perturb over the range−0·3 ≤ δ ≤ 0·1, to evaluate the short-term impact of management over a wide range of investment, whilst keeping vital rates of the perturbed matrix within realistic limits.

Stage | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
---|---|---|---|---|---|---|---|---|---|---|

A_{koala} | 1 | 0 | 0·3026 | 0·1663 | 0·1244 | 0·0891 | 0·0556 | 0·0394 | 0·0226 | 0·0118 |

2 | 0·9908 | 0·5359 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

3 | 0 | 0·4580 | 0·4550 | 0 | 0 | 0 | 0 | 0 | 0 | |

4 | 0 | 0 | 0·5000 | 0·0655 | 0 | 0 | 0 | 0 | 0 | |

5 | 0 | 0 | 0 | 0·7272 | 0·2216 | 0 | 0 | 0 | 0 | |

6 | 0 | 0 | 0 | 0 | 0·4617 | 0·2265 | 0 | 0 | 0 | |

7 | 0 | 0 | 0 | 0 | 0 | 0·3538 | 0·1267 | 0 | 0 | |

8 | 0 | 0 | 0 | 0 | 0 | 0 | 0·4693 | 0·4247 | 0 | |

9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0·1762 | 0·6090 | |

0·135 | 0·330 | 0·225 | 0·120 | 0·080 | 0·065 | 0·035 | 0·010 | 0 | ||

d_{koala} | Long term | 0·3 | 0·5 | 0·5 | 0 | 0 | 0 | 0 | 0 | 0 |

Transient | 0 | 0 | 0·5 | 0·5 | 0 | 0 | 0 | 0 | 0 | |

e_{koala} | Long term | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Transient | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |

Population structure is very close to the stable structure predicted by the model (Fig. 3a), and as such population dynamics closely follow long-term growth from the outset (Fig. 3b). Hence, population inertia is small, with the population becoming at most up to 3% larger than predicted by λ_{max}. A reduction in survival and fecundity using the recommended management regime increases population inertia (Fig. 3d), which is directly antagonistic to management goals. The absolute increase in inertia is small, but it is important to note that recommended management does not result in an immediate halt to growth: the population will continue to grow for 2–3 years before stabilizing (Fig. 3b). Nonlinearity of the transfer function means that the larger the magnitude of perturbation, the less accurate sensitivity analysis is at describing the magnitude of change in inertia. The potential range of transient dynamics is decreased overall, with management decreasing the upper bound on inertia (Fig. 3c) and increasing the lower bound on inertia (Fig. 3e). This indicates that management can decrease the uncertainty surrounding short-term population density, although again these changes are small in their magnitude. Sensitivity analysis would have overestimated this effect: both transfer functions show diminishing returns, which is not captured by sensitivity approximations.

Managing for transient dynamics would suggest that focusing on survival alone is a better option: decreasing fecundity tends to increase inertia, whereas decreasing survival tends to decrease inertia. It appears that targeting stage 3 individuals would have the largest impact (Appendix S2). Decreasing survival of stage 3 individuals does reduce population inertia (Fig. 4d), and it is possible to manipulate this to stop population growth, despite long-term growth being above unity (Fig. 4b). This management option also results in a significantly decreased upper bound on inertia (Fig. 4c) and a barely altered lower bound on inertia (Fig. 4e). However, the effort and cost required to implement this management scheme would be far greater than that recommended by Baxter *et al.* (2006), and within a few years, the population resumes growth. As a long-term management scheme, this proves therefore to be inadequate. However, it may be desirable to first, manage for transient dynamics through translocation of adults, and then phase in contraceptive measures to manage for asymptotic dynamics. This exploits the population reductions that can be achieved through transient dynamics and then maintains those reductions in the long term.

### Discussion

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgement
- References
- Supporting Information

The transfer function for population inertia provides a means of nonlinear perturbation analysis of transient population dynamics. We have illustrated the importance of considering this alongside asymptotic perturbation analyses in population management, which adds to other studies that have demonstrated that an ignorance of transient dynamics can work against management goals (Koons, Rockwell & Grand 2007).

A notable outcome of our analyses is the extreme nonlinearity that can be exhibited in perturbation curves of population inertia. Many of the transfer functions generated in the case studies showed a marked nonlinearity that may render the linear approximations of sensitivity analysis ineffective except for very small perturbation magnitude. Figure 5 illustrates some more examples of extreme nonlinearity exhibited in transfer functions of the cactus and koala models, and Appendix S2 explores these further. Our evaluation of many other PPM models not presented here suggests that such nonlinearity is perhaps the rule rather than the exception. It has been argued that sensitivity analyses of asymptotic growth may be fairly robust to model assumptions, including linearity of perturbation response (Caswell 2001 p.613-615), and this warrants further study. Our preliminary evidence suggests that this may not be the case for transient dynamics. Targets of conservation and management often require large magnitude of perturbation, and so sensitivity may prove too poor an approximation for predicting their impact on transient dynamics.

In the light of this, nonlinear perturbation of transient dynamics could prove to be very important, and working with population inertia and the transfer function framework has a number of benefits. In a previous paper (Stott, Townley & Hodgson 2011), we identified an emerging framework for analysis of transient dynamics with choices to make regarding: (i) standardising dynamics to remove asymptotic effects, (ii) choosing an initial demographic distribution to use in the model and (iii) choosing a timeframe to evaluate. Population inertia (Koons, Holmes & Grand 2007) is an index of transient dynamics that is robust to all of these choices: as it is possible to evaluate upper and lower bounds on inertia (Stott, Townley & Hodgson 2011), there is no need to know the population’s demographic distribution, although it is possible to evaluate sensitivity of case-specific dynamics using a given demographic distribution if desired. Population inertia correlates very well with other measures of transient dynamics (Stott, Townley & Hodgson 2011), indicating that it should be a good proxy index to use in perturbation analyses, especially when it is unclear what timeframe of the projection to analyse. Lastly, in the equation for population inertia, the influence of asymptotic dynamics is removed, so that perturbation analysis of inertia is a standardised approach that may be compared across models, regardless of differences in their asymptotic dynamics. The transfer function framework works well with population inertia: matrix eigendata are easy to express in terms of transfer functions and population inertia is expressed in terms of matrix eigendata. As illustrated, the methods we present are flexible, compared with classic approaches, in their ability to model complex management regimes involving simultaneous perturbation to many matrix elements. We have focussed here on single-parameter, rank-one perturbations, whilst multi-parameter, multi-rank perturbations may offer the option to study yet more complicated perturbation structures (Hodgson, Townley & McCarthy 2006). We are working on the more complicated algebraic expansion for this method.

An alternative approach of nonlinear perturbation analysis is to use matrix determinants to represent the transfer function (and derivatives thereof) in terms of characteristic polynomials (Lubben *et al.* 2009; Miller *et al.* 2011). This affords a more flexible specification of perturbation structure without the need for complex algebraic expansion and offers another benefit in its ability to evaluate polynomials at λ = λ_{max}. However, when using this approach to attempt analysis of population inertia, we discovered numerical calculation and evaluation of polynomials to be inherently unstable, in both R (R Development Core Team 2011) and MATLAB (Mathworks 2011), using the polynom package in R and the ss2tf function in MATLAB. It is possible that the use of symbolic algebra (as advocated in Miller *et al.* 2011) will lessen this instability.

The methods discussed here would benefit from application to other population models. It would certainly be useful to have analytical transient indices and nonlinear perturbation methods for stochastic and density-dependent models. Another important challenge is to migrate transient analyses and nonlinear perturbation analyses to integral projection models (Easterling, Ellner & Dixon 2000). It is argued that these models require less data than PPMs so are better for data-deficient systems (Ramula, Rees & Buckley 2009), and as they do not rely on an arbitrary discretisation of the life cycle, they may be robust to many of the analytical problems associated with matrix dimension (see Easterling, Ellner & Dixon 2000; Salguero-Gómez & Plotkin 2010; Stott *et al.* 2010a), and matrix reducibility (Stott *et al.* 2010b). Stochastic and density-dependent applications exist (Ellner & Rees 2006, 2007), but the study of transient dynamics and use of nonlinear perturbation analysis requires development.

Meanwhile, it is becoming increasingly easy to incorporate complex analyses into studies that use linear, time-invariant PPM models. Many studies rely on such basic models, as parameterisation of stochastic or density-dependent models often requires more data than it is feasible to provide, especially under constraints on time and finance. But, perhaps analytical approaches need to move beyond sensitivity when it comes to informing population management: methods for nonlinear perturbation analysis of both asymptotic and transient dynamics are now freely available. These vastly increase the information that can be provided by simple models, without any demand for further data. Most importantly, as many studies are starting to show, an ignorance of these dynamics may even be detrimental to management goals.

### Acknowledgement

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgement
- References
- Supporting Information

Funding provided by the European Social Fund.

### References

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- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgement
- References
- Supporting Information

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### Supporting Information

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgement
- References
- Supporting Information

**Appendix S1.** Algebraic proofs of formulae.

**Appendix S2.** Matrix multi-plots of transfer functions.

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