Present address: Department of Biological Sciences, University of Plymouth, Drake Circus, Plymouth PL4 8AA, UK.

# Integrating vital rate variability into perturbation analysis: an evaluation for matrix population models of six plant species

Article first published online: 23 JAN 2004

DOI: 10.1111/j.1365-2745.2001.00621.x

Additional Information

#### How to Cite

Zuidema, P. A. and Franco, M. (2001), Integrating vital rate variability into perturbation analysis: an evaluation for matrix population models of six plant species. Journal of Ecology, 89: 995–1005. doi: 10.1111/j.1365-2745.2001.00621.x

#### Publication History

- Issue published online: 23 JAN 2004
- Article first published online: 23 JAN 2004
- Received 31 October 2000 revision accepted 9 May 2001

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### Keywords:

- demographic variation;
- plant demography;
- population growth rate;
- sensitivity analysis;
- variance-standardized perturbation

### Summary

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

- 1Matrix population models are usually constructed by employing average values of vital rates (survival, growth and reproduction) for each size category. Perturbation analyses of matrix models assess the influence of vital rates or matrix elements on population growth rate. They consider the impact of either an unstandardized (sensitivity analysis) or a mean-standardized (elasticity analysis) change in a model component. Certain vital rates are intrinsically more variable than others. This variation can be taken into account in variance-standardized perturbation analysis, which applies changes to vital rates in proportion to their variability.
- 2We applied variance-standardized perturbation analysis to six plant species with different life histories (a forest understorey herb, two tropical forest palms and three tropical forest trees). 1500 random values were drawn from observed frequency distributions of each vital rate in each size category, and population growth rates (λ) were calculated for each of the simulations.
- 3Variability differed widely between vital rates, being particularly high for growth and reproduction. Vital rate variation was negatively correlated with its effect on λ (measured by either sensitivity or elasticity). The variation in λ resulting from the sampling procedure differed between species (with higher values in shorter-lived plants) and vital rates (with particularly high values due to variation in growth rates).
- 4The relationships between λ and vital rates were close to linear. Therefore, the product of sensitivity (or elasticity) and degree of variability of a vital rate was a good estimator of the variation in λ, explaining 95% of the variation in λ in the six study species.
- 5Thus, a reliable estimation of the 95% confidence interval of λ due to variation in one of the vital rates can be calculated as the product of the 95% confidence interval of the vital rate and its sensitivity.
- 6Our results suggest that variance-standardized perturbation analyses are a useful tool to determine the impact of vital rate variation on population growth rate.

### Introduction

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

Matrix population models are a widely used tool of demographic analysis (Caswell 1989a). An important development of these models is the family of techniques known as perturbation analyses, which compare the importance of different model components for model output – usually population growth. The most commonly used perturbation techniques, sensitivity (Caswell 1978) and elasticity (de Kroon *et al*. 1986) analyses, have become standard procedures in a range of research fields (Benton & Grant 1999; de Kroon *et al*. 2000). Sensitivity analysis considers the effect on population growth of a fixed infinitesimal change in a vital rate (*unstandardized* perturbation), whereas elasticity analysis quantifies the influence of an infinitesimal change that is proportional to the size of each vital rate (*mean-standardized* perturbation).

Matrix population models are most often constructed using average values of vital rates (survival, growth and reproduction), and their projections assume that population dynamics are constant over time and space. However, vital rates vary between individuals, between populations and over time, and it is important to evaluate the effect of this demographic variation on (the reliability of) the resulting population growth rates. Several techniques have been developed to assess the impact of variation in time (e.g. Tuljapurkar 1989; Caswell & Trevisan 1994), space (e.g. Alvarez-Buylla & García-Barrios 1993; Alvarez-Buylla 1994; Pascarella & Horvitz 1998), or both time and space (e.g. Caswell 1989b, 1996a; Horvitz *et al*. 1997; Ehrlén & van Groenendael 1998). Most of these techniques consider variation between populations or over different time periods, using sets of transition matrices, each of which represents one (sub)population or one period of time.

Less attention has been paid to the influence of variation *within* one population or during one time period, although its significance is widely acknowledged (Sarukhán *et al*. 1982; van Tienderen 1995, 2000; Silvertown *et al*. 1996; Ehrlén & van Groenendael 1998; Benton & Grant 1999; Bierzychudeck 1999; de Kroon *et al*. 2000). This variation is not explicit in the transition matrix, as the matrix only contains average values. It may result from genetic and phenotypic variation between individuals, from differences in microenvironmental conditions experienced by different individuals and from uncertainty in parameter estimates (Wisdom & Mills 1997; Caswell *et al*. 1998; Hunter *et al*. 2000; Wisdom *et al*. 2000). A first-order approximation of the variance in population growth rate demonstrates how demographic variation leads to uncertainty in population growth (Lande 1988; Caswell 1989a). This concept has been used in the development of methods to determine confidence limits for population growth rate (Caswell 1989a; Alvarez-Buylla & Slatkin 1991, 1993, 1994).

Some vital rates (e.g. seed production) are more variable and thus more liable to change than others (e.g. adult survival). Demographic processes with a low sensitivity value but a very high degree of variation may thus potentially cause more variation in the population growth rate than those processes with high sensitivity and low variability. When the probability of changes in certain vital rates is taken into account, those vital rates with the greatest ‘influence’ on population growth rate may differ from those suggested by sensitivity or elasticity analysis alone. Van Tienderen (1995) has suggested a perturbation technique that specifically incorporates variability in demographic processes, which can be considered *variance-standardized*, since changes imposed are proportional to the variability of each vital rate (e.g. by using standard deviation units). A similar approach has been adopted by Ehrlén & van Groenendael (1998), employing several transition matrices.

We conduct such analyses for six plant species with different life histories using Monte Carlo simulations in which values of vital rates are randomly drawn from observed frequency distributions and population growth rates are calculated for each sampling exercise. The advantages of this simulation procedure over the use of a fixed perturbation unit (e.g. one standard deviation, as in van Tienderen 1995) are (1) that the relationships between population growth rates and vital rates can be visualized and analysed, and (2) that frequency distributions of population growth rates can be obtained, from which the degree of variability in the population growth rate can be derived. The perturbation approach adopted here is comparable to sensitivity and elasticity analyses in that it applies a change in one vital rate and in one size category at a time while keeping all others constant. In the absence of direct information we assume that covariation between vital rates is negligible. Perturbations are carried out directly on the vital rates (survival, growth and reproduction; also called lower-level parameters) determined in field studies, rather than on matrix elements. Such rates are also often the focus for intervention in the conservation and management of natural populations (Caswell 1996a; Mills *et al*. 1999).

Specifically, the goals of our study were: (1) to examine patterns of variability of vital rates for plant species with different life histories; (2) to determine the effect of this variability on population growth; (3) to determine the form of the relationships between population growth rates and vital rates and to test whether the often-assumed linear approximation of these relations holds; and (4) to compare the results obtained from variance-standardized perturbation analysis to those obtained with standard sensitivity and elasticity analysis.

### Materials and methods

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

#### Elements of transition matrices

Matrix population models project the size and structure of populations in time. The basic model is **n**(*t* + 1) = **A n**(*t*), where **A **is a square *m* × *m* matrix with transitions among *m* categories during a certain time interval, and **n** is the population vector containing densities of individuals in *m* size (or stage) categories. The dominant eigenvalue (λ) of matrix **A **is equivalent to the growth rate of the population (Caswell 1989a). In stage-based matrix models (Lefkovitch 1965), elements *a _{ij}* (with

*i*denoting row number and

*j*denoting column number) of transition matrix

**A**can be grouped into those representing stasis (

*P*, the probability of surviving and remaining in stage

_{j}*j*from one recording date to the next), progression (

*G*, the probability of surviving and growing from stage

_{ij}*j*to stage

*i*which contains larger individuals), retrogression (

*R*, the probability of surviving and going back from stage

_{ij}*j*to stage

*i*which contains smaller individuals) and fecundity (

*F*, the number of sexual offspring produced by an individual in stage

_{j}*j*).

Matrix elements are built from underlying *vital rates*– survival (σ_{j}), growth (positive: γ_{ij}; negative: ρ_{ij}) and reproductive output (*f _{j}*) – to which they are related by ,

*G*= σ

_{ij}_{j}γ

_{ij},

*R*= σ

_{ij}_{j}ρ

_{ij}and

*F*=σ

_{j}_{j}

*f*. We employ pairs of subscripts only when several transitions occur (e.g. progression and retrogression), using just the subscript of the source category (

_{j}*j*) for matrix elements and vital rates that involve a single transition (e.g. stasis) or are specific to a stage (e.g. survival).

#### Vital rate sensitivity and elasticity

In its standard form, sensitivity analysis considers the impact of changes in *matrix elements* on population growth rate (λ):

eqn 1

where *s _{ij}* is the sensitivity of matrix element

*a*,

_{ij}*v*and

_{i}*w*are elements of the left (

_{j}**v**) and right (

**w**) eigenvectors associated with λ, and <

**w**,

**v**> denotes the scalar product (Caswell 1978).

However, sensitivities can also be applied to the *vital rates* (lower-level parameters) (Caswell 1989a, 1996a; Brault & Caswell 1993; Mills *et al*. 1999). This is done by tracking the changes in λ resulting from changes in the vital rates implicit in matrix elements *a _{ij}*. For the vital rates mentioned above, the sensitivities of vital rates are (following Caswell 1989a; pp. 126–129 and Caswell 1996a):

eqn 2

eqn 3

eqn 4

eqn 5

Elasticity values can also be calculated for vital rates. Standard elasticity considers the proportional change in λ due to a proportional change in a parameter (de Kroon *et al*. 1986):

eqn 6

where *e _{ij}* is the elasticity of matrix element

*a*, and

_{ij}*s*is its sensitivity. Equation 6 shows that elasticity is obtained by multiplying sensitivity by

_{ij}*a*/λ. In analogy, elasticity values of vital rates can be obtained by multiplying vital rate sensitivity by

_{ij}*x/*λ (Caswell 1989a; p. 135), where

*x*is the value of the vital rate under consideration (σ

_{j}, γ

_{ij}, ρ

_{ij}or

*f*).

_{j}Unlike the values for matrix elements, vital rate sensitivity and elasticity may be negative, but as it is the magnitude of the change that interests us here, rather than its sign, absolute values are quoted throughout the paper. For vital rates, elasticities do not sum to 1 (Caswell 1989a).

#### Variance-standardized perturbation analysis

In order to assess the influence of variability in each demographic parameter on the population growth rate (λ) we consider variation in vital rates (survival, growth and reproduction), which represent a single demographic process, rather than in matrix elements, which combine several processes. We consider variation in *one* vital rate in *one* size category at a time. Demographic variation between individuals may be caused by genetic, phenotypic and/or microenvironmental differences within a category, as well as by temporal variation.

##### Frequency distributions

Frequency distributions, based on variation in vital rates between individuals in each category, were first identified. For observed transitions, positive and negative growth (γ_{ij} and ρ_{ij}) can be described using a binomial frequency distribution (as the basic observations are whether an individual does or does not move into another category). If growth in size is measured for the individual plants and subsequently used to calculate transitions using stage durations, individual growth rates mostly follow a (log)normal distribution. Survival probability (σ_{j}) follows a binomial distribution. Reproductive output (*f _{j}*) may follow either a (log)normal or binomial distribution, or be made up of several parameters, each with a different frequency distribution (e.g. probability of reproducing [binomial] × number of offspring produced per reproductive individual [(log)normal]): values are then drawn from the respective distributions and multiplied together. For each category, the frequency distributions of observed vital rates are described by their mean and (binomial) standard deviation.

##### Sampling of individuals

For each combination of size category and vital rate, 1500 values were drawn at random, with the restriction that all had to fall within the 95% confidence interval of the observed vital rate distribution, and that (bio)logically impossible values (negative γ_{ij} and *f _{j}* values, probability values outside the <0,1> interval) were excluded. For vital rates with a binomial frequency distribution, values were drawn from a normal distribution, as this approximates the binomial distribution fairly well for moderately large samples (Sokal & Rohlf 1995). A matrix was constructed for each of the 1500 sampled values, using unchanged values for all other vital rates of the category under consideration and for matrix elements of all other categories. Population growth rate (λ) was then computed for each matrix.

##### Analysis of simulation results

Mean, standard deviation, 95% confidence interval (calculated using the SD of the distribution; henceforth referred to as CI95) and coefficient of variation (CV) were calculated for each vital rate and the resulting λ-values. The magnitude of changes in λ due to the simulated variation in vital rates was compared between species and vital rates, and related to both their degree of variation and to their effect on population growth (sensitivity and elasticity). The relationship between each vital rate and population growth was also assessed. Absolute (sensitivity, CI95) as well as relative (elasticity, CV) measures of importance and variability of vital rates were calculated.

We applied the perturbation analysis outlined above to six plant species with different life histories and life spans (Table 1), and including small to large-sized transition matrices. Detailed information on demographic studies, matrix construction and vital rates of the study species can be found in Appendix 1.

Species | Family | Life form | λ | Size | Life span | Reference |
---|---|---|---|---|---|---|

Primula vulgaris | Primulaceae | Forest understorey herb | 1.038 | 5 | 10–30 | Valverde & Silvertown 1998 |

Geonoma deversa | Palmae | Understorey palm (ramets) | 1.041 | 6 | c. 37 | Zuidema 2000 |

Euterpe precatoria | Palmae | (Sub)canopy palm | 0.977 | 11 | 106 (35) | Zuidema 2000 |

Duguetia neglecta | Annonaceae | Understorey tree | 1.006 | 12 | 255 (95) | Zagt 1997 |

Bertholletia excelsa | Lecythidaceae | Emergent tree | 1.007 | 17 | 360 (106) | Zuidema & Boot, in press |

Chlorocardium rodiei | Lauraceae | Canopy tree | 0.998 | 15 | 444 (156) | Zagt 1997 |

### Results

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

#### Variability and sensitivity of vital rates

Variability of vital rates differed widely, with coefficients of variation ranging from 0 to 77 (Median CV = 27.1, *n* = 188). Vital rate types differed significantly, both for absolute (CI95; Kruskal–Wallis (KW): χ_{3}^{2} = 42.7, *P* < 0.001) and relative (CV; χ_{3}^{2} = 120.7, *P* < 0.001) measures of variation. Confidence intervals were narrower for survival than for positive growth and reproduction (multiple comparisons KW, *P* < 0.05; Siegel & Castellan 1988). No difference was found between species in vital rate variability, either for absolute (comparing CI95 values of all vital rates between species; KW: χ_{5}^{2} = 8.9, *P* = 0.11) or for relative measures (CV values: χ_{5}^{2} = 3.9, *P* = 0.57).

Considered over all vital rates and species, there was a negative correlation between the variability of a vital rate and its effect on λ, both when measured in absolute (CI95(vr) vs. sensitivity) and in relative (CV(vr) vs. elasticity) terms (Fig. 1, Table 2). It is clear that reproductive output, which spans almost the entire range of values, is mostly responsible for this relationship when absolute measures are considered (Fig. 1c; Table 2: Spearman's *r* = −0.90). Looking at the contribution by species, the points with lowest sensitivity and highest vital rate CI95 values correspond to the reproductive output values of the two longest-living tree species, *Chlorocardium rodiei* and *Bertholletia excelsa* (lower right-hand side of Fig. 1a). In contrast, when considering relative measures (elasticity vs. CVs in Fig. 1b,d), survival rather than reproduction spans the widest range, and this is mostly responsible for the overall correlation (Table 2). Survival in adult categories occupies the high elasticity–low variability end of the relationship, whereas seedling survival occupies the opposite end (KW for CV of drawn values in seedling, juvenile and adult categories: χ_{2}^{2} = 29.4, *n* = 65; *P* < 0.001). When evaluated at the species level, negative correlations between vital rate variability and sensitivity or elasticity of λ were found for most species (Table 2).

Species | Vital rate | Sensitivity vs. CI95(vr) | Elasticity vs. CV(vr) | n |
---|---|---|---|---|

^{}* *P*< 0.05, ***P*< 0.01, ****P*< 0.001, NS = not significant.
| ||||

All | All | −0.54*** | −0.70*** | 188 |

Primula | All | NS | −0.63** | 17 |

Geonoma | All | −0.63** | −0.61** | 18 |

Euterpe | All | −0.61** | −0.73*** | 25 |

Duguetia | All | −0.77*** | −0.72*** | 34 |

Bertholletia | All | −0.72*** | −0.53*** | 39 |

Chlorocardium | All | −0.74*** | −0.75*** | 55 |

All | Survival | −0.47*** | −0.56*** | 65 |

All | Positive growth | −0.27* | NS | 75 |

All | Negative growth | NS | NS | 20 |

All | Reproductive output | −0.90*** | NS | 28 |

#### Variance-standardized perturbation results

The variation in population growth rate (λ) resulting from the variance-standardized perturbations over all vital rate types and species ranged from 0.000 to 0.077 for CI95(λ) and from 0.001 to 1.891 for CV(λ). Significant differences were found between vital rate types, both for CI95(λ) (KW: χ_{3}^{2} = 14.6, *P* < 0.01) and CV(λ) (KW: χ_{3}^{2} = 29.4, *P* < 0.001), with negative growth having less influence on the variation of λ than other vital rates (Fig. 2a; KW: multiple comparisons, *P* < 0.05). Similarly, species differed in the magnitude of variation in λ resulting from vital rate variability (KW: χ_{5}^{2} = 52.3, *P* < 0.001 for CI95(λ) and χ_{5}^{2} = 51.3, *P* < 0.001 for CV(λ)).Variability of λ generally decreased with species life span (Fig. 2b).

The relationship between λ and vital rates is intrinsically non-linear, but the results of our simulations show a low degree of curvature leading to close to linear relations (e.g. Figure 3c). The relationship between λ and either survival or negative growth was usually concave (43 out of 63 cases and 16 out of 20, respectively, as in Fig. 3a). Relationships between λ and either positive growth or reproductive output were mainly convex (67 out of 75 and 19 out of 28, respectively, as in Fig. 3b). Vital rate sensitivity can be determined as the slope of the relationship between λ and the vital rate (vr) evaluated at the mean value (arrows in Fig. 3). By analogy, elasticity is the slope of the curve relating log(λ) to log(vr), also evaluated at the mean vital rate value.

The variation in λ found in our simulations is the result of both variation of vital rates (ranging between the vertical dotted lines in Fig. 3) and the responsiveness of λ to modifications of the vital rates (evaluated by sensitivity or elasticity; slopes in Fig. 3). To check the extent to which these two factors explained variation in λ, we related the magnitude of variation in λ (CI95(λ) or CV(λ)) to that in vital rates (CI95(vr) or CV(vr)) and to sensitivity or elasticity (Table 3). Each species showed a correlation with either one or the other, independently of differences in life history. However, both sensitivity (or elasticity) and vital rate variation were often positively correlated with variation in λ when particular vital rates were considered.

Species | Vital rate | CI95(λ) vs. | CV(λ) vs. | ||
---|---|---|---|---|---|

Sensitivity | CI95(vr) | Elasticity | CV(vr) | ||

^{}* *P*< 0.05, ***P*< 0.01, ****P*< 0.001, NS = not significant; where given, superscript values indicate exact probability level.
| |||||

All | All | 0.66*** | 0.20** | 0.60*** | NS |

Primula | All | 0.73** | NS | 0.78** | NS |

Geonoma | All | NS | NS | NS | 0.45^{0.063} |

Euterpe | All | NS | 0.49* | NS | 0.54** |

Duguetia | All | 0.64** | NS | 0.61** | NS |

Bertholletia | All | NS | 0.47** | NS | 0.30^{0.065} |

Chlorocardium | All | 0.69** | NS | 0.82** | NS |

All | Survival | 0.59*** | 0.31* | 0.53*** | 0.26* |

All | Positive growth | 0.69*** | 0.45*** | 0.90*** | 0.30* |

All | Negative growth | 0.84*** | NS | 0.95*** | NS |

All | Reproductive output | NS | NS | 0.48** | 0.55** |

The fact that the relationships between λ and vital rates were often close to linear implies that the slope of the relationship at the mean vital rate value is a good predictor of the ‘importance’ of the vital rate over its complete range of possible values (see Caswell 2000). We therefore evaluated the extent to which the product of vital rate variability (range of values along *x*-axis in Fig. 3) and the sensitivity (slope indicated in Fig. 3) would predict the variability of λ (range of values along *y*-axis in Fig. 3). For strictly linear relationships (Fig. 3c) this product would provide the exact value of the range of variation of λ. This approach was derived from the basic equation for variation in λ due to variation in matrix elements (equation 77 in Caswell 1989a). In this formula, the first-order approximation of Var(λ) is calculated by multiplying the variance in matrix element *a _{ij}* by its squared sensitivity, and summing these products over all cells in the matrix: Var(λ) = Σ(Var[

*a*] ×

_{ij}*s*

_{ij}

^{2}). We found a strong correlation between (sensitivity (or elasticity) × vital rate variability) and the variability of λ (Fig. 4). This, and the regression analyses on log-transformed data (Table 4) confirm the approximate linearity of the relationship between λ and vital rates (coefficients of determination and slopes obtained from these regressions were close to one, indicating a tight, linear relationship).

Species | Vital rate | CI95(λ) | CV(λ) | ||||
---|---|---|---|---|---|---|---|

R^{2} | Constant | CI95 × Sens | R^{2} | Constant | CV × Elast | ||

- *
*P*< 0.05; - **
*P*< 0.01; - ***
*P*< 0.001.
| |||||||

All | All | 0.96 | −0.23 | 0.95 (0.01)*** | 0.95 | 0.06 | 0.96 (0.02)* |

Primula | All | 1.00 | −0.01 | 1.00 (0.00) NS | 1.00 | 0.01 | 1.00 (0.01) NS |

Geonoma | All | 1.00 | 0.12 | 1.02 (0.02) NS | 0.88 | −0.31 | 0.91 (0.08) NS |

Euterpe | All | 0.91 | −0.55 | 0.90 (0.05) NS | 0.92 | 0.40 | 1.03 (0.06) NS |

Duguetia | All | 0.97 | −0.38 | 0.94 (0.03)* | 0.96 | 0.03 | 0.97 (0.03) NS |

Bertholletia | All | 0.89 | −0.29 | 0.93 (0.05) NS | 0.90 | 0.17 | 0.97 (0.05) NS |

Chlorocardium | All | 0.96 | −0.45 | 0.93 (0.03)** | 0.95 | 0.02 | 0.97 (0.03) NS |

All | Survival | 0.95 | −0.51 | 0.92 (0.03)** | 0.95 | −0.27 | 0.90 (0.03)*** |

All | Positive growth | 0.94 | −0.19 | 0.95 (0.03)* | 0.96 | 0.43 | 1.02 (0.03) NS |

All | Negative growth | 1.00 | −0.19 | 0.97 (0.01)* | 0.99 | 0.04 | 0.99 (0.02) NS |

All | Reproductive output | 1.00 | −0.10 | 0.99 (0.01) NS | 0.94 | −0.52 | 0.87 (0.04)** |

When comparing the results of variance-standardized perturbation analysis with sensitivity and elasticity analysis for the six study species, rather different patterns emerged (Fig. 5). There were marked differences in the relative ‘importance’ of vital rates suggested by the different methods, as shown by the changes in rank. Overall, when comparing elasticity and variance-standardized perturbation, survival decreases in importance whereas growth increases in importance from the former to the latter method. Survival probability is the vital rate with the highest elasticity: 93% of the top-5 ranked vital rates correspond to survival probability. In many cases this implies low mortality in adult categories or slow growth (long residence times) in juvenile stages. Ranks resulting from the variance-standardized perturbation method, on the other hand, show a much lower contribution of survival to the top-5 ranks (30%). Using this technique, positive growth is in many cases (60%) the most important vital rate. Here, the top-5 ranks mainly include growth of juveniles in categories with highly variable growth rate. In particular, the plastic response of juveniles of tropical forest trees (such as *Euterpe precatoria* and *Bertholletia excelsa*) to the large variation in light availability in their forest environments, may be responsible for the large simulated variation in population growth rate.

### Discussion

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

#### Significance of vital rate variation

Variation in vital rates differed widely between the six study species. In general, survival probability was the least variable vital rate, particularly in adult categories (Fig. 1). A negative relationship between vital rate importance (sensitivity or elasticity) and vital rate variability was found for all species and vital rates combined, but also separately for the six species and for most of the vital rates (Fig. 1, Table 2). Similar inverse relationships between demographic variation and sensitivity (or elasticity) have been found for temporal variation in a large number of plant and animal species (Ehrlén & van Groenendael 1998; Pfister 1998). These results have been interpreted to suggest that natural selection promotes population stability by reducing variability in relation to the importance of the life history traits involved (Pfister 1998).

Our simulations show that variation in vital rates as observed in demographic field studies may influence the estimates of population growth rate λ to different degrees (Fig. 2). Variation in survival probability and growth rate had the strongest impact on λ (Table 3), despite the fact that the former was the least variable vital rate. Comparing species, variation in λ appeared to be large for short-lived species and smaller for the longer-lived species (Fig. 2b).

#### Linearity of relationships

The functional relationships between λ and vital rates approached linearity in many instances (e.g. Caswell 1996a,b, 2000; de Kroon *et al*. 2000; but see Huenneke & Marks 1987), despite the intrinsically non-linear relationship between the dominant eigenvalue and either matrix elements or vital rates. This confirms that sensitivities and elasticities perform well beyond the point at which they are calculated (Caswell 2000; de Kroon *et al*. 2000). Given this linearity, an estimate of the variability of λ due to variation in a vital rate is obtained by multiplying the importance of the vital rate (expressed as sensitivity or elasticity) and its variability (expressed as either its 95% confidence interval or its coefficient of variation). When applying this to the six species, very strong relationships were found (Fig. 4, Table 4).

The implication of this result is that the first-order approximation for variance in λ (Caswell 1989a) can be used to obtain a simple and reliable estimate of the variation in lambda due to variation in a certain vital rate. Two readily available parameters can be multiplied to estimate the variation of λ:

CI95(λ) ≈ CI95(vr* _{j}*) ×

*s*eqn 7

_{j}where CI95(λ) is the 95% confidence interval of the population growth rate resulting from variation in a certain vital rate in category *j* (vr_{j}), CI95(vr_{j}) is the 95% confidence interval of vr_{j} and *s _{j}* is the sensitivity of vr

_{j}. When using relative measures, the analogue of eqn 7 is:

CV(λ) ≈ CV(vr* _{j}*) ×

*e*eqn 8

_{j}where CV(λ) is the coefficient of variation of λ due to variation in a vital rate, vr_{j}, in category *j*, CV(vr_{j}) is the coefficient of variation of vr_{j} and *e _{j}* is the elasticity value for vr

_{j}.

Estimates of variability can be obtained from field data or from the literature; sensitivity and elasticity of vital rates can be calculated using eqns 2–5 (Caswell 1989a, 1996a). Whether one applies absolute or relative measures does not influence the predictive strength of the relationship (in the simulations, both explained more than 88% of the variation in ln(λ); Table 4). It should be noted that eqns 7 and 8 resemble the equation used for ‘direct perturbation analysis’ (Ehrlén & van Groenendael 1998), which considers the variation in population growth rate of differences in vital rates between populations and time periods. One difference between their approach and ours is the source of variation: variation between populations or time periods (direct perturbation analysis) vs. that between individuals in a population (variance-standardized perturbation).

#### Covariation between vital rates

Vital rates are likely to covary and this can happen either within the same stage of the life cycle or between different stages. As a consequence, some combinations of vital rates will be more probable than others, and particular combinations may be impossible. In our simulations, we varied one vital rate in one category at a time, while keeping other vital rates in the focal category and all vital rates in all other categories, constant. That is, the variance-standardized perturbations did not adjust for covariance between vital rates. Some unrealistic combinations of vital rates may have resulted from this. However, by drawing values of vital rates only from the 95% confidence interval, we prevented unrealistic values and unrealistic combinations of values from having a large influence on the variability of λ. Our approach is thus similar to the application of large perturbations, as commonly used for the planning of conservation measures (Heppell *et al*. 1996; Mills *et al*. 1999; Caswell 2000; de Kroon *et al*. 2000).

The influence of demographic covariance on variation in population growth has not often been assessed (Brault & Caswell 1993; Horvitz *et al*. 1997; Caswell 2000). Inclusion of covariance in matrix models is probably hampered by two factors. Firstly, the two main techniques that assess the role of covariance have different starting points. In integrated elasticities (van Tienderen 1995, 2000), covariance between vital rates is specifically modelled using observed or assumed trade-off relationships. In life table response experiments (LTRE; Caswell 1989b, 1996a, 2000), on the other hand, covariances appear as an integral part of the quantification of contributions to observed variation in λ, thus providing empirical evidence of the role of covariation. As LTREs consider differences between (sub)populations or observation periods, to a certain extent they disregard trade-offs that only become apparent at the individual level (these may be included, however, in integrated elasticities). Secondly, the use of both integrated elasticity and LTRE is limited by the amount of information gathered in a particular study. Detailed knowledge on covariance structures necessary for integrated elasticities and sets of transition matrices required for LTRE are usually not available (Menges 2000).

#### Comparing perturbation methods

Conceptually, perturbation methods may be grouped into two families (Caswell 1997; Horvitz *et al*. 1997), namely prospective and retrospective analyses. Whereas prospective analyses (e.g. sensitivity, elasticity) explore the functional dependence of λ on either matrix elements or vital rates, retrospective analyses (e.g. life table response experiments: LTREs) determine how an observed pattern of variation has affected variation of λ in the past (Caswell 2000). Variance-standardized perturbation analysis, as applied in this paper, combines considerations of variation in λ due to observed variation in vital rates (an aspect of retrospective analysis) with the functional relationship of λ to that vital rate (a subject of prospective analysis; Fig. 3). It can be valuable because it identifies those vital rates that, as a result of their intrinsic variability (or uncertain estimation) and functional relationship with λ, have a large impact on the variation in population growth rate (see Fig. 5). Such information can be applied for the identification of manipulable traits for the management of endangered or exploited species.

When should different perturbation methods be chosen? This depends on the question posed. Elasticity and sensitivity analyses evaluate the *potential* effect on λ of a change in a vital rate, and therefore answer the question ‘What would happen if a particular vital rate were modified?’. Large modifications of certain vital rates may be used to assess the impact of conservation measures (e.g. Heppell *et al*. 1996; Mills *et al*. 1999; Caswell 2000). Life table response experiments (LTREs) evaluate the importance of demographic variation on variation in λ, addressing the question ‘What are the contributions of different vital rates to observed variation in population growth rate, for example, between (sub)populations or time periods?’. LTREs use a set of transition matrices for different populations or observation periods, and consider the influence of variation between these transition matrices on population growth. They are therefore especially useful for disentangling complex demographic consequences of environmental change. Variance-standardized perturbation analysis asks the question: ‘What are the consequences for population growth of observed variation in vital rates or uncertainty in their estimation?’, addressing the effect of either random variation between individuals or sampling bias on the rate of population growth. It considers the variation that is usually ‘hidden’ in the averaged matrix elements. Although the questions posed by LTREs and variance-standardized perturbation analysis are similar, they differ in starting point and required information. LTREs begin with observed variation in population growth rate of different populations or different time periods, whereas variance-standardized perturbation starts from observed variation in vital rates in one population and during one time period. LTREs therefore require several transition matrices for different (sub)populations or time periods, whereas variance-standardized perturbation requires one transition matrix with information on variability of the vital rates used for matrix construction. However, the sources of demographic variation used for both methods are related and often overlap: variation between populations can be considered as aggregated between-individual variation, spatial variation may be included in one transition matrix or specified in several (patch-specific) matrices and, similarly, demographic data from different years may be combined into one average model or used for several annual models. Thus, the form in which demographic data are presented may determine what method can be applied: LTRE can be used if several transition matrices are available; variance-standardized perturbation analysis can be used if information on variability of vital rates is available. Finally, it should be stressed that, because it addresses a different question, variance-standardized perturbation analysis is an *additional* tool for the analysis of matrix models.

### Acknowledgements

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

We thank Teresa Valverde and Roderick Zagt for allowing us to use their data. Heinjo During, Johan Ehrlén, Carol Horvitz, Xavier Pico and Marinus Werger provided valuable comments on draft versions. PZ acknowledges the Instituto de Ecología, UNAM, Mexico, and the Open University, UK, for their hospitality. Part of this study was financed by grant BO009701 from the Netherlands Development Assistance. MF thanks Conacyt, Mexico, DGAPA, UNAM, Mexico, the Ferguson Trust, and the Open University for support and hospitality during a sabbatical period.

### Supplementary material

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

The following material is available from http://www.blackwell-science.com/products/journals/suppmat/JEC/JEC622/JEC622sm.htm

#### Appendix 1

Blackground information on matrix models for the six study species.

### References

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

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