## Introduction

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.