## Introduction

Fitness, although a fundamental concept in biology, is not straightforward to define (Stearns 1989; Murray 1990; Metz *et al*. 1992; Benton & Grant 2000). The fitness concept in life-history theory has – through the years – evolved considerably (Brommer 2000) and currently hinges on invasibility, the possibility of a rare mutant strategy to replace the strategy played predominantly in the population (Metz *et al*. 1992; Rand *et al*. 1994; Geritz *et al*. 1998; Benton & Grant 2000). However, invasibility is not readily measured in natural populations. Measuring selection – and the response to it – in empirical studies primarily concerns short-term evolutionary change among extant genotypes and requires a robust and quantifiable measure of evolutionary success (Roff 1997; Merilä*et al*. 2001). Although fitness is most accurately described by the representation of an individual's genes – or descendants – far in the future (e.g. Leimar 1996; Murray 1997; Houston & McNamara 1999), many estimates of fitness consider fewer generations, typically one. Lifetime reproductive success (LRS) is a commonly used proxy for individual fitness and can be viewed as the individual-based analogue of the population-wide measure *R*_{0}, the net reproductive ratio (e.g. Clutton-Brock 1988a; Newton 1989). Estimating LRS is an attempt to a complete representation of the lifetime performance of an individual, and it is therefore a better measure of fitness than any single component of fitness such as survival in a particular life-history stage (Endler 1986). The latter can lead to biased estimates of fitness whenever components of fitness face trade-off situations (e.g. early fecundity and longevity, Stearns 1992). Such trade-offs mean that interpreting one component, such as reproductive output in a given season, as a surrogate of overall fitness is too short-sighted.

Caswell (1989, 2001) has strongly argued for a more demographic approach in measuring fitness, which would encompass as many aspects of organism's life history as possible. LRS is, in fact, but one aspect of the performance of a given life history. LRS describes how the expected number of offspring (*R*_{0}) was realized in a particular sample of individuals. A major shortcoming of LRS is that timing of reproduction within the life cycle of an organism is not taken into account, although timing is a major component of fitness (Stearns 1992). In a growing population, reproducing early in life greatly increases the number of descendants left in the future (Houston & McNamara 1999). In addition, the use of a rate-insensitive estimate of individual fitness, such as LRS, contrasts strongly with the majority of theoretical life-history studies, which typically use a rate-sensitive measure of fitness, the population's intrinsic rate of increase λ_{pop} (Stearns 1992). Hence, there is an important gap between theoretical and empirical studies in the quantification of evolutionary forces.

In order to close this gap, McGraw & Caswell (1996) advocated a rate-sensitive estimate of individual fitness, λ_{ind} (see also Lenski & Service 1982). This estimate of individual fitness can be considered analogous to the population-wide intrinsic rate of increase as derived from the Euler–Lotka equation using projection matrices (e.g. Caswell 1989; Stearns 1992). For an age-structured population, where the maximum age is ω and the average survival of individuals from age *x* to *x* + 1 is denoted as *P*_{x} and the average (same-sex) production of offspring at age *x* by *M*_{x}, the population growth rate is given by the dominant eigenvalue of the matrices

The former of these matrices is derived from a so-called pre-breeding census, whereas the latter is derived from a post-breeding census (Caswell 1989). Mathematically, these matrices merely represent two different ways of describing the population growth of the same life history, either counting individuals of age ‘1’ as offspring (pre-breeding census) or individuals of age ‘0’ (post-breeding census). The asymptotic population growth rate λ_{pop} is then given by the dominant eigenvalue of either matrix in eqn 1 (Caswell 1989, 2001).

Equivalently, one can consider age-specific *individual* survival *p*_{x} and reproduction *f*_{x} (Lenski & Service 1982; McGraw & Caswell 1996). The propensity ‘growth rate’ of the individual life history is then given as the dominant eigenvalue of the square matrices

for the pre- and post-breeding census, respectively (analogous to eqn 1). Here, *f*_{x} denotes an individual's production of same-sex zygotes at age *x*, which can also be interpreted – in a diploid species – as half of the total number of zygotes produced in order to incorporate the genetic contribution of parent to offspring. If and only if the survival *p*_{x} for each age *x*, during which the parent has survived, is set at ‘1’, the matrices in eqn 2 reduce to

McGraw & Caswell's (1996) fitness measure λ_{ind} for an individual that lived *k* years is the dominant eigenvalue of the matrix in eqn 3. This fitness estimate's main attraction lies in the combination of two fitness elements – total reproduction (LRS) and timing of reproduction – into a single measure. Each observed individual life history is considered separately and its λ_{ind} is the maximum likelihood estimate of the individual's propensity fitness (McGraw & Caswell 1996). The individual life history with the highest growth rate is the most fit. Because this fitness estimate takes into account when an offspring was produced, it requires data on an individual's life span and age-specific production of offspring.

In this paper, we review how λ_{ind} has been used to study a variety of ecological and evolutionary questions. We then point out the importance of deciding when to count offspring when calculating λ_{ind} and show that this aspect has not been fully realized by those employing λ_{ind}. We conclude by illustrating that LRS and λ_{ind} provide fundamentally different estimates of individual fitness and sketch some important avenues for future research.