The age-structured Price equation we derive contains four terms for each age: a viability selection differential on the trait, a fertility selection differential on the trait, a term describing the average rate of growth or reversion and a term describing the mean difference between offspring and their parents. Each of these terms is weighted by the contribution of age-specific survival and recruitment to population growth. These weighted terms are then summed before being combined with a term describing change in the mean of the trait caused solely by ageing. This biologically intuitive equation has, to our knowledge, not previously been derived, which is surprising as it provides a demography for phenotypic change.
The equation extends the work of Price (1970)
who developed an equation describing the change in the mean of a phenotypic trait over a time step as a function of individual fitness defined as a scalar. The Price equation is typically considered for species with non-overlapping generations where fitness is defined as reproductive success (Frank 1997
). However, if fitness is defined as an individual’s numerical contribution to the population over a time step then the derivation still holds. For example, consider a case where an individual reproduces but also survives. In this case an individual’s numerical contribution will be its reproductive success plus one (itself) (Coulson et al. 2006
). The same logic can easily be extended to cases where genetic contributions are considered. In our derivation of an age-structured Price equation we have decomposed individual fitness into its survival and recruitment components.
This decomposition is illuminating as it provides novel insight into interpretation of the second term of the Price equation. Both Price’s original equation (Price 1970
) and our age-structured form describe differences in the means of trait distribution across populations. The viability selection differentials describe the differences in the trait means between survivors and all individuals within an age class (Falconer 1960
). Clearly, assuming some mortality, not all individuals survive; this means that the population of survivors within an age class is smaller than the population size of the age-class prior to selection. The second term involving survival is the growth and reversion term. This term describes how the mean of the trait changes among survivors between time t and t+1. This term can be calculated across all individuals within an age-class at age t regardless of whether they survive or not. Alternatively, as we demonstrate with our “+” notation, it can also be calculated across survivors only. The choice of notation will depend on how one views the change in trait distribution as occurring. If one considers a two step process, where individuals are selected to survive or die and then survivors change their trait value, then the “+” notation will be most appealing. Similar considerations apply to the recruitment terms.
In the population of red deer we analyzed we found that viability selection changed the mean of the trait distribution as individual’s aged. Fertility selection was weaker. This result suggests that previously reported selection on birth weight using lifetime measures of fitness (Clutton-Brock and Coulson 2002
; Clutton-Brock et al. 1997
; Clutton-Brock and Albon 1982
; Clutton-Brock et al. 1982
; Kruuk et al. 1999
) is generated primarily by viability selection and not fertility selection. In itself this result is not unexpected as Clutton-Brock (1988)
reported a decomposition of variation in lifetime reproductive success across a range of vertebrates showing that longevity was typically the dominant component. However, our results demonstrate that a richer picture of selection and phenotypic change can be achieved by decomposing fitness into its constituent components.
The viability selection we reported is countered by a substantial difference in the mean value of parental and offspring traits; this difference varies with age, being greatest at young and old ages. Parents tend to produce offspring that are more similar to the population mean than they are to themselves. The reasons for this are not clear, but could be a result of developmental, energetic or genetic constraints (Wagner and Altenberg 1996
; Moran 1992
; Smith et al. 1985
) which may, or may not, have additive genetic components. In the Breeder’s equation if selected parents produce offspring with trait values that are more similar to the population mean than they are to their own trait values then the heritability of the trait will be close to zero (Falconer 1960
). Our decomposition is consistent with previous research in this population, which reports heritabilities of birth weight that are much closer to zero than unity (Kruuk et al. 2000
; Coulson et al. 2003
; Wilson et al. 2006
In the red deer study equilibrium of the trait mean is primarily maintained because the viability selection differential is countered by differences between offspring and parental trait values. In Price’s (1970)
original equation this form of equilibrium can arise when the first term is positive and the second term is negative. In the age-structured Price equation we have derived explicit conditions describing other routes to equilibrium. These include selection operating in different directions, possibly mediated by a trade-off, via survival and fertility; selection operating via a covariance between survival and growth rates but in opposite directions at different ages; and the direction, and route, via which selection acts varying with the environment (Lande 1982
; van Tienderen 2000
; Coulson et al. 2003
). Coulson et al. (2003)
demonstrated that for birth weight in this population all of these processes operate to prevent evolution of the trait mean. The framework we develop will enable researchers to examine the processes which lead to stasis in the mean of a trait value despite apparent positive selection, as well as changes in trait value in the apparent absence of selection.
Our derivation of an age-structured Price equation provides a useful step in characterizing the dynamics of the mean of a phenotypic trait in an age-structured population living in a variable environment. However, there are clearly obvious areas for future research. First, because evolutionary ecologists are often interested in changes in the variance in a trait distribution (Turelli and Barton 1994
), as well as the mean, extending our derivations for higher moments of trait distributions is desirable. This work is already underway (extending the work of (Rice 2004
) in the non-age-structured case). Second, in order to generate dynamic equations, each term could be replaced by functions describing how the term is influenced by environmental variables. Such an approach has proved illuminating in understanding the temporal dynamics of population size and population growth and should prove insightful in identifying processes associated with phenotypic change (Coulson et al. 2001
). Finally, in order to use our equations to describe genetic change, our derivation would need to be linked to models describing the genotype-phenotype map. An obvious starting point would be the additive genetic model. Such work should allow unprecedented insights into evolutionary change of fitness-related phenotypic traits in age-structured population living in stochastic environments.
In this paper we have developed a general equation to exactly describe changes in the mean of a phenotypic trait in a population over a time step, and have applied it to detailed data from a population of red deer. This work makes several important advances. First, we do not consider fitness as a scalar, but instead work with fitness components. Second, and a direct consequence of the first advance, is we can consider the contribution of specific demographic classes to observed phenotypic change. Third, by focusing on short time steps we are able to examine how the relative contributions of different demographic processes that generate change vary with time. Finally, we provide the first comprehensive description of the processes generating fluctuations in the mean of a fitness-related phenotypic trait in an age- structured population living in a stochastic environment.