Our aim in this section is to develop general theory linking integral projection models (IPMs), the Price equation, generation length and (biometric) heritability estimates (Jacquard 1983) from mother-daughter regressions. In the derivations below we allow character-demography functions to vary with time. In the empirical example that follows we keep things deliberately simple and parameterize a model for a constant environment. We work with number density distributions that describe the number of individuals within an age class with respect to character values. The area under this distribution is the number of individuals within the age class. We refer to this area as the ‘size’ of the distribution. The sum of the sizes of these distributions across all age classes is the population size.
Integral projection models
Age-stage-structured matrix models provide a general mathematical description (based on accounting identities) of the dynamics of population size and structure (Lefkovitch 1965). Both age and stage (or stages in the multivariate cases) are discrete. For continuous characters like body size, stage classes are constructed by binning characters into discrete stage classes. If age-specific transition rates between stages are known, the population growth rate and change in stage structure over a time step is exactly described. These transition rates are: (i) stage-specific survival; (ii) transition rates of survivors among stage classes; (iii) stage-specific fertility rates; and (iv) the stage classes into which offspring born to parents in a specific age-stage-class are recruited (defined here as reproductive allocation). All of these rates may vary with age as well as stage, and in stochastic models they may also vary with time (Tuljapurkar 1990; Ellner & Rees 2006).
The IPMs are built on functions that describe the associations between a character (or characters in the multivariate case) and survival, fertility, development of the character among survivors and the probability density distribution of offspring character values given parental characters (Easterling et al. 2000). In populations where dispersal rates can be ignored these are the four fundamental relationships connecting characters to demographic rates; they can vary with age and in variable environments with time (Ellner & Rees 2006). Relationships between a character and immigration and emigrations rates need to be considered in cases where dispersal rates cannot be ignored. IPMs are models that describe how number-density is added to, removed from and transformed within a uni- or multivariate character number density distribution. IPMs can accommodate both continuous and discrete traits (Ellner & Rees 2006) and are consequently mathematically more general than matrix models, although results for IPMs carry over naturally to matrix models (Easterling et al. 2000; Ellner & Rees 2006). The use of discrete time requires that age be counted in discrete intervals.
Assume (1) that a population is sufficiently large so demographic stochasticity can be ignored and (2) that relationships exist between a character z and survival S(a,t,z′), fertility R(a,t,z′), ontogenetic development of the character among survivors G(a,t,z | z′), and offspring character values D(a,t,z | z′) within each age class a and at each time t. Additionally, assume that viability selection occurs before ontogenetic development among survivors, and fertility selection (conception) occurs before reproductive allocation determines offspring character values. Models could be formulated such that growth occurs before survival, fertility, and reproductive allocation but such models are not discussed further here. Denote the number density of individuals at age a and character value n(a,t,z). The dynamics of this number density distribution from t to t + 1 can be written,
(eqn 1a) (eqn 1b) (eqn 1c) (eqn 1d)
Definitions of variables are provided in Table 1. Recruitment, or fertility, is defined as the number of offspring born between t and t + 1 that survive to t + 1. Eqn (1a) gives the number density distribution of character values among recruits to be added to the population at time t + 1 as a function of parental character values at time t. The number density distribution of offspring character values produced by each age-class is generated in two steps: a recruitment function R(a,t,z′) produces a number density distribution of parental character values that is then transformed into the number density distribution of offspring character values by the probability density function D(a,t,z | z′). The integral is taken over all parental character values. To obtain the population level number density distribution of newborns, we sum the age-specific number density distributions of offspring characters across all ages. As well as contributing to the offspring number density distribution, each n(a,t,z′) will produce the distribution n(a + 1,t + 1,z). Eqn (1b) describes how first a survival function S(a,t,z′) removes number density from n(a,t,z′) before a probability density function G(a,t,z | z′) describes how ontogenetic development transforms density among survivors.
Table 1. Definition of variables used in the text
|The population mean of variable x|
|σ2(x)||Population variance of x|
|Change in character mean between t and t+1: |
|Δσ2(Z(t))||Change in the variance of the character between t and t+1|
|w(t)||Mean fitness defined as the sum of mean survival and mean recruitment: |
|λ||Predicted mean fitness at equilibrium population structure|
|p(a,t)||Proportion of the population in age-class a at time t|
|n(a,t,z),n(a,t)||Continuous, discrete distribution of character values in age-class a at time t|
|S(a,t,z),S(a,t)||Continuous function, matrix, describing expected survival|
|R(a,t,z),R(a,t)||Continuous function, matrix, describing expected recruitment|
|G(a,t,z|z′),G(a,t)||Continuous function, matrix, describing ontogenetic development kernel|
|D(a,t,z|z′),D(a,t)||Continuous function, matrix, describing the reproductive allocation kernel|
|z||Vector of midpoint character values for each age-character class|
|Va||Additive genetic variance of the character|
|M(a,t + a − 1,z| z′)||Density of offspring with character values z produced by parents with character value z′ when they were aged 1 at time t|
|ML||Lifetime reproductive success|
|N(t)||Female population size in year t|
Eqns (1a) and (1b) describe the dynamics of a continuous character, z. However, it is useful to approximate IPMs in discrete matrix form to aid their analysis (Easterling et al. 2000). When approximated in this way we can write the kernels D(a,t,z | z′) and R(a,t,z | z′) and the functions S(a,t,z′) and R(a,t,z′) as matrices (see ‘Numerical Implementation’ below). Matrices are denoted with boldface font: for example G(a,t). Integral operators provide a powerful notation that covers both kernels and their matrix approximations. We denote integral operators using tildes, for example . For a continuous character the integral operator is a kernel; for a discrete character the integral operator is a matrix. Eqn (1c) rewrites (1a) in integral operator notation; eqn (1d) similarly rewrites (1b). These integral operators are similar to those used in standard IPM theory (Ellner & Rees 2006) but in our development it is vital to keep separate the effects of survival (in ), ontogenetic change (in ), recruitment (in ) and reproductive allocation (in ) as this allows calculation of selection differentials and the biometric heritability of the character. In Table 2 we describe key number density distributions using each notation.
Table 2. Description of distributions and moments of distributions in continuous and discretized forms
|i||n(a,t,z)||n(t)||Character distribution at t|
|ii||S(a,t,z)n(a,t,z)||S(t)n(t)||Character distribution after viability selection|
|iii||R(a,t,z)n(a,t,z)||R(t)n(t)||Character distribution after fertility selection|
|iv||∫dz′G(t,a,z′)S(t,a,z′)n(a,t,z′)||G(t)S(t)n(t)||Character distribution after ontogenetic development|
|v||∫dz′D(t,a,z|z′)R(t,a,z′)n(a,t,z′)||D(t)R(t)n(t)||Character distribution after reproductive allocation|
|vi||n(a,t + 1,z)||n(t + 1)||Character distribution at t + 1|
|Mean of the distribution of z|
|mth moment of the distribution of z|
The IPMs and their matrix approximations can be used to predict population size and structure one time step ahead. IPMs also predict change in means and variances of the character number density distribution over a time step as a function of selection and other processes captured by the age-structured Price equation (Coulson & Tuljapurkar 2008). In stochastic environments the fundamental functions used to construct IPMs vary with time, and the population structure and population growth rate change from one time step to the next. However, the population converges to a stationary number density distribution of population structures and growth rates (a stochastic equilibrium) (Tuljapurkar 1990). Means and variances of the character number density distribution, as well as the population growth rate and structure, converge to equilibrium values in deterministic models, and to a stationary distribution in stochastic models.
From IPMs to characters and Price
The eqns (1a) and (1b) have been used to study population dynamics and the evolution of optimal character values (Childs et al. 2003). In this section we are interested in character dynamics rather than population numbers, and we first show that the same equations provide the tools for tracking moments of character number density distributions. The mean trait value among individuals of age a at time t is just
an equation that applies also if the character is vector-valued – that is multivariate. The mth moment of the character value (if scalar, or of a component, if vector) is
and the variance is then
If z1,z2 are the components of a vector valued phenotypic character, we can track the phenotypic covariance via the joint moments
From here on we focus on a scalar character but the analyses extend to vector-valued characters along the lines of the above equation.
As with character values, we can compute the averages of survival rates,
and similarly averages of ontogenetic change, fertility, and so on. Even more usefully, we can compute covariances between fitness components and characters. Thus for survival rate and character value,
Readers familiar with the analysis of selection on characters will recognize the covariance in (5) as the selection coefficient on the character due to differential survival among individuals aged a at time t. Clearly, the dynamics in (1a) and (1b) make it possible to track selection acting via survival, growth, reproduction and so on. They are therefore a direct link to the fundamental frameworks used to understand the dynamics of phenotypic characters – the Price equation (Price 1970) and the breeders equation (Falconer 1960). These frameworks are both formed in terms of selection differentials.
The age-structured Price equation describes change in the population level mean of a number density character distribution between time t and t + 1 (Coulson & Tuljapurkar 2008). An equation for change in the variance has also been derived (S. Tuljapurkar & T. Coulson, unpublished). Terms in the Price equation describe how survival, recruitment, ontogenetic development and reproductive allocation alter the mean of number density distributions within and between each age-class (Table 2). In addition to these contributions we need two additional quantities to write a form of the age-structured Price equation (Coulson & Tuljapurkar 2008). First, the normalized density at character value z of individuals at age a at time t is
and the fraction of individuals at age a at time t is
Second, the growth rate of the population between t and t + 1 is
In the following equations we denote means and variances of the number density distributions across all ages as and σ2(Zi) for i = i,…,iv (Table 2). Change in the mean, (Coulson & Tuljapurkar 2008), and change in the variance, Δσ2(Z(t)) (S. Tuljapurkar & T. Coulson, unpublished), of the character number density distribution can then be written,
The terms in square brackets in the first two rows of both equations describe differences in the mean (7) and variance (8) between pairs of character number density distributions (Table 2). In both cases the first row describes contributions via survival and ontogenetic development while the second row captures contributions via recruitment and reproductive allocation. The remaining terms deal with contributions to change via fluctuations in the age-structure. In what follows we focus on change in the mean (7), although the interpretation is the same for the variance and for any higher central moment of interest.
The first term on the right in the top row of (7) describes how viability selection shifts the character mean; it is a viability selection differential. The second term on the right of (7) in the top row describes the average rate of ontogenetic development of the character among survivors. In the second row of (7) the first term in square brackets is a fertility selection differential, while the second term describes the average difference between offspring and parental character values (reproductive allocation). The terms outside square brackets in the first two rows of (7) provide the demographic weights needed to average the terms in square brackets across age-classes. These first two rows in (7) describe how within age-class processes change the mean value of the character. The bottom row describes how differences between age-classes alter the character number density distribution. In (7) the first term in the bottom row on the right describes how differences in mean survival rates between age-classes alters the character number density distribution, while the second term describes how differences in reproductive rates between age-classes contribute to change. These terms will be non-zero in populations at equilibrium if there are survival and fertility differences between age-classes that are independent of the character.
Calculating life history and quantitative genetic quantities
Our model can also be used to calculate life history descriptors like generation time and net reproductive rate, as well as an estimate of the character heritability and selection on the character via lifetime reproductive success. To calculate these quantities we first show how to track the performance of cohorts in terms of survivorship and fertility. We use our integral operator notation.
We track cohort dynamics by iterating eqn (1d). Between ages 1 and 2, changes in a cohort are described by the integral operator as in (1c) and (1d). Between ages 2 and 3, the corresponding operator is , and so on. String these together to obtain
(eqn 9) (eqn 10)
is the identity kernel (continuous formulation) or identity matrix (discrete formulation), and is a kernel (continuous formulation) or matrix (discrete formulation) describing survivorship. In the continuous notation of Easterling et al. (2000) the iteration in (10) is as follows,
(eqn 11) (eqn 12)
where δ(z − z′) is the Dirac delta function. In discrete space the Dirac delta function is the Kronecker delta.
The expected number density of offspring with character value z produced at age a by a parent born at time t with a character value z′ is denoted M(a,t + a − 1,z | z′) and is given by
To find the lifetime reproduction of a parent born at time t with character value z′, we add together offspring produced at all ages through some (possibly large) maximum age A,
A cohort born at t with a character number density distribution produces at age a an offspring number density distribution , and a lifetime offspring distribution .
We can use these equations to calculate a number of life history quantities. Such calculations make sense in a constant, density-independent environment, when rates are time-independent. In our model, age-dependence complicates the known methods (Ellner & Rees 2006). We define generation time T by a frequently used identity (Caswell 2001), which is tautological but may be useful. We calculate generation time T by the identity R0 = erT where r is the asymptotic growth rate and R0 is the net reproductive rate. R0 is the dominant eigenvalue of the operator ML in (14): recall that this operator describes lifetime reproduction. The asymptotic growth rate describes the growth rate when the population has a stationary age and character density distribution. In the latter, denote the character number density distribution of newborns (age class 1) by ; recall the integral operators in eqn (13) that describe the number density of offspring of a cohort when that cohort reaches age a. In the stationary state these operators do not depend on time so we can write them simply as . Then r is the solution to the integral equation (U. Steiner, S. Tuljapurkar & T. Coulson, unpublished)
In practice no one is going to solve these integral equations. Instead we use a discrete matrix approximation and turn the integrals into sums, as we illustrate in the next section.
We now turn to the breeder's equation. The breeder's equation has been widely used to understand phenotypic change of heritable characters (Bulmer 1980; Lande & Arnold 1983). Specifically it describes the response to selection defined as the per generation change in the mean of the breeding value distribution. A breeding value of a character describes the additive genetic worth of a parent for that character.
The breeders equation, in the univariate form considered here, contains two terms – a selection differential between the character and lifetime reproductive success () and a character heritability, h2. The heritability is the ratio of the additive genetic variance Va to the phenotypic variance σ2(Z). Heritabilities and additive genetic variances can be estimated in many ways. The classic biometric approach we use here is through a regression of daughter character values measured at age a against maternal character values also at age a. Twice the slope of the regression line is the character heritability (Falconer 1960). This approach is entirely statistical; hence the use of the term biometric heritability. As discussed by Jacquard (1983) the connection between this biometric heritability and any underlying genetic variation is not simple, even though it is often assumed to be so (see discussions in Willis, Coyne & Kirkpatrick 1991; Kruuk 2004).
To estimate the biometric heritability of body mass measured at age 1, we need to consider parents born into different size classes and track the number density distribution of the stage classes of offspring born to these parents at each age in the life course. To do this we start with a cohort of newborns who progress through the life cycle to become parents. This cohort of newborns is described by a number density over character values, , which we iterate forwards to track the number density distribution of offspring produced. We consider all offspring produced over a lifetime. From eqn (14) we see that the joint number density distribution of offspring character value x and parental character value y must be proportional to
In a one sex model, the regression of offspring trait value Zo on parental trait value Zp has a slope that equals half the heritability,
From the joint number density distribution we have
(eqn 17) (eqn 18) (eqn 19) (eqn 20)
The first two equations above yield the mean character values of parents and offspring, respectively. The third, eqn (19), is the important new relationship here and shows that the character number density distributions from the model can be used to compute the parent-offspring covariance that is the key to determining biometric heritability. Eqn (20) yields the variance among parents. We can now compute half the heritability h2 using eqn (16).
It is important to note that the value of h2 does not depend on the number of parents that we start with as newborns, only on their character distribution [because the number cancels out of the ratio in (16)]. Several aspects of biometric heritability can be explored using our analysis. First, we can compare the equilibrium heritability obtained by starting with a stable character density distribution of newborns (i.e. ) with time-dependent values for cohorts who are observed over their reproductive lives. Such a comparison will illuminate the effects of environmental change on life history transitions. Second, we could compute age-dependent biometric heritabilities at each age a by repeating the covariance calculation in eqn (19) but considering only offspring produced at age a– we will do this in future work. This would allow us to examine the effects of age, a variable that is often factored out of the usual models for estimating heritability by treating age as a covariate.
The equations above show how a wide range of population biology quantities can be calculated from IPMs. Implementation of the model requires us to approximate the model in matrix form (Easterling et al. 2000). In the next section we explain how this is done.
The continuous character-demography functions on which IPMs are built are identified through statistical analyses (Easterling et al. 2000). Predicted values from these continuous functions can be calculated for very small-width discrete bins, and these values used to construct a high dimensional transition matrix. An IPM will only accurately capture the dynamics of a population and a character distribution if the statistical functions used to construct the model accurately capture observation. As with any statistical analysis, the identification of accurate and appropriate functions requires good data and biological knowledge of the system under study. Let us assume that functions have been identified. How would our IPM [(1a) and (1b)] look in matrix form?
To write our discretized age-stage IPM as one large matrix (Lebreton 1996) requires a note on notation. The age-specific continuous number density distributions are combined into one vector n(t). The ith element of this vector represents the number of individuals in age-class a and character class j. Each possible character class is included within each age even if no individual of age a can have that value. For example, it may be impossible for a new recruit to have an average adult body mass. However, there is an element in n(t) at this impossible character value but this element will always be zero. If we have nine age-classes and 100 character classes n(t) will consequently be of length 900, with elements 1–100 representing age class 1 (n(1,t)), elements 101–200 representing age class 2 (n(2,t)) etc.
A set of square matrices is used to iterate n(t) to n(t + 1) with matrix elements equal to predicted values of S(a,t,z), R(a,t,z) (both diagonal matrices), G(a,t,z | z′) and D(a,t,z | z′) calculated at the mid point value of each stage class. We also define a vector z consisting of these mid-point values of each stage-class (Easterling et al. 2000). In our age-character model z consists of the list of mid-point values for phenotypic classes repeated by the number of age-classes. The vector z is required to calculate quantities describing character dynamics (Table 2).
An age-structured IPM is now approximated in matrix form as,
Each of the matrices S(t), R(t), G(t), D(t), Ψ and Γ are square ‘block’ matrices consisting of an array of age-specific matrices defined above. An age-specific sub-matrix of this large matrix is described with indices (a,t). The S(t) and R(t) matrices are diagonal describing survival and recruitment rates of individuals in each age-character class – they are discretized versions of S(a,t,z′) and R(a,t,z′). Each sub-matrix G(a,t) describes transition rates between stage classes within an age-class among survivors, except it does not yet age the survivors by 1 year (see below). Each sub-matrix D(a,t) describes the transition rate from maternal stage to offspring stage, except that it does not yet place offspring into the age-class of new recruits. See the online appendix for a figure displaying the form of these matrices.
The matrices Γ and Ψ describe age transitions. Γ moves offspring out of the maternal age class into the new recruit class (aged 1) – the top row of sub-matrices of Γ are all identity matrices while all other sub-matrices contain only zeros. Ψ acts in a similar manner to Γ but ages survivors. Ψ and Γ are time invariant. The functions D(a,t,z | z′) are approximated by ΓD(t), while the functions G(a,t,z | z′) are approximated by ΨG(t). We will report quantities at equilibrium, so now drop the index t.
The dominant eigenvalue of (21), λ, is the population growth rate (Lebreton 1996), and the left and right eigenvectors associated with λ are respectively the reproductive value and stable age-character distribution. Our integral operator notation demonstrates it is straightforward to rewrite our equations for the age-structured Price equation and the breeders equation in matrix form. Having calculated quantities in these equations we next explore how they are related to one another. We first construct a matrix and calculate key quantities at equilibrium. Next, we independently perturb parameter values in the character-demography functions and examine how each perturbation alters each of the quantities. This is a form of sensitivity analysis (Caswell 2001). We perturb function parameters rather than specific matrix elements because we believe that environmental variation and evolution will change these functions. We chose not to centre functions prior to perturbation because most published character-demography functions are not centred.
The population of Soay sheep Ovis aries living in the 250 ha Village Bay catchment of the Island of Hirta in the St. Kilda archipelago, Scotland, has been studied in detail since 1985 (Clutton-Brock & Pemberton 2004). There are no sheep predators on the island, and no interspecific competition for forage from other large herbivores meaning the population is only food limited. Each spring newborn individuals are caught and uniquely marked with ear tags within hours of birth. Mortality tends to occur during the winter months. Regular mortality searches during this period result in the majority of carcasses being found, normally within a day of death. Since 1986, each spring, summer and autumn, 10 censuses of the population are attempted. Over 95% of individuals seen in these censuses are identified – the unidentified individuals tend to be transients seen on the edge of the study area (Coulson, Albon, Pilkington & Clutton-Brock 1999). The birth, death and census data are used to provide a list of which individuals are living permanently within the study population each August. The population size in each year is consequently known accurately. The population exhibits periodic crashes when up to 70% of the population can die (Coulson et al. 2001). Maternity is inferred from observations of birth or suckling. Each August a team catches as many individuals as possible – on average 50% of the resident population. Any unmarked individuals that are caught are marked. Each time an individual is caught blood samples, faecal samples and a range of phenotypic data including body mass are collected. Observed body mass means and variances both within and across age-classes are given in Table 3.
Table 3. Observed and predicted quantities. Predictions obtained assuming equilibrium age-character structure
|12·55||12·11||σ2(Z(a = 1))||6·17||7·07|
|17·41||17·08||σ2(Z(a = 2))||5·92||5·91|
|22·67||22·70||σ2(Z(a = 3))||7·66||7·01|
|23·95||23·79||σ2(Z(a = 4))||8·03||6·75|
|Other terms in (7)||1·25||1·17||other terms in (8)||1·08||0·81|
We used data on life history and body masses collected from the female component of the population between 1986 and 2008. Although we focus on body mass, any morphological, physiological, genetic or behavioural character could be used in our approach. We consider four age-classes identified from previous analyses of survival rates: lambs, yearlings, prime-aged adults aged 2–6 years, and senescent individuals aged over seven (Coulson et al. 2008). A detailed description of data collection protocols is provided elsewhere (Clutton-Brock & Pemberton 2004).
Individual body mass, survival, fertility and offspring mass data were used to identify functions required for parameterization of integral projection models. We fitted generalized linear models with a binomial error structure to annual individual survival, reproduction, and, for those individuals that bred, whether one or two recruits were produced. We defined reproduction as whether an individual bred and produced offspring between t and t + 1 that survived to enter the population in the August after birth (t + 1) (Coulson et al. 2001). The litter size (twinning) function described the number of recruits each female produced (1 or 2). Growth rate functions were estimated by using multiple linear regression models of mass in year t + 1. Reproductive allocation functions were estimated through multiple linear regression of the mass at t + 1 of offspring produced between t and t + 1 that recruited to the population in t + 1. There is considerable temporal variation in demographic rates (Coulson et al. 2001). We fitted year class as a categorical variable in all models to correct for this variation. Obviously body mass was fitted as an independent term in all models. We fitted separate models for each of the four age classes.
The resultant character-survival functions S(a,t,z) are of the form exp (α + βZ)/(1 + exp (α + βZ)) where α and β are obtained from logistic regressions for survival. The functions R(a,t,z) are obtained by combining the reproduction and litter size (twinning) functions, both which are of the same form as the logistic models for survival. If we define the twinning functions as φ(a,t,z) and the fertility functions as F(a,t,z) then R(a,t,z) = F(a,t,z)(1 + φ(a,t,z)). To estimate growth kernels G(a,t,z | z′) it is necessary to combine the function describing mean body size at year t + 1 given body size in year t with a function describing the variance around these associations and scaling so that all transition rates out of an age-stage class sum to unity. The variance function is identified by regressing the squared residuals around the mean body mass function against body mass (Easterling et al. 2000). We found no compelling evidence for nonlinearity in these functions so used linear regressions. We define the intercept and slope of the linear regression of body mass in year t + 1 against body mass in year t as α and β and the intercept and slope of the variance function as α and β. If we next define and μ(z) = α + βz, the probability density function describing transition rates between z and z′ is,
The same logic is used to define the D(a,t,z | z′) functions for each age-class. We next define the integration limits in (1a) and (1b) that also provide the smallest and largest values of z. The smallest observed August female sheep mass was a recruiting lamb weighing 2·9 kg and the largest was from a 34·2 kg adult. When constructing S(t), R(t), G(t) and D(t) we generated 100 phenotypic classes ranging from 0 to 37·5 kg as this provided a good approximation to the continuous functions (Easterling et al. 2000): decreasing bin size further had no influence on all quantities calculated to 2 decimal places. We ensured that transition probabilities out of class j for the kernels D(a,t) and G(a,t) summed to one by dividing each element by the sum of estimated transition probabilities. All statistical analyses and model construction and analysis were conducted in (R Development Core Team 2009). Code for constructing IPMs utilized functions provided by Ellner & Rees (2006).
For the matrix approximation of the IPM we calculated the following quantities at equilibrium (we consequently drop the index t): asymptotic population growth, λ; mean character value at the population level and within each of our four age classes for a = 1,…,4; variance in character value at the population level σ2(Z) and within each of our four age classes σ2(Z(a)) for a = 1,…,4; the contributions of viability selection, fertility selection, growth among survivors, reproductive allocation and the demographic weights to and Δσ2(Z) summed over age-class (Table 3); generation length T; heritability of body mass h2; the selection differential between body mass as a recruit to the population and lifetime reproductive success. We compared these model predictions to the same quantities calculated from the individual-based data or with previous published estimates (Table 3).
Intercepts and slopes of each of the statistical functions used to parameterize the IPMs were independently perturbed by 1% and new IPMs and matrix approximations constructed. The direction of each perturbation was chosen so as to increase λ. The perturbed matrices were then used to calculate the the proportional change in the equilibrium values listed above. Because the S(a,t,z) and R(a,t,z) functions were on the logistic scale we rescaled perturbations to the same scale as the G(a,t,z | z′) and D(a,t,z | z′) functions.