An individual's phenotype will usually influence its probability of survival. However, when evaluating the dynamics of populations, the role of selective mortality is not always clear. Not all mortality is selective, patterns of selective mortality may vary, and it is often unknown how selective mortality compares or interacts with other sources of mortality. As a result, there is seldom a clear expectation for how changes in the phenotypic composition of populations will translate into differences in average survival. We address these issues by evaluating how selective mortality affects recruitment of fish populations. First, we provide a quantitative review of selective mortality. Our results show that most of the mortality during early life is selective, and that variation in phenotypes can have large effects on survival. Next, we describe an analytical framework that accounts for variation in selection, while also describing the amount of selective mortality experienced by different cohorts recruiting to a single population. This framework is based on reconstructing fitness surfaces from phenotypic selection measurements, and can be employed for either single or multiple traits. Finally, we show how this framework can be integrated with models of density-dependent survival to improve our understanding of recruitment variability and population dynamics.
The dynamics of populations can be affected by both the quantity and quality of the individuals within them. That is, the demographic processes that drive the dynamics of populations are influenced by both the number of individuals and the phenotypes of those individuals. Classic theory has emphasised numerical effects such as density-dependent regulation as the major, endogenous force driving the dynamics of populations (reviewed by Murdoch 1994; Cappuccino & Price 1995). However, the phenotypic composition of a population can also exert a strong influence on population dynamics, especially if the phenotypes under study are closely related to demographic components of fitness (reviewed by Gaillard et al. 2000; Hairston et al. 2005; Schoener 2011). Despite these observations, in practice it can be difficult to predict how changes in the distribution of phenotypes within a population will actually translate into variation in population dynamics (reviewed by Saccheri & Hanski 2006; Kokko & López-Sepulcre 2007; Metcalf & Pavard 2007). This is particularly true for processes such as population replenishment, which may depend on complex, non-linear relationships among phenotype, mortality and density.
Although it has long been recognised that the phenotypic composition of populations will influence their dynamics, an overall appreciation of the magnitude of these effects (especially relative to environmental factors as a source of variability) has advanced more slowly (reviewed by Thompson 1998; Benton et al. 2006; Kokko & López-Sepulcre 2007; Pelletier et al. 2009). In part, this delay has been because classical demographic analyses and models (in which individuals are treated as identical) are simple and accessible, whereas the quantitative tools for analysing phenotypic variation (and its relative influence on dynamics) require more data and have taken time to develop (e.g. Lande 1976; Lande & Arnold 1983; Bjørnstad & Hansen 1994; Van Tienderen 2000; Smallegange & Coulson 2013). A major advance in this field came when Hairston et al. (2005) presented a general framework for disentangling the influences of both phenotypic trait change and environmental variation on population dynamics (also see Ellner et al. 2011 for extensions of this approach). In this framework, the partitioning of environmental and phenotypic contributions to population dynamics is accomplished by analysing time-series observations of population-level responses (e.g. population size), and average values of both phenotypes and environmental variables of interest. This framework has been applied to several empirical studies (e.g. Ezard et al. 2009; also see examples in Hairston et al. 2005; Ellner et al. 2011), and can be quite successful at illustrating how phenotype change can affect dynamics. However, a strong (and potentially limiting) assumption of the methods proposed by Hairston and colleagues is that any effects of changes in the distribution of phenotypes are a function of changes in the mean value. This assumption may hold if the relationships between phenotypes and population response variables (e.g. per capita growth rate, survival, fecundity, etc.) are linear (Ellner et al. 2011). However, if such relationships are not linear, then through non-linear averaging (e.g. Ruel & Ayres 1999) both the means and variances of phenotype distributions can play an important role in population dynamics, and other approaches are needed. In this study, we present a framework that can accommodate strong non-linearities in the relationships between phenotypes and fitness and thus can be very useful in analysing how phenotypic variation can affect population dynamics, even in the presence of strong density dependence (another non-linear relationship). We apply this framework to studies of selective mortality in fish populations and use it to shed light on how variation in phenotypes may drive variability in recruitment.
Replenishment of fish populations is notoriously variable. In part, this is because most fishes have life histories in which numerous offspring are produced, but very few survive to adulthood. A consequence of having a life history in which early stages can be extremely abundant (often outnumbering adults by several orders of magnitude) is that even small changes in mortality during early life can have large effects on recruitment (Houde 1987). Because seemingly small variations in mortality easily can be responsible for order-of-magnitude fluctuations in recruitment, scientists have long been interested in the characteristics of individuals that drive variation in mortality during early life (e.g. Anderson 1988; Miller et al. 1988, Bailey & Houde 1989; Houde 2008).
The relationship between individual phenotype and relative survival during early life can be examined through the lens of phenotypic selection analysis (Lande & Arnold 1983). Phenotypic selection analysis has long been used to understand evolutionary trajectories, but a main feature is that it separates selection from inheritance. When such analyses are made without information on heredity, they cannot be used to predict evolution. However, they may still be used to understand selection as a within-generation, demographic process.
The last few decades have seen a rise of studies examining how the relative survival of fishes is influenced by phenotypic characteristics. The realisation that fish otoliths (accretions of calcium carbonate around protein matrices in the inner ear) can carry a permanent record of traits related to size, growth rate, and condition has enabled studies of selective mortality and yielded important insight into why certain individuals survive and others do not (Sponaugle 2010). Recent reviews of this field indicate that selection (i.e. the change in the phenotype distribution that is generated by differential mortality) is typically very strong during early life phases of fishes (Sogard 1997; Perez & Munch 2010). Both of these reviews noted a substantial amount of variability in selection, although neither study examined selective mortality at the scale of the population, and neither study explored how variation in selection may affect population dynamics. Selection during early life can be summarised using selection coefficients, but it is important to keep in mind that survival during early life is only a component of fitness. From an evolutionary viewpoint, one might expect early mortality to be strongly selective with respect to traits such as size and growth (which are commonly measured traits in this field). Mothers may trade offspring size for offspring number, and the maximisation of fitness may come at the expense of offspring survival (Vance 1973; Smith & Fretwell 1974). As a result, the relationship between early life traits and a more complete measure of fitness (e.g. lifetime reproductive success of mothers) may indicate stabilising, rather than directional, selection because reproductive success is a combination of both the number of offspring and their survival (e.g. Smith & Fretwell 1974; see Einum & Fleming 2000; Johnson et al. 2010; for empirical examples in fishes). A consequence of this phenomenon is that even though early life-history traits may be evolutionarily stable, there can still be very large differences in the relative probability of survival of individuals during early life (as evidenced by many empirical studies; Sogard 1997; Perez & Munch 2010).
That selection is widespread and often strong provides evidence that an individual's phenotype can strongly influence its relative probability of survival. However, at the level of the population, understanding how phenotypic variability affects recruitment is much more complicated. First, it can be unclear whether the amount of selective mortality incurred is substantial enough to be important to the dynamics of populations. The amount of selective mortality that a population or a cohort (i.e. a group of similarly aged individuals) experiences, and how that compares to total mortality, is not often quantified. Second, selection may be highly variable. Even when one considers selection on a single trait in a focal population, measurements often vary substantially (e.g. Meekan & Fortier 1996; Good et al. 2001; Rankin & Sponaugle 2011), making it difficult to discern any general patterns. Finally, it remains unclear how selective mortality compares or interacts with density-dependent mortality – a major factor influencing early survival and recruitment (reviewed by Myers & Cadigan 1993; Rose et al. 2001; Hixon & Webster 2002). Box 1 outlines a general approach to quantifying selective mortality and to partitioning population-level mortality into selective and non-selective components.
To understand how phenotypic variation affects mortality within cohorts and therefore recruitment to populations, we need to study selection within a quantitative, analytical framework. Here, we review empirical studies of selection in fishes as a first step towards evaluating the overall importance of selective mortality to recruitment. Next, we describe an analytical framework that accounts for variation in selection, while also describing the amount of selective mortality experienced by cohorts. This framework is based on empirical reconstruction of fitness surfaces from selection measurements, and can be employed for either single, or multiple traits. Finally, we show how this framework can be integrated with models of density-dependent survival to improve our understanding of recruitment variability.
Selection, demographic costs and recruitment variability in fishes
Selection implies a demographic cost to the population. When phenotype distributions change through differential mortality, stronger selection implies greater mortality overall. Demographic costs of selection can therefore be important in the context of recruitment where dynamics are driven by variation in both cohort size and cohort mortality. All else being equal, cohorts that experience stronger selection must incur greater mortality (i.e. greater costs), and if cohorts that recruit to a population vary in the degree of selection (and selective mortality), then selective mortality may be an important source of variability in total mortality, and thus variability in recruitment (Box 1).
Box 1. Components of mortality
Consider a population in which individuals risk mortality from multiple sources, some of which are selective with respect to phenotype, and others that are not. Let nz,t describe the number of individuals of phenotype z at time t. This distribution will change through time as individuals are removed from the population through both selective and non-selective mortality. The rate of selective mortality, fz, is a function of phenotype and the rate of non-selective mortality, α, is a constant. Therefore,
From the previous expression, the distribution of phenotypes at time t + 1 can be obtained:
and population size (total abundance) at time t + 1 is then
Factoring in the distribution of phenotypes at time t yields
which can be expressed as:
At the level of the population (or cohort, or group), mortality can be separated into two components: selective mortality (1-), and non-selective mortality .Total mortality (MTOT) is 1 − e−α and may range from 0 to 1. From these relationships we can conclude that when the selective component of mortality is large, the non-selective component must be small. We can also conclude that because selective mortality is a component of total mortality, variation in selective mortality among cohorts will translate to variation in the total amount of mortality that cohorts experience. This translation will be direct if the rate of non-selective mortality (α) is constant among cohorts (or at least random with respect to selective mortality).
In many cases, patterns of selection (i.e. changes in the distributions of phenotypic values) are driven by selective mortality. Even if mortality is not measured directly, the amount of selective mortality a population experiences can be inferred from selection measurements. Empirical measurements of selection often focus on calculating the selection differential, i.e. the mean phenotypic value after selection minus the mean before selection (Falconer & Mackay 1996). The selection differential (S) can also be expressed in terms of the (normalised) distribution of phenotypes before selection and the selective mortality function. Specifically,
If the selection differentials and phenotype distributions are known, and if a simple functional form (e.g. truncation selection) is chosen to describe the rate of selective mortality, then the parameters defining fz can be solved for and the amount of selective mortality can be calculated as described above.
As a first step towards evaluating variability in selective mortality and its potential role in generating recruitment variability, we compiled published estimates of selection on early life traits in fishes. For many fishes, synchrony in reproduction and/or settlement from the plankton produces discrete cohorts that settle at approximately the same age. Selection measurements are typically made within cohorts and selection measurements can be used to infer mortality (Box 1). To evaluate how selective mortality affects recruitment to particular populations, we need to examine variability in selective mortality among cohorts because it is among-cohort variation in mortality that contributes to recruitment variability. In our literature search, we considered only those studies that estimated selection on the same focal trait(s) for two or more separate cohorts. Our search yielded a total of 136 selection estimates for 33 species (Table S1, also see Appendix S1 in Supporting Information). Although very few studies of selection have actual estimates of total mortality, we were able to derive estimates of selective mortality from measurements of selection.
Selection is typically measured by the selection differential, i.e. the mean phenotypic value of the population after selection minus the mean before selection (Falconer & Mackay 1996). For each cohort, we estimated the amount of selective mortality that would be necessary to generate the observed selection differential, given the assumption that before selection, traits were normally distributed with the observed means and variances. For this analysis, selection was assumed to be generated by truncation mortality (e.g. all individuals with phenotypic values above a certain threshold (zt) survived and all below died – this pattern could be reversed if smaller phenotypes were favoured). For example, fz = 0 for z > zt and fz = ∞ for z ≤ zt. To calculate truncation mortality, we used a simple optimisation routine to find the truncation point on a standard normal distribution that would produce a truncated distribution of ‘survivors’ with a mean value that would produce the same standardised selection differential as the one observed for each cohort. We then calculated selective mortality as 1 minus the integral of the truncated distribution. We chose truncation as the functional form of selection because it is the most efficient form of selection, and therefore provides the most conservative estimate of selective mortality (Van Valen 1965). However, other functional forms of selection are possible. To examine how sensitive the conclusions of this analysis would be to choice of functional form, we repeated this analysis assuming that selection can be described by a less efficient, Gaussian function (Appendix S1). We recognise that the functional form of selection may vary among cohorts, but choosing a single functional form to describe selection allows for a standardised comparison among cohorts, species, etc. This analysis is therefore meant as a broad-stroke summary of variation in selective mortality.
On average, the amount of selective mortality experienced by cohorts of fish during recruitment was substantial (the median of the within-species average amount of selective mortality was 0.52; Fig. 1a). Noting that truncation mortality provides a conservatively low estimate of selective mortality, and that total mortality is less than 1, these results suggest that much of the mortality during early life is selective mortality (i.e. median, within-species average non-selective mortality < 0.48). If selection is assumed to follow a modified Gaussian form, then selective mortality was expected to be greater (median, within-species average = 0.79, implying non-selective mortality < 0.21). Importantly, the range of selective mortality values within species (i.e. the difference between the highest and lowest values) also tended to be large, and did not differ much based on choice of functional form. For both truncation and modified Gaussian mortality, median, within-species range of selective mortality was 0.40 (Fig. 1b). Again, these estimates may be conservative because the analyses assume no variation in the functional form of selection. Although these data represent rough estimates, they do suggest that variability in selective mortality can lead to substantial variation in total survival. This, in turn, suggests that selective mortality can be a large source of variation in recruitment.
Why do selection measurements vary?
Measurements of selection (e.g. selection differentials) can vary across cohorts for multiple reasons. First, we need to recognise the effects of measurement error, which can cause apparent differences in the strength (and even direction) of selection (e.g. Mitchell-Olds & Shaw 1987; Siepielski et al. 2009; Morrissey & Hadfield 2012). Inference about selection always involves sampling, and often involves comparing two samples (one before and one after selection). Sampling error will inflate the apparent variability in selection when comparing among cohorts. Strictly speaking, sampling error cannot be avoided, but large sample sizes will help minimise the effects of sampling error. A second form of measurement error stems from the fact that survival is often influenced by multiple, correlated traits. Even if selection on a particular trait of interest remains constant across cohorts, differences in selection on other, correlated traits, and/or differences in the correlation between traits can cause selection differentials to vary. Analysing selection within a multivariate framework allows one to measure the direct effects of selection on each trait and can greatly minimise the variability associated with correlation among traits (Lande & Arnold 1983; see Johnson et al. 2012 for applications to before- and after-selection samples).
Real differences in selective mortality across cohorts can be caused by two phenomena. First, the fitness surface (in the case of recruitment, the relationship between phenotypic value and expected survival) may vary among cohorts. Differences in the biotic and/or abiotic environment may change the shape, location or form of the fitness surface. There are many well-known examples where environmental changes clearly alter the fitness surface. For example, fish that are experimentally separated from major predators may experience changes in selection on traits such as colouration (e.g. Endler 1980), defensive morphology (e.g. Marchinko 2009) and maturation time (e.g. Reznick & Bryga 1987). In other cases, fitness surfaces may change in response to the number of competitors, and selection may thus be density dependent (Einum et al. 2008). However, in many ecological scenarios (e.g. different cohorts of larvae settling to the same habitat over time with no obvious changes in the environment) fitness surfaces may change little, if at all.
A second reason why selection may vary is that the distribution of phenotypes often varies across cohorts. Different cohorts may have developed under dissimilar environmental conditions and/or originated from genetically different source populations, leading to variation in the distributions of phenotypes. Such variation in phenotype distributions can be a major source of systematic variation in selection (e.g. Endler 1986; Weis et al. 1992; Steele et al. 2011). Central to this phenomenon is the fact that fitness surfaces are typically non-linear. For example, expected survival is bounded between 0 and 1, so relationships between phenotype and relative survival can be strongly non-linear. Moreover, it is frequently observed that optimal survival often occurs at intermediate, rather than extreme phenotypes (Janzen & Stern 1998). On a non-linear fitness surface that remains constant, cohorts with different distributions of phenotypes will ‘sample’ different areas of the fitness surface, resulting in differences in the magnitude, and sometimes even the direction of selection (Fig. 2).
Fitness surfaces are not necessarily constant across cohorts, as changes in the environment may cause fitness surfaces to vary. However, we believe that an essentially constant fitness surface is not necessarily a rare scenario. Rigorously evaluating whether fitness surfaces actually differ in space and time requires multiple estimates of selection and an appropriate null model for selection variation. When estimates of fitness surfaces (or summaries such as selection coefficients) vary, it may be tempting to conclude that the underlying fitness surfaces are different. However, we believe that such differences should be considered apparent differences in fitness surfaces until the effects of sampling different regions of a non-linear, but constant fitness surface can be ruled out.
In the sections below, we discuss how a constant fitness surface may provide a parsimonious model to explain variation in selection. When data from multiple cohorts are available, and their distributions of phenotypes vary, systematic variation in selection measurements can provide evidence of whether the fitness surface can be treated as constant across cohorts. Because cohorts of fish can vary substantially with respect to the distribution of their phenotypes (in our review, median CV of cohort means = 7.6%, median CV of within-cohort standard deviations = 23.0%), the effects of such variation on selective mortality and recruitment may be large. We also discuss how the assumption of a constant fitness surface and the techniques described below may be appropriate for other taxa and ecological scenarios (see 'Discussion' and Appendix S2 in Supporting Information).
Reconstructing fitness surfaces to evaluate recruitment
If fitness surfaces are constant and non-linear, then cohorts that differ with respect to their distribution of phenotypes will experience different patterns of selection. However, it is often the case that fitness surfaces are unknown (especially across a broad range of phenotypic values). Rather, investigators typically know the means and variances of trait values before and after a period of selective mortality, and can therefore estimate selection. When multiple estimates of selection are available, one can look for a systematic relationship between trait distributions and selection estimates as evidence of a constant fitness surface. If the fitness surface can be assumed to be reasonably constant, then one can use selection estimates to reconstruct the fitness surface over a broad range of phenotypic values. Describing the fitness surface provides a framework for inferring differences in selective mortality among cohorts. This framework can be extremely useful in that it (1) offers an explanation for differences in observed measures of selection, and (2) describes differences in relative survival among cohorts, even when direct measures of mortality are unavailable.
To illustrate this approach, we examined variation in selection on pelagic larval duration (PLD) among cohorts of a common, demersal blenny (Lipophrys trigloides). Data come from Macpherson & Raventos' (2005) study of post-settlement, selective mortality. These authors measured selective mortality on PLD for eight different cohorts. Samples of recent settlers were collected and PLD was measured from otoliths. Juveniles of the same cohorts were collected ~45 days later. Distributions of PLD values were compared between initial settlers and surviving juveniles in a standard, before-after approach. Note that Macpherson & Raventos (2005) also measured selection on other traits (including size at hatching and size at settlement). Here, we concentrate on a single trait (PLD) to illustrate the analytical approach in the univariate case. A multivariate approach, which explicitly considers mortality that is selective with respect to multiple traits, is described in a subsequent section.
Selection differentials associated with PLD tended to be negative (mean = −1.01, range = −2.37 to 0.46 when standardised to a common measure of variance) suggesting that within cohorts, fish that had shorter PLDs tended to survive better (shorter PLD is likely related to faster growth and/or better condition). Among cohorts, the magnitude of selection changed systematically with average phenotypic value (Fig. 3a). Cohorts that had shorter average PLDs experienced weaker selection, and therefore less selective mortality.
Selection differentials describe the slope of the fitness surface at the cohort mean (Phillips & Arnold 1989). If the selection differentials change linearly with mean phenotypic value (and variances are similar among cohorts), it suggests that over a broad range of phenotypes the fitness surface (i.e. the relationship between phenotypic value and expected fitness) may be reasonably described by a Gaussian function. Defining the fitness function (Wz) as Gaussian, then
where θ is the optimal phenotypic value, ω indicates the width of the curve describing fitness and α specifies the magnitude of non-selective mortality. A Gaussian fitness function has the desirable property that fitness cannot be negative. Moreover, the Gaussian function is somewhat flexible. Depending on parameter values and the phenotypic range of interest, a Gaussian function can describe several forms of selection (e.g. concave up, approximately linear, or concave down). If the distribution of trait values before selection, pz, is normal with mean and variance σ2, σ selection differential, S, can be calculated as
(Lande 1981). Note that the value of selection differentials will change with both the mean and variance of phenotypes before selection.
If selection can be measured for multiple cohorts, then one can estimate parameters of the underlying fitness surface by finding the parameters of eqn (2) that are most likely to have produced the observed values of S, given values of and σ2. The degree of fit between predicted and observed values of S can provide a heuristic indication of how well selection can be described by a constant fitness surface. Once the fitness surface (Wz) is estimated, the mean fitness (survival) of a cohort can also be estimated as ∫ pzWzdz, where pz is the phenotype distribution specified by the mean and variance of phenotypic values before selection.
We used maximum likelihood to fit a non-linear model (eqn (2)) to Macpherson & Raventos' (2005) data on selection differentials on PLD. Our likelihood function was specified by assuming that the residual variation of the selection differentials was normally distributed with equal variance across observations. Mean values were specified by eqn (2), and variance was estimated from the data. We used the optim function in R (R Development Core Team 2013) and a Nelder-Mead non-linear optimisation algorithm to find the parameter values of eqn (2) that were most likely to have produced the observed selection differentials. As a measure of explanatory power, we used a pseudo r2 value, defined as . Note that although a value of 1 is possible only when there is absolutely no measurement error in observed values S, this statistic provides a useful approximation for the explanatory power of non-linear models.
A Gaussian surface did a reasonable job of describing variation in selection differentials (pseudo r2 = 0.61). This estimate of the fitness surface suggested that shorter PLDs were favoured, with an optimum suggested to be somewhere in the vicinity of 2 SD below the overall, among-cohort mean PLD (Fig. 3b). One can estimate selective mortality for each cohort by calculating the area of overlap between the distribution of phenotypes and the fitness surface. This estimates the minimum amount of mortality a cohort experiences, since it does not include non-selective mortality (nor does it include mortality that is selective, but independent of the focal trait). Estimates of selective mortality associated with PLD ranged from 0.35 to 0.96. Assuming non-selective mortality is similar across cohorts, these are large differences in relative survival.
A multivariate example
When estimates of means, variances, covariances and selection differentials are available for multiple traits and multiple cohorts, a similar procedure can be used to estimate a multivariate fitness surface. Although multivariate fitness surfaces can be more difficult to display, they are likely to be more accurate because selection gradients are likely to be more accurate than selection differentials (Lande & Arnold 1983). Also, explicitly accounting for trait covariances and correlational selection will provide a more accurate accounting of selective mortality.
To illustrate how selection measurements can be used to reconstruct multivariate fitness surfaces over a broad range of phenotypic values, we used data from a study of selective mortality on early life-history traits in a reef-associated wrasse (Thalassoma bifasciatum; Grorud-Colvert & Sponaugle 2011). Within this dataset, measurements of selection were made for eight cohorts of fish settling to similar reef habitats in the Florida Keys, USA. Larvae of this species settle and bury into sand and rubble habitat for 3–5 days while they undergo metamorphosis. The before-selection sample represents fish that were captured immediately after emergence (i.e. fish whose ages were 0–4 days post emergence). The after-selection sample represents fish of the same cohort that were captured as surviving juveniles (ages > 9 days post emergence). For this example we analysed selective mortality on larval growth rate (estimated from width of otolith increments during the larval stage), and width of the metamorphic band deposited in the otolith during metamorphosis (an indicator of settlement condition and energy reserves; Hamilton 2008).
As a first step in evaluating whether variation in multivariate selection can be explained by phenotypic variation on a constant, multivariate surface, we examined relationships between distributions of trait values and strength of selection. For each cohort, we calculated selection gradients on both larval growth and condition using the procedures outlined by Johnson et al. (2012). Briefly, for each cohort we used the differences in means before and after selection to calculate selection differentials, and then multiplied this vector by the inverse of the observed variance–covariance matrix to convert selection differentials to selection gradients. Selection gradients measure direct selection on each trait and provide a measure of the slope of the fitness surface along the direction of the focal phenotype (Phillips & Arnold 1989). These slope estimates are averaged across the observed distribution of phenotypes, and if the fitness surface is non-linear, selection gradients will depend on trait variances. However, examining relationships between selection gradients and mean trait values can still provide a useful, first look at the fitness surface. The results of such exploratory procedures can then be used to inform subsequent analyses.
For condition, the selection gradients varied strongly with mean phenotypic value (Fig. 4). These data suggest a fitness surface that is constant and that becomes less steep as condition increases, but levels out (i.e. the slope approaches zero) at ~0.5 SD above the mean. Selection gradients varied much less for larval growth (Fig. 4b) and many of the values were near zero, suggesting a low slope in the vicinity of the observed mean values. Variances in the after-selection samples tended to be smaller than those in the before-selection samples (mean values of the variance ratios were 0.74 for condition, 0.82 for larval growth), suggesting a concave-down fitness surface. Taken together, these data suggest that a Gaussian function may be a reasonable model for the fitness surface (Lande 1981).
Assuming that fitness can be described as a multivariate, Gaussian function of phenotypic values then
where the vector θ indicates the location of the phenotypic optimum, the matrix ω describes the dispersion of fitness values about the optimum and α describes non-selective mortality. Assuming that the distribution of phenotypes before selection can be described as multivariate normal with means and covariance matrix P, then the distribution of phenotypes after selection is also a multivariate Gaussian distribution with a vector of means, , and a covariance matrix, P*, described as
Note that in the multivariate case, the values of the selection differentials can change with the distribution of the focal phenotype and with the distribution of other, correlated traits. In other words, selection differentials will depend on the means and covariances of the traits before selection.
To fit a Gaussian function to the data on selection in bluehead wrasse, we estimated the values of θ and ω that were most likely to have produced the observed changes in the means and (co)variances of phenotypes for each cohort. Our likelihood function was specified by assuming that for each cohort the observed means after selection were distributed as multivariate Gaussian. Expected values for the means after selection were described by eqn (4a). The dispersion of these means was described by the expected covariance matrix after selection (eqn (4b)) divided by the number of individuals in the after-selection sample for each cohort. Observed covariances after selection were assumed to follow a Wishart distribution with the degrees of freedom specified as the number of individuals in the after-selection sample minus one (Press 2012). The joint likelihood was calculated as the product of likelihoods for the means and the covariances, and the product of this quantity was taken across all cohorts. We used the optim package in R (R Development Core Team 2013) and a Nelder-Mead algorithm to obtain maximum likelihood estimates of θ and ω.
Modelling the multivariate fitness surface as a constant, Gaussian function explained much of the observed variation in selection. When considering the entire data set of fitted values (2 after-selection means + 3 elements of an after-selection covariance matrix for each of eight cohorts = 40 fitted values), the pseudo r2 for the relationship between predicted and observed values was 0.57. The estimated fitness surface indicates that there was strong selection favouring faster larval growth and greater condition (Fig. 4c). The data also suggest selection for a positive combination of these traits. The fitness peak in the upper right quadrant suggests that individuals that were in good condition and grew fast were individuals that had especially high survival probabilities.
In addition to explaining variation in selection measurements, by reconstructing a fitness surface, we can estimate differences in relative survival rates. The amount of selective mortality a cohort experiences can be estimated by how much the distribution of phenotypes (in the before-selection sample) overlaps with the fitness surface. Specifically, selective mortality can be calculated by integrating the product of the multivariate density function describing the distribution of phenotypes, pz1,z2, and the multivariate fitness surface, Wz1,z2. That is, selective mortality = . For example, cohort 4 has a mean that is near the optimum of the fitness surface (Fig. 4c). This cohort is expected to have relatively low selective mortality (0.61), whereas cohort 10 is centred in an area of low fitness and is expected to have high selective mortality (0.84). The amount of overlap, and therefore the average mortality within a cohort also depends on the (co)variances of the traits involved. For these data, estimated (selective) mortality within cohorts ranged from 0.38 to 0.84. Similar to our univariate example, these data suggest that under some circumstances fitness surfaces may be reasonably constant and that variation in phenotypes can cause substantial variation in survival and recruitment.
Density dependence, selective mortality and recruitment
Although for many fishes, post-settlement mortality is strongly selective (reviews by Sogard 1997; Perez & Munch 2010), post-settlement mortality may also depend strongly on density (reviews by Myers & Cadigan 1993; Rose et al. 2001; Hixon & Webster 2002; Osenberg et al. 2002). For example, high densities of fish within a habitat may result in increased competition among individuals and/or an increased response by predators, which can lead to greater mortality rates (reviewed by Hixon & Jones 2005). Because survival may be strongly influenced by both density and phenotype, it is useful to consider how these attributes combine to influence recruitment.
Simple, density-dependent models of recruitment [e.g. Beverton & Holt (1957) and Ricker (1954) models of stock–recruitment relationships] capture the basic property of regulation, but typically provide a poor fit to real data (e.g. Iles 1994; Myers 2001). A limitation of such models may be that they treat the underlying relationship between density and mortality as constant. Recent studies suggest that density-dependent recruitment is a complex process in which the strength of density dependence may change with several factors, including characteristics of individuals (Shima et al. 2006; Johnson 2008). Given the strong and pervasive effects of selective mortality during larval and juvenile phases (Fig. 1), we believe it would be useful to consider models of recruitment where an individual's probability of surviving depends on both phenotype and density.
To illustrate how density and phenotype may jointly affect recruitment, we consider a population where selection has been well studied and recruitment is strongly regulated by density-dependent mortality. The data describe recruitment of steelhead trout (Oncorhyncus mykiss) in Snow Creek, Washington, USA (Seamons et al. 2007). First, we evaluated density dependence in recruitment by examining the relationship between number of spawning adults and number of returning offspring (sampled when they returned to spawn; Fig. 5). These data, like many stock–recruitment curves, illustrate two points: (1) the presence of strong density dependence, and (2) relatively poor explanatory power (pseudo r2 = 0.16).
Seamons et al. (2007) also measured selection on adult body size. By tagging, measuring and sampling the tissue of virtually all returning adult fish, the authors were able to use genetic parentage analysis to count how many returning offspring each adult fish produced. The relationship between adult body size and lifetime reproductive success provides a measure of the strength of selection (Lande & Arnold 1983). Here, we analysed variation in selection on male body size. Note that although this example focuses on traits of adults (rather than offspring, as in our previous examples) and measures recruitment as the number of returning adults (rather than juveniles), the principles are entirely the same. Moreover, one of the reasons male body size affects recruitment in salmonids may be because of correlations between adult size and offspring size and performance (Heath et al. 1999; Smoker et al. 2000). Because the within-cohort variance differed substantially among cohorts, we plotted the relationship between selection differentials and both the cohort mean and variance (Fig. 6). Note that for some of the cohorts in the original study, sample sizes to estimate selection were very small. To avoid imprecision associated with small samples, we restricted this analysis to selection estimates from cohorts with n > 10. Although taken individually, these sample sizes are small, our analyses focused on variation among selection measurements across the 15 separate cohorts. Selection differentials changed substantially with both mean and variance in phenotypic values, suggesting a non-linear fitness surface. Again, using maximum likelihood, we fit eqn (2) to the selection differentials as described above. A Gaussian fitness surface provided a good fit to the selection data (pseudo r2 for the relationship between predicted and observed selection differentials was 0.56, n = 15).
The data for selection on body size suggest that phenotypic variation and selection on a constant fitness surface can lead to substantial variation in relative survival rates. But do these inferences hold true when we look at more direct estimates of survival, knowing that density dependence is strong? To evaluate this, we can compare a simple, density-dependent model of recruitment to a model in which individual survival probability may be affected by both phenotype and density. To describe density-dependent recruitment, we use a Ricker model:
where NR is the number of recruits, NS is the number of spawners, a describes the rate of offspring production (a combination of fecundity and non-selective, density-independent mortality) and b describes the rate of density-dependent mortality. This model of density-dependent recruitment is appropriate for many salmonids because a strong mechanism of density dependence results from greater disturbance of nests and subsequent egg mortality when spawners are abundant (Ricker 1954; McNeil 1964; Fukushima et al. 1998). However, survival during later stages (especially juveniles) can depend on body size (which is linked to parental body size; Heath et al. 1999; Smoker et al. 2000; Carlson & Seamons 2008). Following a similar procedure as in Box 1., the change in the number of individuals (and distribution of phenotypes) over time can be described as an outcome of three processes. Specifically,
where a, b and NS are as in the density-dependent model, and fz is a function describing the rate of selective mortality. The distribution of phenotypes at time t + 1 is
and total abundance at time t + 1 is
If body sizes are normally distributed within each cohort, then nz,t can be described as the product of the number of spawners (NS) and a normal probability density function (pz). Defining fz as a scaled quadratic function (fz = (z − θ)2/2ω2) yields a Gaussian fitness surface and integrating the former expression yields our model for recruitment:
where θ is the optimal phenotypic value, ω indicates the width of the curve describing fitness, and and σ2 are the mean and variance of the cohort before selection. Other symbols are as in eqn (5). Note that in this model, the function describing the rate of selective mortality was not affected by density. This is because in this system we expect the main mechanism of density dependence (egg mortality because of nest site disturbance at high spawner density) to be largely independent of the main source of selective mortality (size-selective predation during the juvenile phase; Ward et al. 1989), which is traceable in our model due to the correlation between size of offspring and the size of male parents in salmonid populations (Heath et al. 1999; Smoker et al. 2000; Carlson & Seamons 2008). However, if selection is expected to depend on density (or any other measurable environmental factor), it is straightforward to incorporate these effects into fz (e.g. by modelling selective mortality as a function of both phenotype and density; Appendix S2).
We fit both the density-dependent model (eqn (5)) and the combined phenotype- and density-dependent model (eqn (6)) to the observed recruitment values. Our likelihood functions were specified by assuming that the observed estimates of recruitment were Poisson distributed with mean values specified by either eqn (5) or eqn (6). In each case we used the optim function in R (R Development Core Team 2013) and a Nelder-Mead algorithm to find the parameter values that were most likely to have produced the observed pattern of recruitment. Fitting both models to the data suggested that the phenotype- and density-dependent model fits better (∆AICc = 4.5), and that adding phenotypic variation as a predictor more than doubled the explanatory power of the stock–recruit relationship (pseudo r2 = 0.39). These results suggest that even in a strongly density-regulated system, phenotypic variation and selective mortality can be an important source of variation in survival and recruitment (Fig. 7). These results also suggest that within this population, large body size of males provides more than just a competitive mating advantage. That cohorts with larger mean body sizes produced more surviving offspring supports a previously hypothesised link between male body size and overall quality and/or quantity of offspring.
Understanding how phenotypic variation interacts with selective mortality and density dependence can greatly improve our understanding of recruitment. It is clear that selective mortality can be substantial, with episodes of selection frequently removing more than half of the individuals from a cohort. Moreover, variation in selective mortality among cohorts can also be large, suggesting considerable effects on recruitment variability. Here, we have described a framework to evaluate how phenotypic variation can result in systematic differences in selective mortality. This framework can be extremely valuable for making sense of the apparent variability in selection and for understanding variation in population replenishment.
Our examples illustrate that the relationships between phenotypes and early life survival may be relatively constant. In contrast, the distributions of phenotypes that affect survival appear to differ substantially among cohorts. It is the latter phenomenon that may be responsible for much of the variation in selective mortality. A major implication is that if fitness surfaces can be estimated, and one has information on the distribution of phenotypes that affect survival, then recruitment may be reasonably predictable. In fact, information on phenotypic variability and selective mortality can be used in a manner similar to stock–recruitment models: measured attributes of the system (e.g. phenotype distributions and fitness surfaces) can be used to predict average values of recruitment.
Although fitness surfaces can change with environmental conditions, situations in which fitness surfaces are essentially constant may not be particularly rare. However, if fitness surfaces are commonly non-linear, they may be difficult to describe. The challenge for biologists will be to conduct studies with enough replicate measures of selection to reconstruct fitness surfaces and rigorously evaluate whether fitness surfaces may be treated as constant. Although we acknowledge that for fishes, such studies are rare, evidence of constancy in fitness surfaces has regularly been observed for other taxa. For example, systematic and strong relationships between phenotype distributions and selection measurements have been observed in well-replicated studies of selection in populations of insects (e.g. Weis et al. 1992), birds (e.g. Reed et al. 2006; Brommer & Rattiste 2008; Charmantier et al. 2008), and mammals (e.g. Wilson et al. 2006). These observations suggest that the general methods described in this study may be widely applicable, even if constant fitness surfaces are not the norm. Moreover, when it is clear that fitness surfaces do vary with other features of the local environment (e.g. density or any other measurable characteristic), such effects may be incorporated into the function describing the rate of selective mortality (e.g. fz can become fz,N in the case of density-dependent selection; Appendix S2).
When mortality is density dependent, information on phenotype distributions and selective mortality can add another dimension to traditional stock–recruitment models. Simple, density-dependent models usually do a reasonable job of describing non-linear relationships between density and recruitment. However, such models typically do a poor job of explaining variability in real data (Iles 1994; Myers 2001). In other words, density-dependent models capture the central tendency of recruitment, but they cannot explain why certain cohorts deviate from this central tendency. Several studies have implied that accounting for some forms of phenotypic variation (particularly changes in the age/size composition of the spawning population) in stock–recruitment models can improve their performance (Marteinsdottir & Thorarinsson 1998; Scott et al. 1999; Lucero 2009; Brunel 2010). However, both density dependence and the relationship between phenotype(s) and survival are often strongly non-linear. Because of this, simple approaches to analysing the effects of phenotype and density on recruitment (e.g. multiple regression) are unlikely to be adequate. For example, on curvilinear fitness surfaces, differences in the variances of cohorts can lead to large differences in recruitment (eqn (6)). For fishes, within-cohort variance is highly variable among cohorts (from our review, median CV = 23%, range = 2.6–64%), and likely to be a persistent source of recruitment variation. By starting with individual survival probabilities, and accounting for the distributions of individuals within cohorts, we can generate models of density-dependent recruitment that properly account for the non-linear effects of phenotype on survival.
Our review and synthesis of selection measurements suggests that for recruiting cohorts of fishes, most mortality is selective mortality. The amount of selective mortality is also highly variable among cohorts, suggesting that if we have a better understanding of why selective mortality varies, then we can improve our understanding of recruitment. Through several empirical examples where selection has been well studied, our analyses show that fitness surfaces may be relatively constant, but strongly non-linear. These results suggest that much of the variation in mortality during the recruitment process may be predictable and a direct consequence of variation in the distribution of phenotypes that are important for survival.
The true utility of a framework that describes the combined effects of phenotypic variation, selective mortality and density dependence (where applicable) on recruitment is that it will allow one to evaluate both the short- and long-term consequences of variation in phenotypes. For fishes, among-cohort variation in the distribution of early life traits can be influenced by several factors including temperature (Macpherson & Raventos 2005; Sponaugle et al. 2006), parental effects (Heath et al. 1999; McCormick 2006; Johnson et al. 2011), location of larval development (Hamilton et al. 2008; Shima & Swearer 2009) and/or source population (e.g. Post & Prankevicius 1987). In the short term, knowing the shape of the fitness surface will yield insight into how variation in these environmental factors ultimately translates into variation in recruitment. In the long term, an accurate description of the fitness surface will be useful in that it will allow investigators to predict the effects of long-term changes in characteristics of larvae and juveniles. For example, it is expected that early life traits will change in response to factors such as fishery selection (Munch et al. 2005; Walsh et al. 2006; Johnson et al. 2011) and climate change (e.g. Munday et al. 2009; Franke & Clemmesen 2011; Baumann et al. 2012). By understanding how such changes in phenotypes will ultimately affect recruitment, we will be able to anticipate how changes in the abiotic and biotic environment will affect both the quality of individuals in the population, and the population's capacity for resistance and/or resilience to environmental change.
We thank Robert Warner, Todd Seamons, Liz Pásztor and two anonymous referees for their helpful comments on an earlier draft of the manuscript. This work was conducted while D.W.J. was supported by a postdoctoral fellowship from Scripps Institution of Oceanography. During the preparation of this manuscript, both S.S. and B.X.S. were supported in part by grants from the National Oceanic and Atmospheric Administration (NA11NOS4780045 to S.S., NA10OAR4320156 to B.X.S.).
D.J., K.G.C., S.S. and B.X.S. conceived the study. D.J. wrote the first draft of the manuscript, and all authors contributed substantially to the revisions.