Appendix S2. Reconstructing fitness surfaces when selection is density dependent
If selection is density dependent, then the rate of selective mortality can be described as
a function of both phenotype (z) and density (N). The change in the number of individuals (and
distribution of phenotypes) over time can be described as an outcome of two processes.
Specifically,
𝜕𝑛𝑧,𝑡
𝜕𝑡
= 𝑛𝑧,𝑡 (𝛼 − 𝑓𝑧,𝑁 ),
Where nz,t is the distribution of phenotypes at time t, α is a baseline rate of change that is
independent of both density and phenotype, and fz,N is a function describing the rate of
selective mortality. Generally, fz,N can represent the relationship between phenotype, density,
and any fitness measure, but in keeping with the main focus of the paper, here we discuss
selection as generated through mortality (i.e., fz,N has a negative sign). For analytical
tractability, we consider models in which selective mortality depends on initial density (N0). The
distribution of phenotypes at time t+1 can therefore be expressed as
𝑛𝑧,𝑡+1 = 𝑛𝑧,𝑡 𝑒 𝛼−𝑓𝑧,𝑁0
and total abundance at time t+1 is
𝑁𝑡+1 = 𝑒 𝛼 ∫ 𝑛𝑧,𝑡 𝑒 −𝑓𝑧,𝑁0 𝑑𝑧.
(𝑧−𝜃+𝑏𝑁 )2
0
To formulate a specific model for a density-dependent fitness surface, let 𝑓𝑧,𝑁0 =
. In
2𝜔 2
this model, the rate of selective mortality is elevated for those phenotypes that are distant
from the optimal phenotype (θ), but the location of the optimal phenotype changes with
density. b is a coefficient describing the rate at which the fitness peak (i.e., optimal phenotype)
changes with density.
If we describe nz,t as the product of initial density (N0) and the normal density function
(pz), then
𝑁𝑡 = 𝑁0 𝑒 𝛼 ∫ 𝑝𝑧 𝑒
(𝑧−𝜃+𝑏𝑁0 )2
2𝜔2
−
𝑑𝑧.
Essentially, future abundance is related to the overlap between the distribution of phenotypes
(pz) and the density-dependent fitness surface (𝑊𝑧 = 𝑒
expression yields
𝑁𝑡 = 𝑁0 𝑒
𝜔
𝛼
√𝜔2 +𝜎2
𝑒
−
−
(𝑧−𝜃+𝑏𝑁0 )2
2𝜔2
(𝑏2 𝑁2 +2𝑁𝑏(𝜃−𝑧̅)+(𝑧̅−𝜃)2 )
2(𝜔2 +𝜎2 )
). Integrating the former
.
(S2.1)
̅ ) as
This result allows us to express mean fitness (𝑊
𝑁𝑡
̅ = =𝑒
𝑊
𝑁
0
𝛼
𝜔
√𝜔2 +𝜎2
𝑒
−
(𝑏2 𝑁2 +2𝑁𝑏(𝜃−𝑧̅)+(𝑧̅−𝜃)2 )
2(𝜔2 +𝜎2 )
. (S2.2)
̅)
𝜕ln⁡𝑊
Noting that selection gradients can be calculated as 𝜕𝑧̅ (Lande 1976, 1979), then from
equation S2.2 we can generate an expression for selection gradients under this densitydependent model. Specifically,
𝛽=
̅
𝜕ln⁡𝑊
𝜕𝑧̅
=
(𝜃−𝑧̅ +𝑏𝑁0 )
(𝜔 2 +𝜎2 )
.
(S2.3)
If one has multiple estimates of selection gradients, initial density, and the means and variances
of phenotypes before selection, then one can use equation S2.3 to estimate parameters of a
density-dependent fitness surface. Also note that this general approach could be used to
model selection as a function of other environmental variables (including abiotic variables such
as temperature, rainfall etc.). It could also be used to model certain types of frequencydependent selection. For example, rate of selective mortality could be modeled as a function
of both the individual’s phenotypic value, and the mean phenotypic value for the cohort.
An empirical example
We know of no studies of fish populations that have both measured selection many
times and were conducted in an ecological context in which we would expect selection to be
density dependent. However, such data can be found for many other taxa, and the methods
outlined above are general. As an illustrative example, we apply our model of densitydependent selection to a long-term study of the relationship between laying date and lifetime
reproductive success in a population of mute swans (Cygnus olor). Charmantier et al. (2006)
measured selection on laying date for each of 20 breeding years. These authors also recorded
information on population abundance, and the means and variances of laying date within each
year.
For mute swans, an early laying date can be advantageous because early breeders get
access to higher quality nesting sites, which leads to greater survival of offspring (Nummi &
Saari 2003, Charmantier et al. 2006). For this species, breeding success also depends strongly
on density, and nest site limitation is a major mechanism underlying this density dependence
(Nummi & Saari 2003). Consequently, selection on laying date may be density dependent
because high-quality nest sites fill up more quickly when population density is high. In other
words, individuals can get a high quality nest by being slightly earlier than average when
population density is low, but to get a nest of similar quality when population density is high,
individuals need to nest much earlier than average. This type of density-dependent selection
may be described by the “peak shift” model outlined above.
For the data reported by Charmantier et al. (2006), mean lifetime reproductive success
declined with abundance (Fig S2a), though there was considerable variation about this trend
(exponential model of decline: p = 0.05, r2 for a linear model on ln-transformed data 0.19). To
examine whether this density-dependent trend might also be reflected in the selection data, we
first fit a Gaussian fitness function to the observed variation in selection gradients. This was
accomplished by setting b = 0 in equation S2.3 and using nonlinear least squares regression to
fit this equation to the selection gradients reported by Charmantier et al. (2006). Note that the
reported selection gradients were scaled to a common measure of variance (the average of the
within-year variance values). Because residuals from this model indicated a systematic change
in selection gradients with density (Fig S2b), we fit the full version of equation S2.3 to the data.
This “peak shift” model fit much better than a density-independent, Gaussian surface (F test:
F1,17 = 7.483, p = 0.014). Coefficents of S2.3 were θ = 0.731 (0.576 SE), ω = 1.68 (0.679 SE), and
b = -0.026 (0.012 SE). The negative value of b indicates that as density increased, the optimal
laying date was earlier. The relationship between the observed and predicted values was
centered on a 1:1 line (indicating good fit), but the explanatory power of this model was
moderate (r2 for the 1:1 line = 0.31).
To validate this model further, we can use the estimated fitness surface parameters to
calculate mean fitness using equation S2.2. The ‘predicted’ values of mean fitness could then
be compared to the empirical estimates of mean fitness observed in the Charmantier et al.
(2006) study (unpublished data provided by A. Charmantier). Fitting equation S2.3 to selection
gradients does not allow estimation of α, the baseline rate of change that is independent of
phenotype and density. We cannot use equation S2.2 to predict absolute fitness, but we can
compare values of relative fitness. To calculate relative fitness, both observed and predicted
values were divided by their respective means. This procedure will factor out α for the
predicted values and it will factor out the average value of α for the observed values. Variation
in α among years will still contribute to variation in the observed values of relative mean
fitness. The relationship between predicted and observed values of relative mean fitness was
centered on a 1:1 line (Fig S3), indicating a direct relationship between predicted and observed
mean fitness. The r2 value for the 1:1 line was 0.17. Given that many other factors are known
to affect variation in mean reproductive success, these results suggest that even when selection
is strongly density dependent, phenotypic variation can still account for substantial variation in
lifetime reproductive success and population dynamics.
Fig S2. Effects of population density on both mean fitness and strength of selection on laying
date. (a) Relationship between population abundance and mean lifetime reproductive success.
(b) Relationship between abundance and residuals of selection gradients, which were
estimated after accounting for effects of variation in phenotype distributions.
Fig S3. Relationship between predicted and observed values of mean lifetime reproductive
success for a population of mute swans (Cygnus olor). Mean fitness values for each year were
relativized by dividing by the overall mean (i.e., across all 20 years). Predicted mean fitness was
calculated from the fitness surface that was reconstructed from observed selection
measurements. Line represents a 1:1 relationship.