Predicting the Evolutionary Consequences of
Trophy Hunting on a Quantitative Trait
Department of Zoology, University of Oxford, South Parks Road, Oxford OX1 3PS, United Kingdom
SUSANNE SCHINDLER, Department of Evolutionary Biology and Environmental Studies, University of Zurich, Winterthurer Str. 190, CH-8057
LOCHRAN TRAILL, Research Centre in Evolutionary Anthropology and Palaeoecology, John Moores University, James Parsons Building, Byrom
Street, Liverpool L3 3AF, United Kingdom
BRUCE E. KENDALL, Bren School of Environmental Science & Management, University of California, Santa Barbara, CA 93106-5131, USA
ABSTRACT Some ecologists suggest that trophy hunting (e.g., harvesting males with a desirable trait above a
certain size) can lead to rapid phenotypic change, which has led to an ongoing discussion about evolutionary
consequences of trophy hunting. Claims of rapid evolution come from the statistical analyses of data, with no
examination of whether these results are theoretically plausible. We constructed simple quantitative genetic
models to explore how a range of hunting scenarios affects the evolution of a trophy such as horn length. We show
that trophy hunting does lead to trophy evolution deﬁned as change in the mean breeding value of the trait.
However, the fastest rates of phenotypic change attributabl e to trophy hunting via evolution that are theoretically
possible under standard assumptions of quantitative genetics are 1–2 orders of magnitude slower than the fastest
rates of phenotypic change reported from statistical analyses. Our work suggests a re-evaluation of the likely
evolutionary consequences of trophy hunting would be appropriate when setting policy. Our work does not
consider the ethical or ecological consequences of trophy hunting. Ó2017 The Wildlife Society.
KEY WORDS hunting, integral projection models, quantitative genetics, selection.
Trophy hunting that is well managed, and based on robust
monitoring protocols, can be a useful conservation tool in
areas where there is increasing demand for land from
growing human populations (Lindsey et al. 2006, Di Minin
et al. 2016). The logic of the approach is that selectively
hunting a small proportion of males with large horns, antlers,
or body size will have few ecological and evolutionary
consequences because species with sexually selected charac-
ters usually exhibit a polygynous mating system in which
males are not limiting (Milner-Gulland and Mace 1998,
Dickson et al. 2009). However, a debate on the ethics, use,
and consequences of trophy hunting is underway (Lindsey
et al. 2016, Nelson et al. 2016, Ripple et al. 2016), including
an ongoing fast or slow evolution discussion on hunted
bighorn sheep (Ovis canadensis; Coltman et al. 2003, Traill
et al. 2014, Pigeon et al. 2016). We contribute further to the
trophy hunting debate by constructing and analyzing general
quantitative genetic models of the effect of trophy hunting
on phenotypic evolution.
Proponents of trophy hunting argue that selling the rights to
selectively hunt individuals with desirable attributes is a useful
way to raise money (Rodrıguez-Mu~noz et al. 2015). The
argument is that if wildlife populations can be moneterized,
they have value, and this worth makes the area in which the
population lives more easily protected from competing land use
interests (Lindsey et al. 2007). Proﬁt generated from hunting
can be invested in conservation, habitat improvement, or in
local communities, and any ecological and evolutionary
consequences of selective hunting on males is likely to be a
small cost worth paying (Crosmary et al. 2015).
Those opposed to the approach argue either that trophy
hunting is unethical, or that money raised from trophy
hunting rarely gets invested in local communities or in
conservation. For example, in Africa, monies raised from
selling hunting rights can get subsumed into government
coffers, and proﬁts made by outﬁtters do not always make it
back to the local area or communities (Lindsey et al. 2014). In
addition, the ecological outcomes of hunting may be negative:
in east Africa, unregulated trophy hunting inﬂuenced a
localized extirpation of lion (Panthera leo) populations (Packer
et al. 2011), and unethical lion hunting practices in Hwange
National Park in Zimbabwe resulted in 72% of research
animals being killed, including 30% of males <4 years old that
had yet to breed (Loveridge et al. 2007). Furthermore, hunting
may lead to evolution of selected traits as has frequently been
speculated for some sheep populations (Festa-Bianchet et al.
2014, Douhard et al. 2016, Pigeon et al. 2016).
One reason why the ecological and evolutionary consequen-
ces of trophy hunting have received recent interest is that
biologists have found that evolution can be observed on
ecological timescales (Hairston et al. 2005). This has spawned
Received: 20 September 2016; Accepted: 31 January 2017
The Journal of Wildlife Management; DOI: 10.1002/jwmg.21261
Coulson et al. Evolutionary Consequences of Trophy Hunting 1
the ﬁeld of eco-evolution (Schoener 2011). There is
compelling empirical evidence of rapid, joint phenotypic
and ecological change from a number of systems (Hairston
et al. 2005, Ozgul et al. 2010), but evidence of genetic change is
much less widespread (Yoshida et al. 2003), partly because it is
harder to demonstrate. Quite frequently, phenotypic change
is attributable to evolution without supporting evidence of
genetic change (Hendry 2016), or without examining whether
the rates of evolutionary change reported are theoretically
plausible (Coltman et al. 2003).
Coltman et al. (2003) report rapid phenotypic change in the
face of hunting that was predominantly attributed to evolution.
Based on longitudinal data for the Ram Mountain bighorn
population in Canada, Coltman et al. (2003) used statistical
quantitative genetics to argue that selective hunting of, on
average, 2 rams/year from a population of, on average,
approximately 150 bighorn sheep resulted substantially
contributed to a 30% decline in horn size over 5 generations.
In a second paper, Pigeon et al. (2016) reported a new analysis
that also concludes evolution plays a key role in the observed
phenotypic declines. These papers (Coltman et al. 2003,
Pigeon et al. 2016) have become inﬂuential as opponents of
trophy hunting argue that the activity has rapid detrimental
consequences on hunted populations. However, no papers
have yet examined whether the rates of change observed by
Coltman et al. (2003) and Pigeon et al. (2016) are plausible
using the quantitative genetic theory that motivated their
statistical analyses, even though skepticism has been raised as
to whether the phenotypic changes observed can be attributed
to evolution (Traill et al. 2014).
We developed novel, general theory to examine the likely
evolutionary consequences of selective harvesting on a single sex
in a sexually reproducing species. We worked in the quantitative
genetics framework because the genetic architecture of trophy
traits is rarely known (Kruuk et al. 2002). We start with a brief
summary of quantitative genetic theory that motivated our
models, and which is widely used to examine the evolution of
phenotypic traits of unknown genetic architecture in free-living
populations (Meril€aet al. 2001). We then describe the models
we used, along with the parameter values we selected.
We use the following notation. Expectations and variances of
the distribution of N(x,t) are denoted E(x,t) and V(x,t),
respectively. A subscript, either of for m, is used to identify
distributions or moments of distributions taken over only
females or males, respectively. If this subscript is absent, the
distribution is taken over both sexes. We use a superscript R
to identify distributions, or moments of distributions, that
have been operated on by selection.
A Quantitative Genetic Primer
Quantitative genetics assumes that an individual’s phenotype
Zconsists of the sum of various components. These
components include a breeding value Aand the environ-
mental component of the phenotype E, with contributions
from epistasis and non-additive genetic effects also
sometimes included in the sum (Lynch and Walsh 1998).
Only Aand Eare considered here. An individual’s breeding
value describes the additive genetic contribution to its
phenotypic trait value. But what does this mean?
If alleles at a locus have an additive effect on a phenotypic
trait, each allele can be assigned a value that describes the
contribution of that allele (in any genotype at that locus) to
the phenotype. For example, consider a bi-allelic locus with 3
genotypes, aa,aA, and AA. Allele ahas a value of 1 g and
allele Aa value of 2 g. The breeding value of each genotype to
body mass will be aa ¼1þ1¼2, aA ¼1þ2¼3, and
AA ¼2þ2¼4. Breeding values can be summed across
genotypes at different loci to generate breeding values for
multi-locus genotypes. Under the additivity assumption, the
dynamics of breeding values is identical to the dynamics of
alleles; this is the not always the case when non-additive
genetic processes like heterozygote advantage and epistasis
are operating (Falconer 1975).
Many applications of quantitative genetics use the
inﬁnitesimal model (Fisher 1930). This assumes that an
individual’s breeding value for a phenotypic trait is made up
from independent contributions from a large (technically
inﬁnite) number of additive genotypes, each making a very
small contribution to the phenotypic trait. There is no
interaction between alleles at a locus (dominance) or
interactions between genotypes at different loci (epistasis).
In additive genetic models used to predict evolutionary
change, it is usually assumed that Eis determined by
developmental noise. An individual’s environmental compo-
nent can be considered as a random value drawn from a
Gaussian distribution with a mean and a constant variance:
norm(0,V(E,t)). Aand Eare consequently independent. Thus,
The distribution of breeding values is also assumed to be
VðZ;tÞ ¼ VðA;tÞ þ VðE;tÞ:
These assumptions mean that, on average, the breeding value
can be inferred from the phenotype—the phenotypic gambit.
The aim of statistical quantitative genetics is to correct the
phenotype for nuisance variables so the phenotypic gambit
assumption is appropriate for the corrected phenotype
(Lynch and Walsh 1998).
Next, quantitative genetic theory makes the assumption
that the mean of Aamong parents is equal to that in
offspring: for example, E(A,tþ1) ¼ER(A,t). In 2-sex
models, this requires that the expected value of Ain an
offspring is the mid-point of the breeding value of its parents.
Given this assumption,
DEZð Þ ¼ EZ;tþ1ð Þ EZ;tð Þ ¼ EA;tþ1ð Þ EA;tð Þ
where DEðZÞis the difference in the mean of the phenotype
between the offspring and parental generations, SðZÞis the
selection differential on Zand VA;tð Þ
VZ;tð Þ the heritability (h
) of a
2 The Journal of Wildlife Management 9999()
trait, and trepresents generation number. The selection
differential describes the difference in the mean value of the
character between those individuals selected to reproduce
and the entire population prior to selection (Price 1970).
Equation (3) is the univariate breeders equation (Falconer
If all assumptions of the univariate breeders equation are
met, it will accurately predict evolution of a trait assuming
that the selection differential and the additive genetic and
phenotypic variances have been appropriately estimated. One
exception where it can fail is if there are genetically correlated
characters that have not been measured, and which are under
selection (Lande and Arnold 1983).
Lande and Arnold (1983) developed a multivariate form of
the breeders equation that states:
DEðZÞ ¼ GP1S
where DEðZÞis a vector describing change in the mean of
each of the phenotypic traits from the parental to the
offspring generation, Sis a vector of selection differentials on
each character, Gis a genetic variance-covariance matrix, and
Pis a phenotypic variance-covariance matrix. If 2 traits are
genetically correlated, and both are under selection, to
understand how 1 of the traits evolves it is necessary to
understand how the 2 traits are genetically and phenotypi-
cally correlated, and how strong selection is on each of the
In both the univariate and multivariate breeders equations,
the selection differentials capture total selection (Lande and
Arnold 1983). This means that both equations accurately
capture selection on the trait(s) even in the presence of
unmeasured genetically correlated characters. Genetically
correlated characters inﬂuence predictions of evolution in
the breeders equations through their impact on estimates of
the heritability (in the univariate case) and the Gmatrix (in
the multivariate case).
A limitation of the breeders equation is it is not dynamically
sufﬁcient—it should not be used to make predictions across
multiple generations, particularly when evolution is sufﬁ-
ciently strong that it alters genetic variances and covariances
(Lande and Arnold 1983). To construct a dynamic model, it
is either necessary to make assumptions about the genetic
variance (it is sometimes assumed to be constant; Lande
1982) or to track the dynamics of the entire distributions of A
and E(Coulson et al. 2017) or Aand Z(Childs et al. 2016).
We model the dynamics of Aand E.
A Generic Model to Explore the Effects of Trophy
Hunting on Evolution
We developed a 2-sex, dynamic, quantitative genetic model
to explore how hunting on one sex inﬂuences phenotypic
evolution. We iterate the population forwards on a per-
generation time step.
We assume that in the absence of hunting, the trophy is not
under selection in either sex and is consequently not evolving.
This provides us with a baseline scenario in the absence of
hunting with which to compare results from a range of hunting
scenarios. We deﬁne a bivariate distribution NðA;E;tÞof
breeding values Aand the environmental component of the
phenotype Ein generation t. At time t¼0, we assume the
distribution NðA;E;tÞis bivariate normal with means mand
(co)variances S. The 2 components of mare E(A,t) and E(E,t) at
t¼0. Sis a variance-covariance matrix,
Variances can be chosen to determine the heritability h
time t¼0 .
We assume that males and females have the same distribution
of phenotypes and breeding values at birth, and that the birth
sex ratio is unity: NfðA;E;tÞ ¼ NmðA;E;tÞ ¼ NðA;E;tÞ
Next we impose selection. There is no direct selection on
females and the number of recruits they produced is set to 2,
the replacement rate, to ensure the female population
remains the same size over time and the population growth
rate l¼1. This assumes males are not limiting. The
distribution of females selected to reproduce is consequently
fðA;E;tÞ ¼ 2NfðA;E;tÞ. The same function for males is
used in the absence of hunting.
When males are selectively hunted, we remove individuals
from the distribution before assigning male reproductive
success. We then scale the resulting distribution of males to be
the same size as the distribution of females. For example, if all
males of above mean trophy size are culled, the matings they
would have had are redistributed across those males that were
below the mean trophy size and not hunted. In the case of a
Gaussian distribution of the trophy, their lifetime reproductive
success would increase proportionally to the number of males
culled. The proportion pis calculated and the post-selection
mðA;E;tÞ ¼ 1
culated where SmðA;E;tÞis the function describing selection
on the male trophy. The distribution NR
distribution of the components of the phenotype of those
males selected to be fathers.
We impose selection on males by culling a proportion aof
individuals that are above average size,
SmðZ;tÞ ¼ ð1aÞZ;ifZ>EðZ;tÞ
This generates a distribution of fathers NR
equal in size to the distribution of mothers NR
We now have distributions of maternal and parental
characters that are the same sizes and sufﬁcient for the female
population to replace itself with some males reproducing with
multiple mothers. We assume random mating and calculate
the distribution of parental midpoint breeding values
NRðA;tÞby convolving NR
2. X represents
the convolution in equations (5), (6), and (8). To generate the
distribution of offspring breeding values, we convolve this
distribution with a distribution of the segregation variance,
deﬁned as a Gaussian distribution with a mean of 0 and a
variance equal to half the additive genetic variance of the
Coulson et al. Evolutionary Consequences of Trophy Hunting 3
2(Barﬁeld et al. 2011). Effects
of increases in the additive genetic variance via mutation, or
from other sources of genetic variation being converted to
additive genetic variance, can be captured by increasing the size
of the segregation variance. Finally, we generate a distribution
of the environmental component of the phenotype for each
value of Ain the offspring distribution that is proportional to a
Gaussian distribution with a mean of 0 and an environmental
variance that is the same as that in the previous generation. We
now have the bivariate distribution of the components of the
phenotype in offspring N(A,E,tþ1).
Taken together this gives the following recursion,
NfðA;E;tÞ ¼ NðA;E;tÞ
NmðA;E;tÞ ¼ NðA;E;tÞ
fðA;E;tÞ ¼ 2NfðA;E;tÞ ð3Þ
mðA;E;tÞ ¼ 1
NRðA;tÞ ¼ NR
NðA;tþ1Þ ¼ NRðA;tÞ normð0;VRA;tÞ
NðE;tþ1Þ ¼ normð0;VðE;tÞÞ ð7Þ
NðA;E;tþ1Þ ¼ NðA;tþ1Þ;NðE;tþ1Þ½ ð8Þ
Analysis of the Multivariate Breeders Equation
When evolutionary predictions fail to match observation, the
existence of correlated unmeasured characters is often assumed
(Meril€aet al. 2001). However, the potential impact of
correlated characters on evolution assuming selection differ-
entials have been appropriately measured is rarely investig
ated. We used the multivariate breeders equation to examine
how such characters can inﬂuence evolution, and in particular,
whether they can generate rapid evolution in directions
opposite to those predicted by selection differentials which
measure total selection on a trait (Lande and Arnold 1983).
We assume 2 traits Z1and Z2. We predict 1 generation
ahead, so we do not use tfor time to simplify notation. We
deﬁne bivariate Gaussian distributions of the traits’ breeding
values A1and A2(norm(m(A), S(A))) and environmental
components of the phenotype (norm(m(E), S(E))). From this,
we construct a bivariate Gaussian distribution of the phenotype
norm (m(Z), S(Z)) ¼(m(A), S(A)) þ(m(E), S(E)).
We now impose selection on the phenotype with the
following ﬁtness function WðZ;tÞ ¼ b0þb1Z1þb2Z2.
We estimate selection differentials on the 2 phenotypic traits
as S¼P1ðZÞbwhere b¼ ðb1;b2ÞTwhere Tis the vector
transpose and Sis a vector containing the selection differentials
. We also calculate the univariate ﬁtness functions
WðZÞ ¼ b
1Z1and WðZÞ ¼ b0
from instrumental variable analyses (Kendall 2015, Coulson
et al. 2017). From these functions, we calculated the univariate
selection differentials s
2. We calculate univariate
heritabilities using the relevant additive genetic variances and
phenotypic variances for each trait. We then compare
predictions of evolutionary change between the multivariate
breeders equation and the 2 univariate breeders equations.
We set m¼[70 cm, 0]. The value of 70 cm is approximately
the mean horn length of 4-year-old rams that are at risk of
being shot by hunters. The value of zero is the mean of the
environmental component of the phenotype as is usually
assumed in quantitative genetics (Falconer 1975).
To explore the effects of hunting on the evolution of a
trophy, we ran simulations with a range of initial genetic,
environmental, and phenotypic variances. We conducted
simulations to demonstrate the effects of altering the additive
genetic variance and the total phenotypic variance. For
example, the simulations reported in Figures 1 and 2 both
have identical initial additive genetic variances of V(A,1) ¼3,
but they have different environmental variances of
Vðℰ;1Þ ¼ 2 and Vðℰ;1Þ ¼ 0:1, respectively. The simu-
lations reported in Figure 3 demonstrate the effect of
increasing the phenotype variance by increasing the additive
genetic variance compared to those simulations reported in
Figures 1 and 2: V(A,1) ¼5 and V(E,1) ¼2.
We also examined the consequences of injecting additional
genetic variance into the population at each time step by setting
the segregation variance to the constant initial value chosen at
the beginning of the simulation. For all parameter sets, we
explored the effect of removing 25%, 50%, 75%, and 100% of
males of above average horn size (e.g., a¼[0.25, 0.5, 0.75, 1]).
To demonstrate how correlated characters affect phenotypic
evolution over a single generation, we ran a number of
simulations of the multivariate breeders equation. In each
simulation, we set WðZ;tÞ ¼ 0:3þ0:1Z1þ0:1Z2and m
(Z)¼(6,6). These values are arbitrary in that any values could
be used to reveal the effects we demonstrate. We then ran 12
simulations. In each simulation V(A1,t¼0) ¼2 and V(A2,
t¼0) ¼2. We then examine 3 genetic covariance structures
within 4 different distributions of the environmental
components of the phenotype. The ﬁrst assumes no genetic
covariance, the second a negative genetic covariance of 1.41,
and the third a positive genetic covariance of 1.41. We chose
the second and third values because they are the 2 limits that
the covariance can take to ensure the variance-covariance
matrix is positive-deﬁnite. The 4 distributions of the
environmental components of the phenotype are selected
such that phenotypic variances and covariances are dominated
by the additive genetic variances and covariances, and for cases
where approximately half of the phenotypic variances and
covariances are attributable to the additive genetic variances
and covariances. We then explored the effects of positive and
negative covariances between the environmental components
of the phenotypes on evolutionary dynamics. Code to run
models and draw ﬁgures is available from https://github.com/
4 The Journal of Wildlife Management 9999()
Selective trophy hunting led to an evolutionary response in
all of our simulations (Figs. 1–3). In our initial simulation
with a starting heritability of 0.6, the phenotypic mean
declined from a initial value of 70 to between 57 and 62.5
depending upon the proportion of the population culled.
There was relatively little difference in the mean phenotype
Figure 1. The effect of different trophy hunting regimes on the dynamics of the phenotype and the heritability. The dynamics of the mean (A), the variance (B),
and the heritability (C) all depend upon the proportion of males of above average trophy (e.g., horn) size that are culled (numbers next to lines). The dotted gray
horizontal line represents 1.96 standard deviations from the initial mean trophy size. The dashed gray horizontal line is the mean phenotype Coltman et al.
(2003) reported 5 generations later. The near vertical dashed gray line represents the rate of the change in the phenotypic mean they report. These lines are for
illustration only, as our model is not parameterized with data from Coltman et al. (2003). In these simulations, the initial additive genetic variance was set at 3.0,
and the environmental variance at 2.0. We also report the dynamics of the mean phenotype when 25% of above-average trophy sizes are harvested as a function
of increasing additive genetic variance and the heritability (D). In each of the 4 simulations reported in (D), we set the initial phenotypic variance at 5 by using
values for the initial additive genetic variances as (4.99, 3.75, 2.5, 1.25) and for the environmental variances as (0.01, 1.25, 2.5, 1.75). These give initial
heritabilities of 0.99, 0.75, 0.5, and 0.25 (values next to the lines).
Figure 2. The effect of different trophy hunting regimes on the dynamics of the phenotype. The dynamics of the mean (A), and the variance (B) for cases when
the phenotype is determined almost entirely by the additive genetic variance. In each simulation, the initial additive genetic variance was set to 3.0 and the
environmental variance to 0.1. The dashed gray horizontal line is the mean phenotype Coltman et al. (2003) reported 5 generations later. The near vertical
dashed gray line represents the rate of the change in the phenotypic mean they report. These lines are for illustration only, as our model is not parameterised with
data from Coltman et al. (2003). The dotted gray horizontal line represents 1.96 standard deviations from the initial mean trophy size.
Coulson et al. Evolutionary Consequences of Trophy Hunting 5
after 100 generations when 50%, 75%, or 100% of males of
above average trophy value were harvested; all simulations
achieved a decline from 70 to 57 over 100 generations. In
contrast, evolution was notably slower when only 25% of
above average trophy sizes were culled per generation
(Fig. 1A). The phenotypic variation and heritability showed
similar rates of change. This is expected because variation in
the environmental component of the phenotype at birth is
constant across generations. The rate of loss of phenotypic
variation and decline in the heritability scaled with harvest-
ing rate (Fig. 1B and C). When all males above the mean
trophy value were harvested, additive genetic variance was
initially rapidly eroded, before starting to decline more
slowly. This change was reﬂected in the dynamics of the
phenotypic variance (Fig. 1B). These rates of change in the
variance affected the dynamics of the mean phenotype.
Although the initial rate of evolution correlated with
harvesting pressure, over the course of 100 generations
evolution was fastest when 75% of above average males were
harvested. None of our scenarios predicted phenotypic
change at the rate reported by Coltman et al. (2003). In our
initial simulations, it took between 40 and 100 generations
before the mean phenotype evolved to a value that would be
signiﬁcantly different from its initial value (regardless of
sample size). Finally, altering the initial heritability by
reducing the initial additive genetic variance slowed the rate
of evolutionary changed as expected. In contrast, as the
additive genetic variance and consequently heritability
increased, so too did the rate of evolution (Fig. 1D).
In our second simulation, we increased the initial heritability
by reducing the environmental variation. This had a relatively
small impact on the rates of evolution (Fig. 2A), although the
reduction in the phenotypic variance (Fig. 2B) did reduce rates
of evolution at the highest levels of off-take (Fig. 2A).
Increasing the additive genetic variance, and consequently the
phenotypic variance, also increased rates of evolutionary
change slightly (Fig. 3A and B), although rates of phenotypic
change were still between 1 and 2 orders of magnitude slower
than reported by Coltman et al. (2003) and Pigeon et al.
(2016). The time series of selection differentials estimated
across males and females for these simulations are given in
In all simulations, setting the segregation variance to a
constant value generated linear selection because selection
does not rapidly erode the additive genetic variance (Fig. S2).
However, even when all males of above average horn size are
culled, the rate of phenotypic change is still >5 times slower
than that reported by Coltman et al. (2003).
We next compared evolutionary dynamics predicted by the
univariate and bivariate breeders equation to examine
whether correlated characters could lead to rapid evolution
in the opposite direction to selection, or to evolutionary
stasis. The degree of correlation between 2 characters
increased the rate of evolution when the sign of the
phenotypic covariance (/þ) was the same as the sign of
the product of the selection differentials on each trait
(Fig. 4A–D). As the proportion of phenotypic variation
attributable to additive genetic variation tended to unity,
predictions from the univariate and bivariate breeders
equation converged (Fig. 4A). Similarly, although not
reported, at the other limit, as the proportion of phenotypic
variance attributable to additive genetic variance tended to
zero, no evolution was predicted by either the univariate or
bivariate breeders equation and predictions converged.
Departures between the 2 equations were greatest when
intermediate proportions of the phenotypic variances and
covariances were attributable to the additive genetic
variances and covariances (Fig. 4B–D). Both additive genetic
covariances, and covariances in the environmental compo-
nent of the phenotype, could lead to divergence between the
univariate and bivariate breeders equation (Fig. 4B–D).
Although covariances between S(E) and S(A) could affect
rates of evolution, when selection differentials were large,
covariances could not generate stasis or lead to evolution in
the opposite direction to that predicted by selection (Fig. 4,
blue lines). However, as selection got weaker, correlated
characters could prevent selection, and even lead to very small
evolutionary change in the opposite direction to that
Figure 3. The effect of different trophy hunting regimes on the dynamics of the phenotype. The dynamics of the mean (A) and the variance (B) to demonstrate
the effect of a high heritability and large phenotypic variance. In each simulation, the initial additive genetic variance was set to 5.0 and the environmental
variance to 2.0. The dashed gray horizontal line is the mean phenotype Coltman et al. (2003) reported 5 generations later. The near vertical dashed gray line
represents the rate of the change in the phenotypic mean they report. These lines are for illustration only, as our model is not parameterized with data from
Coltman et al. (2003). The dotted gray horizontal line represents 1.96 standard deviations from the initial mean trophy size.
6 The Journal of Wildlife Management 9999()
predicted by evolution (Fig. 4, red lines). However, effect
sizes were small and would be challenging to detect without
large quantities of data.
Our simulations show that selective harvesting can alter the
evolutionary fate of populations, and can result in declines in
trophy size. However, even under intensive trophy hunting,
it is expected to take tens of generations before the mean
trophy size has evolved to be signiﬁcantly smaller than it was
prior to the onset of selective harvesting (see also Thelen
1991, Mysterud and Bischof 2010). Our results also show
that although correlated characters can have impacts on
phenotypic evolution, they cannot be invoked to explain
rapid phenotypic change in the opposite direction to that
predicted from univariate selection differentials.
Our models are kept deliberately simple and make a number
of assumptions. First, we iterate the population forwards on a
per-generation step. This means there is no age structure, and
that a single breeding value determines trophy size throughout
life. For some traits, there is evidence of age-speciﬁc breeding
values (Wilson et al. 2005), and these could inﬂuence
evolutionary rates (Lande 1982). Males are typically shot
once they have reached adulthood, which means direct
selection via hunting does not occur in younger ages. The
indirect effect of trophy hunting at older ages on phenotypes
and ﬁtness at younger ages is determined by genetic
correlations across ages. As we show in our analysis of the
multivariate breeders equation, evolution is most rapid when
the genetic correlations are close to the limit and align with the
direction of selection. Given trophy sizes typically experience
positive selection at all ages (Coltman et al. 2002, Preston et al.
2003), this means that the rate of evolution will be greatest
when genetic correlations are close to unity. At the limit, this
would mean that the same breeding value would determine
trophy size throughout life—an assumption of our model. Our
model consequently likely predicts faster rates of evolution
than would be predicted from a model with age-structured
breeding values and the same selection regime that we assume.
A second assumption we make is that the trait is not subject
to selection before selective harvesting is imposed. Trophy
size positively correlates with ﬁtness in species that are not
harvested (Preston et al. 2003). Trophies may consequently
be expected to be slowly evolving to be larger in the absence
of selective hunting. If that were the case, then the effect of
trophy hunting would have to be greater than in our models
to lead to evolution of smaller trophies at the rates we report.
This is because selective harvesting would have to counteract
evolution for larger trophies in the absence of harvesting,
before then leading to a reduction in trophy size. Our model
would over-estimate the evolutionary impact of trophy
hunting in such a case.
Figure 4. A comparison of the dynamics of the multivariate and univariate breeders equation for different degrees of additive genetic and environmental
variances and covariances. Each ﬁgure reports 3 simulations: no genetic covariance (black lines), strong positive genetic covariances that reinforce selection (blue
lines), and strong negative genetic covariances that oppose selection (red lines). Solid lines represent selection differentials on each trait and dotted lines represent
responses to selection. Horizontal and vertical dot-dashed lines show predictions of evolution from the univariate breeders equation for each trait. The farther the
right hand end of the dashed lines are from the intersection of the horizontal and vertical dot-dashed lines, the greater the disparity between predictions from the
univariate and multivariate breeders equation. We simulated that all phenotypic variation is attributable to genetic (co)variances (A), approximately half of
phenotypic variance is attributable to additive genetic variance (B), and the effect of a positive (C), and negative (D) covariance in the environmental components
of the phenotypes on rates of evolution. The genetic and environmental (co)variance used in each simulation can be found in Table S1.
Coulson et al. Evolutionary Consequences of Trophy Hunting 7
Males in sexually dimorphic species with trophies form
dominance hierarchies (Pelletier and Festa-Bianchet 2006).
If a dominant male with large trophies is shot, it may be
reasonable to assume that surviving males with large trophies
that are toward the top of the dominance hierarchy would
secure the reproductive success the shot male would have
enjoyed. We do not model this process. Instead, we
redistribute the reproductive success across all remaining
males. This egalitarian redistribution of reproductive success
likely exaggerates the evolutionary consequences of trophy
hunting because individuals with small trophies are
beneﬁting from those with large trophies being shot. Our
model, although simple, has consequently been formulated
to likely exaggerate the consequences of trophy hunting on
When predictions from simple models like ours fail to match
with observation, the existence of genetically correlated
unmeasured characters is often invoked as an explanation
(Meril€aet al. 2001). Changing the degree of generic
covariation between 2 characters can signiﬁcantly alter
selection differentials on both characters (e.g., Fig. 4).
However, this does not mean that the failure to measure a
correlated character will lead to incorrect estimates of a
selection differential on a trait. In fact, the failure to measure
a correlated character will have no impact on the estimate of
a selection differential of a focal character (Lynch and Walsh
1998, Kingsolver et al. 2001). Estimates of selection differ-
entials on a univariate character will consequently always give
an upper limit on the rate of evolution of a character that
conforms to the assumptions of the phenotypic gambit.
Genetic and environmental covariation with unmeasured
characters can affect the response to selection. The effect is
most likely to be strongest when characters have heritabilities
in the vicinity of 0.5 and covariances are close to their limits.
The further from this proportion that variance and covariances
get, the less biased predictions of evolution in the presence of
unmeasured correlated characters becomes. Large covariances
that act to reduce the strength of selection can lead to low rates
of evolutionary change in the opposite direction to selection,
but the effect is small and could only be detectable in very large
data sets. We consequently conclude that if the phenotypic
gambit is assumed and signiﬁcant selection on a trait is
observed, then unmeasured correlated characters can act to
slow, or increase, rates of evolution compared to those
predicted by the univariate breeders equation, but they cannot
result in evolutionary change that is greater than the univariate
selection differentials, or lead to evolutionary stasis. We
conclude that although our models on the effect of hunting
on a trophy are simple, they will not be too wide of the mark.
Although our models are simple, they provide some novel
insights. In particular, our strongest selection regimes result
in initial increased rates of evolution. However, they erode
the additive genetic covariance more quickly than less
stringent hunting regimes, rapidly slowing the rate of
evolution. Over longer periods, evolutionary rates are highest
at intermediate rates of hunting compared to higher hunting
rates. These results show how important it is to track the
dynamics of the additive genetic variance when predicting
evolution in the face of strong selection over multiple
generations (see also Lande 1982, Barﬁeld et al. 2011, Childs
et al. 2016, Coulson et al. 2017). Assuming a constant
additive genetic variance in the face of strong selection
would lead to predictions of elevated rates of evolution over
In most of our simulations we assume that the directional
selection we impose erodes the additive genetic variance as is
often assumed in quantitative genetics (Falconer 1975). We
do this by constraining the segregation variation to be equal
to half the additive genetic variance among parents (Barﬁeld
et al. 2011, Childs et al. 2016). However, we also relax this
assumption by maintaining a constant segregation variance
that is not eroded in the face of selection. This mimics
processes, including mutation, that generate additive genetic
variance. By doing this we linearize the longer-term response
to selection, such that evolution continues to alter the trait
value at a greater evolutionary rate over a longer period of
time than is possible when selection erodes the additive
genetic variance. However, even under these circumstances,
statistically signiﬁcant evolution is predicted to take between
10 and 20 generations even under strong selection when all
males of above average horn size are culled.
What do our results contribute to bighorn management?
To appreciate this, it is helpful to understand how Coltman
et al. (2003) reached their conclusions. They report a
heritability of horn length of approximately 0.7. This means
that 70% of differences in horn length between individuals is
due to differences in their breeding values. Because variation
in birth and death rates between individuals with different
phenotypes (selection and drift - the underpinning of
evolution) are the processes that alter the mean breeding
value within the population, the decline in horn length
observed is consequently attributed predominantly to
evolution by Coltman et al. (2003). It is this logic that led
them to their conclusion that trophy hunting generated
rapid, undesirable, evolutionary change. Coltman et al.
(2003) also estimated trends in mean breeding values in an
attempt to quantify evolution, but the methods they used
have subsequently been shown to be unreliable (Hadﬁeld
et al. 2009), which means their breeding value trends should
be treated with skepticism. Pigeon et al (2016) may have
estimated appropriate trends in breeding values, but these
trends suggest only a small evolutionary decline in mean
breeding value of just over 1 cm/generation over only 2
generations. Such a decline seems consistent with the small
evolutionary effects we predict, and at odds with the
conclusions of Coltman et al. (2003).The primary contribu-
tion of our results is to suggest that very fast phenotypic
change of quantitative characters that is sometimes observed
in these populations cannot be due to rapid evolution, at least
not under the assumptions of quantitative genetics, for two
reasons. First, the upper rates of phenotypic change reported
(Coltman et al. 2003) are approximately two orders of
magnitude faster than evolutionary models of intensive
selective harvesting can achieve. Second, the traits that are
hypothesized to evolve, horn length and body size, are subject
to positive selection at some ages, even in the presence of
8 The Journal of Wildlife Management 9999()
harvesting (Traill et al. 2014), yet body and horn size have
become smaller (Coltman et al. 2003). Unmeasured
correlated characters cannot explain this. So what causes
the rapid phenotypic change that is sometimes observed?
There are a number of possibilities.
First, the environment may have deteriorated rapidly,
leading to a change in the mean of the environmental
component of the phenotype (Meril€a et al. 2001, Kruuk et al.
2002), perhaps in a similar manner as reported in a desert
bighorn sheep population (Ovis canadensis nelsoni; Hedrick
2011). Second, the phenotypic gambit on which statistical
quantitative genetic analyses are based may be violated
(Hadﬁeld et al. 2007). This could occur if genotype-
by-environment interactions, dominance variation or epis-
tasis have contributed to the observed phenotypic trends
(Falconer 1975). Quantitative genetics theory and empirical
methods exist to deal with each of these processes (Lynch
and Walsh 1998), but statistical methods to estimate these
processes either require large population sizes or additional
data that may not be available for this population. Third, the
association between body size and horn length and ﬁtness
may not be causal (Meril€a et al. 2001), but both may reﬂect an
individual’s ability to extract resources from the environment.
Individuals that are good at doing this grow to large sizes,
produce large trophies, and have high ﬁtness. If the ability to
extract resources from the environment is not determined by
a simple additive genotype–phenotype map, then neither will
be the association between body size and horn length and
Although our models reveal that very rapid evolution
attributable to selective hunting is not a plausible explanation
for the observed phenotypic declines, our models are not
parameterized for bighorn sheep. Ideally the theoretical
quantitative genetic approach we use here and in Coulson et al.
(2017) should be parameterized for bighorn sheep before any
management recommendations are made. The only data set
we are aware of that may be sufﬁcient to parameterize models
within our framework are from the bighorn sheep population
at Ram Mountain (Coltman et al. 2003, Pigeon et al. 2016).
These data have not been made publicly available, and the data
in Pigeon et al. (2016) are embargoed until 2026. In the ‘Data
archiving statement’ in Pigeon et al. (2016) they state the data
“are also available upon request to anyone who wishes to
collaborate with us or repeat our analysis” which preclude new
independent analyses of these valuable data. In addition, 1
coauthor on Coltman et al. (2003) and 2 coauthors on Pigeon
et al. (2016) are signatories on Mills et al. (2015), which argues
against making long-term individual-based data open access.
Given it seems unlikely that these valuable data will be made
publicly available any time soon, we suggest that Coltman and
colleagues to use their data to construct and analyze the class of
model we use here. Until this is done, we recommend that the
conclusions of Coltman et al. (2003) and Pigeon et al. (2016)
are not used to inform wildlife management policies given
their conclusions are not theoretically plausible.
Quantitative genetics theory is powerful, elegant, and based
on irrefutable logic (Falconer 1975, Lande and Arnold
1983). The statistical methods used to estimate evolutionary
change are also extremely powerful when assumptions that
underpin the analyses are met (Lynch and Walsh 1998). We
recommend that when evolution is inferred from these
statistical analyses, quantitative genetic theory based on the
assumptions that underpin the analyses is used to check that
reported patterns are plausible. For example, could a
correlated character that results in the same selection
differential that is observed on the trait generate the
observed patterns? This is particularly important when
statistically identiﬁed rates of evolution are very rapid, or
occur in the opposite direction to that predicted. If patterns
from these statistical analyses are not theoretically possible,
some key assumption underpinning the statistical analysis
has been violated, and conclusions from the statistical
analyses are unreliable. Simple quantitative genetic models
rarely provide predictions that match with observation in the
wild (Meril€a et al. 2001). When this happens, and
predictions and observation cannot be reconciled, the use
of phenotype-only models (Ellner et al. 2016), or models
with more complex genotype–phenotype maps (Yang 2004),
can provide useful insight into causes of phenotypic change,
particularly when these models capture observed dynamics
accurately, as they frequently do (Coulson et al. 2010, Merow
et al. 2014).
Our work suggests that highly selective trophy hunting will
result in evolutionary change, but that it will not be
particularly rapid. Evolutionary change would be more rapid
if both sexes were selectively targeted as is unfortunately the
case for African elephant (Loxodonta africana) populations in
some countries (Selier et al. 2014). When harvesting is less
selective, or coupled with habitat change, the evolutionary
consequences of selective harvesting may be harder to detect
(Garel et al. 2007, Crosmary et al. 2013, Monteith et al.
2013, Rivrud et al. 2013). Our work does not tackle the ethics
or ecological consequences of trophy hunting, nor do we
account for potential economic beneﬁts of hunting for local
communities, whether these be in Canada (Hurley et al.
2015) or in the developing world (Lindsey et al. 2007). These
issues should be given considerably more weight when
designing population management and conservation strate-
gies compared to the likelihood of rapid evolution.
We thank 2 anonymous reviewers for useful comments on an
earlier version of this manuscript. TC also acknowledges
support from the Natural Environment Research Council
via a standard grant.
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Coulson et al. Evolutionary Consequences of Trophy Hunting 11