Should hunting mortality mimic the patterns of natural mortality?

Abstract

With growing concerns about the impact of selective harvesting on natural populations, researchers encourage managers to implement harvest regimes that avoid or minimize the potential for demographic and evolutionary side effects. A seemingly intuitive recommendation is to implement harvest regimes that mimic natural mortality patterns. Using stochastic simulations based on a model of risk as a logistic function of a normally distributed biological trait variable, we evaluate the validity of this recommendation when the objective is to minimize the altering effect of harvest on the immediate post-mortality distribution of the trait. We show that, in the absence of compensatory mortality, harvest mimicking natural mortality leads to amplification of the biasing effect expected after natural mortality, whereas an unbiased harvest does not alter the post-mortality trait distribution that would be expected in the absence of harvest. Although our approach focuses only on a subset of many possible objectives for harvest management, it illustrates that a single strategy, such as hunting mimicking natural mortality, may be insufficient to address the complexities of different management objectives with potentially conflicting solutions.

Full text

Biol. Lett. (2008) 4, 307–310
doi:10.1098/rsbl.2008.0027
Published online 21 February 2008
Population ecology
Should hunting mortality
mimic the patterns of
natural mortality?
Richard Bischof
1,
*, Atle Mysterud
2
and Jon E. Swenson
1,3
1
Norwegian University of Life Sciences, PO Box 5003,
1432 A
˚s, Norway
2
Centre for Ecological and Evolutionary Synthesis (CEES ),
Department of Biology, University of Oslo, PO Box 1066,
Blindern, 0316 Oslo, Norway
3
Norwegian Institute for Nature Research, 7485 Trondheim, Norway
*Author for correspondence (richard.bischof@umb.no).
With growing concerns about the impact of selec-
tive harvesting on natural populations, research-
ers encourage managers to implement harvest
regimes that avoid or minimize the potential
for demographic and evolutionary side effects.
A seemingly intuitive recommendation is to
implement harvest regimes that mimic natural
mortality patterns. Using stochastic simulations
based on a model of risk as a logistic function of a
normally distributed biological trait variable, we
evaluate the validity of this recommendation
when the objective is to minimize the altering
effect of harvest on the immediate post-mortality
distribution of the trait. We show that, in the
absence of compensatory mortality, harvest
mimicking natural mortality leads to amplifi-
cation of the biasing effect expected after natural
mortality, whereas an unbiased harvest does
not alter the post-mortality trait distribution
that would be expected in the absence of harvest.
Although our approach focuses only on a subset of
many possible objectives for harvest manage-
ment, it illustrates that a single strategy, such
as hunting mimicking natural mortality, may
be insufficient to address the complexities of
different management objectives with potentially
conflicting solutions.
Keywords: demography; life history; simulation;
management; selection; vulnerability
1. INTRODUCTION
There is growing concern regarding potential demo-
graphic side effects and evolutionary consequences of
selective harvesting on wildlife populations (Harris et al.
2002;Coltman et al. 2003). Perturbations of a popu-
lation’s demographic structure (Mysterud et al. 2002;
Milner et al. 2006) and short- and long-term changes of
morphological traits or life-history strategies due to
artificial selective pressures (Festa-Bianchet & Apollonio
2003) are some of the processes through which selective
hunting may affect populations beyond more obvious,
direct effects on population size and growth rate through
the removal of individuals. In search of management
strategiesthatminimizethedemographicsideeffects
and are ‘evolutionarily enlightened’ (Gordon et al.
2004), it has been suggested that harvesting regimes
should mimic natural mortality patterns (e.g. Milner
et al. 2006;Loehr et al. 2007;Bergeron et al. 2008).
Surprisingly, the general applicability of this recommen-
dation has received little theoretical or empirical
evaluation. Recently, Proaktor et al. (2007) presented
model-based evidence that selection for lighter weight
at first reproduction in ungulates could be a conse-
quence of harvest and that harvest pressure is more
important in driving this adaptive response than the
degree of harvest selectivity. It seems plausible that this
would apply to other situations in which the benefits of
more and earlier reproduction eventually outweigh its
costs (e.g. lower quality offspring), possibly due to higher
overall mortality and consequently a greater chance of
not reproducing later. To our knowledge, this is the only
strong argument thus far in support of the statement
that harvest selectivity patterns should mimic natural
mortality, because a harvest biased towards younger
(and lighter) individuals could minimize the aforemen-
tioned adaptive response. Even in such a case, simply
targeting small (i.e. young) individuals may lead
to further decreases in the size at, and time to,
maturation as recent literature on fisheries-induced
evolution suggests (Kuparinen & Merila¨ 2007 and
references therein).
In this article, we are specifically concerned about the
lack of scrutiny of the statement with regard to the
immediate disruption caused due to demographic or
other changes as a result of biased harvest. To avoid
ambiguity, we identify a clear objective for harvest
management with respect to selectivity, namely that
harvesting and natural mortality acting on a population
should result in a post-mortality population structure (or
biological trait distribution) that is identical or at least
very similar to the structure that would be expected in
the absence of harvest (see also Harris et al. 2002). With
this objective in mind, we ask the question: should
hunting mortality mimic natural mortality in order to
limit the potential for disruptions caused by demo-
graphic or trait-distribution changes?
The effects of selection on trait distributions are now
relatively well understood (e.g. Lynch & Walsh 1998).
Particularly relevant to our work is a paper by Vaupel
et al. (1979), which explores the effects of viability
selection on the distribution of a trait over time and age
cohorts. The authors termed this trait ‘frailty’ to high-
light the fact that it is related to an individual’s risk of
mortality, and assumed that the probability density
function of frailty follows a gamma distribution. Vau p e l
et al. (1979) further assumed that the force of mortality
(a measure of an individual’s risk) is a function of time,
age and frailty. Although our basic approach is similar,
we develop a slightly different model and explore the
outcome through the simulations. We also extend Vau p e l
et al. (1979) by discriminating between two mortality
causes and by investigating how altering the shape of the
viability selection function affects the post-mortality trait
distribution.
2. MATERIAL AND METHODS
(a)Model
For model construction, we assume a normally distributed random
variable x(with mean mand variance s) that represents a certain
trait of individuals in the population, with associated probability
density function
fðxÞZ
1
sffiffiffiffiffiffi
2p
pexp KðxKmÞ2
2s2

:ð2:1Þ
Received 18 January 2008
Accepted 4 February 2008
307 This journal is q2008 The Royal Society

We then assume that risk pis a logistic function of trait x
(figure 1), where the relationship between pand xcan be expressed as
pxZ
1
1CeKðaCbxÞ:ð2:2Þ
Here, aand bare the intercept and slope of the linear regression
(with the logit link), respectively. Although the assumptions behind
equations (2.1) and (2.2) oversimplify a world where risk probably is
a complex function of multiple variables (morphology, age, experi-
ence, behaviour, space use, etc.), the approximation is sufficient for
our purposes. The above approach centres on a logistic relationship
between risk and a normally distributed continuous feature of the
population, but this representation also allows the incorporation of
discrete or factor variables, as well as other distributions. Following
the precautionary principle and because strong compensation can be
expected to occur only rarely (Lebreton 2005), we assume that
mortality is additive. Furthermore, we ignore potential density-
dependent effects that in real populations may, for example, positively
affect the growth rate of individuals exhibiting trait values that are
selectively targeted.
An interesting finding of Vaupel et al. (1979) was that, as
individuals age, their force of mortality increases more rapidly
than the average force of mortality of the age cohort they belong
to, because the removal of frail individuals decreases the average
frailty of the surviving cohort. The mechanism underlying this
phenomenon applies also to our model, although for simplicity we
did not include an age term. While surviving individuals retain
their original trait value as they move from pre- to post-mortality,
the average trait value shifts towards the less vulnerable end of the
trait spectrum (assuming no recruitment within that time step).
(b)Simulations
We investigated changes in the probability density distributions of
trait x(e.g. size) in a heterogeneous hypothetical population with
two groups with different mean trait values (e.g. females and males)
resulting from exposure to harvest followed by natural mortality.
We evaluated the effect of different shapes of the logistic function
linking harvest risk and trait value on both the post-mortality trait
distribution and the ratio of the two groups in the population. We
used three main expressions of the logistic function based on its
shape (‘mimic’, ‘inverse’ and ‘unbiased’; figure 2) relative to the
risk associated with natural mortality, by altering the intercept and
slope in the logistic function (equation (2.2)).
We conducted stochastic simulations using R v. 2.5.0
(R Development Core Team 2007). We repeated simulations with
the same settings 100 times and calculated bias and 95% CI limits
from 1000 bootstrapped replicas of the mean parameter values. We
note that, although we chose to illustrate the effect of viability
selection using simulations, the effects of multiplying a distribution
with a function can also be evaluated analytically, e.g. through the
use of conjugate priors within a Bayesian framework ( Fink 1997).
3. RESULTS
For the case of harvest preceding natural mortality,
simulation results (figure 2,table 1) indicate that (i)
inverse harvest risk prior to natural mortality
diminishes and in extreme cases reverses the biasing
effect of natural mortality on the density distribution
of the biological trait, (ii) unbiased harvest risk keeps
the biasing effect of natural mortality unchanged,
and (iii) mimicking harvest risk amplifies the biasing
effect of natural mortality on the density distribution
of the biological trait. Biased natural mortality alters
the ratio of the two groups in the population, with
additional changes in the ratio due to mimic and
inverse harvest mortalities, but no further alterations
if harvest is unbiased. The altering effect of biased
harvest on the trait distribution and the ratio of the
two groups in the population increases with increas-
ing harvest rate (table 1). Because we assume no
density-dependent effects and, if harvest mortality
is limited by a quota, the above patterns, at least
qualitatively, also hold true for har vest following
natural mortality.
4. DISCUSSION
The general statement that harvest mortality should
mimic natural mortality in order to avoid demographic
disturbance or evolutionary consequences is not yet
sufficiently supported, and needs to be qualified. We
found that, for the specific objective of maintaining
0
0.010
0.020
(a) (i) (i) (i)
(ii) (ii) (ii)
(iii) (iii) (iii)
(b)(c)
density
0 50 100 150 200 250 300
0
0.4
0.8
trait value
risk
0 0.2 0.4 0.6 0.8 1.0
0
2
4
6
risk
density
0 50 100 150 200 250 300
trait value
0 0.2 0.4 0.6 0.8 1.0
0
1.0
2.0
risk
0 50 100 150 200 250 300
trait value
0 0.2 0.4 0.6 0.8 1.0
0
1
2
3
4
5
risk
Figure 1. Illustration of the link between the density distribution of risk and a normally distributed biological trait x(s.d.Z20;
hashed lines: 2!s.d. boundaries, arbitrary unit), with risk being a logistic function of x. Shifts in the mean trait value ((a(i)–(iii))
100, (b(i)–(iii)) 150 and (c(i)–(iii)) 200) of a hypothetical population (nZ5000) change the density distribution of risk in the
population.
308 R. Bischof et al. Hunting mortality and natural mortality
Biol. Lett. (2008)

unchanged post-mortality distributions of a trait (or
demographic feature), hunting mortality should be
unbiased. This holds true regardless of whether hunting
occurs prior to or after natural mortality. Therefore, in
the absence of strong compensation in mortalities and
until further supporting evidence emerges, we would
limit recommending that hunting mortality should
mimic natural mortality patterns to the following cases.
(i) Natural mortality regimes have been altered, e.g.
as a result of extermination of natural predators.
(ii) The objective is to minimize selective pressure for
earlier reproduction driven by increased overall
mortality as a result of adding harvest.
(iii) An amplification of the biased outcome of natural
mortality is desired.
(iv) The main objective is to minimize the negative
direct impact of harvest on population growth by
targeting those demographic groups whose survival
has the lowest elasticity/sensitivity.
(v) Natural mortality is unbiased.
In our example, increasing overall mortality (whether
the latter is biased or not) by a constant (e.g. adding
unbiased harvest) does not alter the selective pressure
on traits directly linked to risk. We emphasize that
different objectives, such as (i) minimizing the effects of
harvest on the distribution of traits or demographic
features or (ii) limiting the selective pressure for lower
age and size at first reproduction, may have conflicting
solutions, as well as different temporal scopes (see also
Law 2001).
0
0.2
0.4
0.6
0.8
1.0(a) (i) (i) (i)
(ii) (ii) (ii)
(b)(c)
risk
50 100 150 200
0
0.005
0.010
0.015
0.020
trait value
density
50 100 150 200
trait value 50 100 150 200
trait value
Figure 2. Changes in trait distributions as a result of various patterns of hunting mortality relative to biased natural mortality for
a simulated heterogeneous population (two cohorts with nZ1000 each, s.d.Z30, 100 and 150). (a–c(i)) show natural and
hunting risks as a logistic function of the normally distributed biological trait x(arbitrary unit; lines: red, hunting; black, natural).
(a–c(ii)) show distributions of the biological trait before and after mortality (lines: grey dashed, before mortality (groups
separate); grey solid, before mortality (joint); black, after natural mortality without hunting; red dashed, after hunting and
natural mortality). Risk associated with hunting mortality either (a) mimics natural mortality, is (b) unbiased (slope and intercept
of the logistic function set to 0), or is (c) inverse to natural mortality. Harvest rate was set at 30% of the initial population size.
Table 1. Bootstrapped estimates of the mean trait value (m, arbitrary unit) and ratio (r) of groups (group 1 : group 2) of
surviving individuals in a hypothetical population after hunting followed by natural mortality and after natural mortality in
the absence of hunting mortality (m
0
,r
0
) from 100 simulation runs for each of the three shapes of the risk function (see text
and figure 2) and two different harvest rates. (The initial population consisted of two groups (nZ1000 each) with mean trait
value mZ100 and 150, respectively, and s.d.Z30. Natural mortality was modelled as a logistic function of x(see text), with
intercept aZK5 and slope bZ0.04.)
harvest
risk shape
harvest
rate mCIL
a
(m)m
0
CIL(m
0
)rCIL(r)r
0
CIL(r
0
)
mimic 0.25 103.95 103.82, 104.09 109.12 108.99, 109.25 2.10 2.08, 2.12 1.74 1.72, 1.75
0.5 96.65 96.52, 96.78 109.05 108.90, 109.20 2.82 2.78, 2.86 1.77 1.75, 1.78
unbiased 0.25 108.92 108.72, 109.12 109.10 108.96, 109.24 1.77 1.75, 1.79 1.75 1.73, 1.77
0.5 109.14 108.88, 109.38 109.14 109.01, 109.27 1.76 1.73, 1.78 1.74 1.73, 1.76
inverse 0.25 114.47 114.25, 114.70 109.01 108.87, 109.17 1.45 1.43, 1.47 1.77 1.75, 1.78
0.5 123.97 123.67, 124.25 109.01 108.86, 109.17 1.05 1.04, 1.07 1.75 1.73, 1.77
a
Ninety-five per cent CI limits from 1000 bootstrapped estimates.
Hunting mortality and natural mortality R. Bischof et al. 309
Biol. Lett. (2008)

We focused on the potential of selective harvest to
alter the post-mortality distribution of a single trait
from the distribution that would be expected if natural
mortalityoccurredintheabsenceofharvesting.Awider
scope is required to evaluate all important ecological
and evolutionary consequences of harvesting and to
answer the questions about optimal harvesting strategies
comprehensively. Such models may include age effects
on trait values and risk, density-dependent effects, and
environmental and demographic stochasticity. Further-
more, empirical exploration into how various harvesting
strategies in concert with biased natural mortality affect
trait distributions of natural populations are required to
validate what theory suggests.
We thank S. J. Hegland, A. Ordiz, O. G. Støen, A. Zedrosser
and two anonymous reviewers for their comments. This
manuscript benefited substantially from T. Coulson’s advice,
for which we are grateful. Funding for this project came
from the Norwegian University of Life Sciences (R.B.) and
the Research Council of Norway (A.M.).
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