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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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Biol. Lett. (2008)

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