Analysis of harvest and effort data for wild populations influctuating environments

A.L. Jensen *

School of Natural Resources, University of Michigan, Ann Arbor, MI 48109-1115, USA

Received 1 November 2001; received in revised form 17 May 2002; accepted 28 May 2002

Abstract

An approach is developed for analysis of harvest and effort data that uses a surplus production model modified for

analyses in a fluctuating environment, and that maximizes harvest with as small an effect on the intrinsic rate of increase

as possible. Harvest and hunter effort data that are suitable for analysis with a surplus production model have been

collected for several wildlife species for many years. I used data for the mallard duck, lesser scaup duck, white-tailed

deer, and black bear. The parameters were estimated with harvest and hunter data using non-linear least squares. For

each species both the relation between the maximum sustainable harvest (MSH) and effort and a more conservative

optimal level of harvest and effort were obtained under a range of environmental conditions. The mallard duck and

white-tailed deer were both over harvested for short periods in the past, but only the black bear population appears

over harvested today. # 2002 Published by Elsevier Science B.V.

Keywords: Deer; Ducks; Black bear; Harvest; MSY; Optimum harvest; Surplus production

1. Introduction

The concept of a maximum sustainable harvest

(MSH) has been widely used for assessment of

fisheries, but it is based on deterministic models

and many fish populations harvested at their

maximum sustainable level have been over har-

vested (e.g. Mace, 2001), and simulation studies

have shown that harvesting at the MSH in a

random environment leads to over harvesting and

extinction (e.g. Beddington and May, 1977). In

this study optimal rates of harvest were deter-

mined using a surplus production model modified

for fluctuating environments. A conservative opti-

mal harvest also was determined where the harvest

was as close to the MSH as possible while at the

same time the intrinsic rate of increase, which

decreases with increase in the rate of harvest, was

as close to the maximum as possible. The harvest�/

effort curve is relatively flat on top, so a small

decrease in harvest could result in a large increase

in the intrinsic rate of increase.

Fisheries catch and effort data often have been

analyzed with surplus production models, (e.g.

Schaefer, 1954; Ricker, 1975; Quinn and Deriso,

1999), but Hjort et al. (1933) applied a surplus

production model for study of a harvested black

bear population in Norway. Harvest and hunter* Tel.: �/1-313-763-6280

E-mail address: ajensen@umich.edu (A.L. Jensen).

Ecological Modelling 157 (2002) 43�/49

www.elsevier.com/locate/ecolmodel

0304-3800/02/$ - see front matter # 2002 Published by Elsevier Science B.V.

PII: S 0 3 0 4 - 3 8 0 0 ( 0 2 ) 0 0 1 7 5 - 8

effort data for several wildlife species have beencollected by Canadian, United States Federal, and

State natural resource agencies for many years,

and these data can be applied to examine optimal

harvests in fluctuating environments. I used data

for the mallard duck (Anas platyrhynchos), lesser

scaup duck (Aythya affrinis ), white-tailed deer

(Odocoileus virginianus) and American black bear

(Ursus americanus ) (Dexter, 2000). The mallardduck is the most abundant duck in the Mississippi

flyway, and it is the duck most commonly

harvested (Dexter, 2000). The lesser scaup duck

is also commonly harvested in the Mississippi

flyway. Both mallard and scaup harvest and the

number of duck hunters have been estimated for

many years for the Mississippi flyway (Dexter,

2000). Extensive data also were available for thewhite-tailed deer and black bear harvests for many

different states in the USA; I used the harvest and

hunter data for the State of Minnesota (Dexter,

2000). The Minnesota deer season was cancelled in

1973 following severe winter weather, and the data

applied here were for a recovering population

from 1974 to the present. The dynamics of the

different species were sufficiently different toprovide some interesting insights into the perfor-

mance of the surplus production model.

2. Logistic surplus production model

The logistic surplus production model has been

widely applied for assessment of fisheries (e.g.

Graham, 1935; Schaefer, 1954; Pella and Tomlin-

son, 1969; Quinn and Deriso, 1999). Surplus

production is the rate at which individuals can be

removed from a population without change in

population size. Surplus production models as-sume that a population’s capacity to increase is a

function of population density, and that popula-

tion density will not change if members are

removed at the same rate as the population’s

capacity for increase. The logistic surplus produc-

tion model,

dH=dt�kN (1)

dN=dt�rmaxN�rmaxN2=K�kN (2)

is based on the logistic equation, where H isharvest at time t , k is a harvest coefficient (yr�1),

N is population abundance at time t , rmax is the

intrinsic rate of increase for a sparse population

(yr�1), and K is the environmental carrying

capacity. The harvest coefficient k can be written

as k�/qE , where q is a catchability coefficient and

E is hunting effort in terms of number of hunting

licenses, hunter days, or some other suitablemeasure. Eq. (1) and Eq. (2) assume that harvest

is proportional to the product of numbers and a

harvest coefficient and that in the absence of

harvesting population growth is logistic.

At equilibrium, where dN /dt�/0 and harvest is

He�/kNe, harvest can be written in terms of the

parabola

He�rmaxNe�rmaxN2e =K ; (3)

and equilibrium abundance,

Ne�K(rmax � k)

rmax

; (4)

can be obtained from Eq. (2) with dN /dt�/0.Population size will. asymptotically approach an

equilibrium for any value of k such that 0B/k B/

rmax. The abundance at which the MSH occurs is

NMSH�/K /2. The MSH itself, which is the max-

imum annual harvest that the population can

sustain in a deterministic environment, is found

by substitution of NMSH into the equilibrium

harvest equation to give MSH�/rmaxK /4. Theinstantaneous harvest mortality that results in

the MSH is kMSH�/MSH/NMSH�/rmax/2, which

is one-half the population’s maximum intrinsic

rate of increase, and the effort at the MSH is

EMSH�/rmax/2q .

3. Maximum harvest with minimum effect on rmax

The logistic population growth equation for a

harvested population (Eq. (2)) can be written as

dN=dt�rN�rN2=A; (5)

where r�/rmax�/k is the intrinsic rate of increase

of the harvested population and A�/K (1�/k /rmax)

is the adjusted carrying capacity of the harvested

A.L. Jensen / Ecological Modelling 157 (2002) 43�/4944

population. Harvesting decreases the population’srate of increase and the carrying capacity, and to

reduce the risk to a harvested population it could

be harvested so that r was as close to rmax as

possible while at the same time the harvest was as

close to MSH as possible. As k increases, r

decreases, and harvest increases, and at some

intermediate level of harvest

d(r=rmax)=dk��d(He=MSH)=dk; (6)

and this is the level of harvest where r is as close to

rmax as possible while at the same time the harvest

is as close to the MSH as possible. Harvesting

under the condition where Eq. (6) is true will be

called the conservative optimal harvest. From the

above equations,

r=rmax�1�k=rmax (7)

He=MSH�4k=rmax�(2k=rmax)2; (8)

and

d(r=rmax)=dk��1=rmax (9)

d(He=MSH)=dk�4=rmax�8k=r2max; (10)

The conservative optimal level of harvest ob-tained from Eq. (6), Eq. (9), and Eq. (10) is kopt�/

3rmax/8. In terms of kMSH, which equals rmax/2, the

conservative optimal level of harvest becomes

kopt�/0.75kMSH. Abundance at the conservative

optimal harvest is (from Eq. (4))

Nopt�K(rmax�kopt)=rmax�5K=8; (11)

as compared to K /2 at the MSH. The conservative

optimal harvest itself is

Hopt�koptNopt�15rmaxK=64

� (15=16)MSH; (12)

and the conservative optimal effort is

Eopt�3rmaxq=8�0:75EMSH: (13)

4. Harvesting in a fluctuating environment

In a fluctuating environment the equation for

the equilibrium harvest equation is

He�rmaxNe�rmaxN2e =K�QNe; (14)

where Q is an environmental purturbation whichmay change from year to year. Beddington and

May (1977) examined the effects of random

fluctuations on harvest, but environmental fluc-

tuations are not always random, and often there

will be trends in the abundance of a harvested

species such as there was for the recovering deer

population of Minnesota. In a fluctuating envir-

onment the rate of increase and the carryingcapacity change from one year to the next and

the relation between equilibrium yield and effort

will be represented by a family of parabolas

indexed by Q . The MSH and the effort at the

MSH in a fluctuating environment (from Eq. (14),

via calculus) are:

MSH�(rmax�Q)2K=4rmax (15)

EMSH�(rmax�Q)=2q (16)

Both MSH and EMSH are functions of Q , and

elimination of the environmental purturbation Q

from Eq. (15) and Eq. (16) gives the relation

MSH�(Kq2=rmax)E2MSH (17)

between the MSH and effort. The relation between

optimal harvest and the square of effort at the

optimal harvest is invariant under environmental

variation, and this relation can be applied to

identify over harvested populations in an environ-

mental that fluctuates over time.

The relation between harvest and effort at aconservative optimal level of harvest in a fluctuat-

ing environment also can be obtained. When

harvest is maximized such that the effect on the

rate of increase is minimal, Eq. (15) and Eq. (16)

become

MSH�(15=16)(rmax�Q)2K=4rmax (18)

EMSH�(3=8)(rmax�Q)=2q; (19)

and elimination of Q gives

MSH�(15 Kq2=9rmax)E2 (20)

Harvest and effort data for a population being

harvested at the conservative optimal level should

be to the left of the curve described by Eq. (17) and

fluctuate randomly about the curve described by

Eq. (20).

A.L. Jensen / Ecological Modelling 157 (2002) 43�/49 45

5. Results and discussion

The parameters rmax (yr�1), q (number per unit

effort per year), and K (number) were estimated

with harvest and hunter data using the non-linear

least squares approach described by Pella and

Tomlinson (1969). The number of hunters is a

measure of effort when it is assumed that abun-

dance of the animals is proportional to the harvestper hunter, that the amount of hunting on average

is the same for all hunters, that all hunters have the

same skill level, and that the bag limit does not

change. These assumptions are only approxi-

mately true.

Deer are hunted with rifles and bows. Bow

hunters were converted to an equivalent number of

rifle hunters because most of the deer wereharvested by rifle hunters, and the total number

of rifle hunter equivalents was calculated from the

rifle and archery hunter data as

hunters�rifle hunters(total harvest=rifle harvest):

The non-linear least squares parameter esti-

mates were a carrying capacity of K�/1 250 000

deer, intrinsic rate of increase rmax�/0.60 yr�1,

and catchability coefficient q�/3.75�/10�7 (Table

1). The model fit the observed trend in the harvestdata well with R2�/0.55; i.e. 55% of the variation

in the deer harvest was related to variation in the

rate of harvest, but the estimated harvests do not

pick up the year to year fluctuations in harvest

(Fig. 1A). The maximum sustainable harvest was

MSH�/rmaxK /4�/187 500 deer with an effort of

EMSH�/rmax/2q�/800 000 hunters and a harvest

rate of kMSH�/rmax/2�/0.30 yr�1. The conserva-tive optimum harvest was Hopt�/(15/16)MSH�/

175 781 deer with an effort of Eopt�/0.75

EMSH�/600 000 hunters and a harvest rate of

kopt�/0.75kMSH�/0.225 yr�1.

The estimated parameters for black bear were a

carrying capacity of K�/24 000 bears, intrinsic rate

of increase rmax�/0.50 yr�1, and catchability

coefficient q�/0.23�/10�5 (Table 1). Again, the

model fits the trends in the observed harvest data

well with R2�/0.53, but the annual fluctuations in

harvest were not well described (Fig. 1B). The

maximum sustainable harvest was MSH�/3000

bears with EMSH�/10 870 hunters and kMSH�/

0.25 yr�1. The conservative optimum harvest

was Hopt�/2813 bears with an effort of Eopt�/

8153 hunters and a harvest rate of kopt�/0.1875

yr�1.

The estimates of parameters for the mallard

were K�/1.44�/107, rmax�/0.5.5 yr�1, and q�/

3.3�/10�7 (Table 1); the coefficient of determina-

Table 1

Estimates of the intrinsic rate of increase r (yr�1), carrying

capacity K (numbers), and catchability coefficient q (number

per unit effort per year) for mallard, scaup, white-tailed deer

and black bear

Parameter Mallard Scaup Deer Bear

r 0.55 0.72 0.60 0.50

K 1.44�/107 1.85�/106 1.25�/106 2.4�/104

q 3.30�/

10�7

2.60�/

10�7

3.75�/

10�7

2.30�/

10�5

Fig. 1. (A) Minnesota deer harvest (stars) and harvest esti-

mated with surplus production model. (B) Minnesota black

bear harvest (stars) and harvest estimated with surplus produc-

tion model. (C) Mississippi flyway mallard duck harvest and

harvest estimated with surplus production model. (D) Mis-

sissippi flyway scaup duck harvest and harvest estimated with

surplus production model.

A.L. Jensen / Ecological Modelling 157 (2002) 43�/4946

tion was 0.35. The predicted harvests described thetrends in the observed harvest, but again they did

not pick up the short term fluctuations and the

fluctuations from 1 year to the next are large (Fig.

1C). The parameter estimates give MSH�/

1 980 000 mallards, kMSH�/0.28 yr�1, and

EMSH�/833 333 hunters. The conservative opti-

mum harvest was Hopt�/1 856 250 mallards with

an effort of Eopt�/625 000 hunters and a harvestrate of kopt�/0.21 yr�1. The parameter estimates

for the lesser scaup were K�/1.85�/106, rmax�/

0.72 yr�1, and q�/2.60�/10�7 and the coefficient

of determination was 0.28 (Fig. 1D). There appear

to be short term cyclic fluctuations in the scaup

data that are not described by hunting and

probably arise from environmental variation.

The parameter estimates give the MSH�/333 000scaup, kMSH�/0.36 yr�1, and EMSH�/1 380 000

hunters. The conservative optimum harvest was

Hopt�/312 188 scaup with an effort of Eopt�/

1 035 000 hunters and a harvest rate of kopt�/

0.27 yr�1. For each species, harvesting at the

conservative optimal level instead of at the MSH

increased the intrinsic rate of increase considerably

(i.e. decreased the harvest rate) with a compara-tively small decrease in harvest.

Environmental variation causes both short and

long term fluctuations in harvest; for example, the

Minnesota deer population was recovering from a

severe winter during the entire period for which

data are available, but there also were sizable

fluctuations from year to year. For all species the

surplus production model picked up the trends inthe harvest data but did not pick up the year to

year fluctuations (Fig. 1A�/D), which indicates

that the source of the year-to-year fluctuations was

not a result of harvesting but rather was a result of

some environmental variation. The year-to-year

fluctuations are especially large in the duck

populations. Greenwood et al. (1995) have dis-

cussed the effects of environmental factors on thedynamics of harvested duck populations, and it

has long been known that duck abundance is

correlated to numbers of ponds in the northern

plains of the USA and Canada (e.g. Watt, 1968).

For each species, the equilibrium relations

between harvest and effort at the MSH were

calculated with Q�/�/0.20rmax, �/0.10rmax, 0,

0.10rmax, and 0.20rmax using Eq. (14), and the

resulting series of parabolas represent a sample of

all the possible equilibrium relations between

harvest and effort under different environmental

conditions, and in a variable environment the

population moves from curve to curve over time

(Figs. 2�/4, and Fig. 5).

For each species the relation between the MSH

and effort under different environmental condi-

tions (Eq. (17)), was also plotted (Curve B in Figs.

2�/4, and Fig. 5), and the area to the right of Curve

B in each Figure is the area of over harvest,

Finally, for each species the conservative optimum

harvest and effort under different environmental

conditions (Eq. (20)) were plotted for each species

(Curve A in Figs. 2�/4, and Fig. 5). For a

population harvested at the conservative optimum

level the harvest and effort data fluctuate ran-

domly about Curve A and remain to the left of

Curve B.

The results indicate that for the deer population

there was little variation in effort but substantial

Fig. 2. Minnesota deer harvest (stars), equilibrium sustainable

harvest (parabolas; Eq. (14)) with from the bottom up Q�/�/

0.20rmax, �/0.10rmax, 0, 0.10rmax, and 0.20rmax, non-equilibrium

MSH (Curve B), and non-equilibrium conservative harvest

(Curve A).

A.L. Jensen / Ecological Modelling 157 (2002) 43�/49 47

variation in abundance as the population recov-

ered from a severe Minnesota winter (Fig. 2); a

substantial part of the variation in the deer harvest

was not random. The deer population was never

over harvested (Fig. 2). In recent years the black

bear appears to have been over harvested (Fig. 3).

Only a limited number of licenses were issued for

bear hunting, and the number issued depended on

estimates of bear abundance and on the desire to

control bear abundance. During the past few years

much larger numbers of licenses were issued for

bear hunting, and this has led to high variation in

harvest per unit effort and over harvesting. The

analyses indicate that the mallard population was

over harvested in the late 1960’s, but in recent

years the level of mallard harvest has been less

than the MSH (Fig. 4). The scaup has never been

over harvested (Fig. 5); however, there was very

high year to year variation in the scaup harvest

and hunter data (Fig. 5).

Many models have been developed for assess-

ment of wildlife populations (e.g. Geis et al., 1969;

Anderson, 1975; Cowardin and Johnson, 1979;

Johnson et al., 1987; Starfield and Bleloch, 1991,

and Greenwood et al., 1995), but none of these

models use the readily available harvest and

hunter data. Important characteristic of the logis-

tic surplus production model are its simplicity and

its use of data that have been collected for many

years over a wide area by wildlife agencies. The

surplus production model describes the effects of

hunting on all four populations reasonably well

and it identified the relative level of harvest with

few assumptions and little data. Although the

simple logistic surplus production model applied

here describes the impact of harvesting under

simplified and ideal conditions the models did

not pick up the short-term fluctuations. More

complex forms of the surplus production model

are available when age structured or sex structured

data are available (e.g. Jensen, 1995, 2000), and

environmental, data also can be incorporated into

the model (e.g. Jensen and Marshall, 1983).

Harvest data for different states and for different

Fig. 3. Minnesota black bear harvest (stars), equilibrium

sustainable harvest (parabolas; Eq. (14)) with from the bottom

up Q�/�/0.20rmax, �/0.10rmax, 0, 0.10rmax, and 0.20rmax, non-

equilibrium MSH (Curve B), and non-equilibrium conservative

harvest (Curve A).

Fig. 4. Mississippi flyway mallard duck harvest (stars), equili-

brium sustainable yield (parabolas; Eq. (14)) with from the

bottom up Q�/�/0.20rmax, �/0.10rmax, 0, 0.10rmax, and

0.20rmax, non-equilibrium MSH (Curve B), and non-equili-

brium conservative harvest (Curve A).

A.L. Jensen / Ecological Modelling 157 (2002) 43�/4948

management units within states may be useful for

assessing the effects of environmental factors.

Acknowledgements

I thank Mr Roger Lake of the Minnesota

Department of Natural Resources in St. Paul,Minnesota, for giving me the data.

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Fig. 5. Mississippi flyway scaup duck harvest (stars), equili-

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A.L. Jensen / Ecological Modelling 157 (2002) 43�/49 49