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MICROCOPY RESOLUTION TEST CHART
NATIONAL BUREAU OF STANDARDS - 1963

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LEGAL NOTICE,
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M
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1
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ORNL-P-262 Drie-P
SEP 2 1 1969
MASTER
EFFECTS OF POPULATION DENSITY ON GROWTH RATES
P
.
OF ANIMAL POPULATIONS
JAMES T. TANNERI
Radiation Ecology Section, Health Physics Division
Oak Ridge National Laboratory, Oak Ridge, Tennessee
TAI.
.
EZAT
RUNNING HEAD: Population density and growth rates

This pappen lite guance E. c. bobed, the
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prior since thens been pu in Micro-
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prior to the issuance date of this Micro-
card. Since the U.S.A.E.C. has no evi-
dence that it has been published, the pa-
per is being distributed in Microcard
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via da Corunluulon, or Me employment nuk mca contractor.
.
1
1
Permanent address: Department of Zoology, University of
.
Terinessee, Knoxville, Tennessee.
:1
Parts of this research were sponsored by the U. S. Atomic Energy
Commission under contract with the Union Carbide Corporation.
=
.
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Abstract. The growth rate of a population 18 defined as the
change in numbers per vnit population at the time of the change per
time (r+ = dN/Nat). The objective of this study was to determine
whether the growth rate 18 or is not a function of the population
.
.
:
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32
-
density.
:-
In most mathematical models of populations whose changes re-
semble those of actual populations, the growth rate is a decreasing
function of population density. The relation between the growth rate
and population density of actual animal populations was determined from
the records of 111 different populations representing 71 species. Growth
rates were calculated for each interval of time; regression methods were
then used to calculate a coefficient of growth rate on population density
and to test whether or not this coefficient was significantly different
i
.
from zero.
Of the 71 species represented, 7'were eliminated from further
analysis because their census records were not significantly different
from a series of random numbers, 42 had negative coefficients signifi-
cantly different from zero, 21 had coefficients not significantly
different from zero (all but one of these estimated coefficients were
negative), and one, the world's human population, had a positive
coefficient significantly different from zero.
There were no differ-
A
ences between taxonomic groups (insects, other invertebrates, fish,
t
*
birds, mammals).
The primary conclusion 18 that in most animal species
a population's growth rate is a decreasing function of density. This
.
3
explains the relative stability of animal populations, which rarely
continue to increase at rates their fertility would allow and rarely
decrease to extinction.
Tentative conclusions are presented regarding the processes
regulating population numbers. Populations of herbivorous insects at
low and moderate levels are regulated by predators and parasites causing
mortality that is an increasing function of density
Favorable con-
ditions may allow an increase in insect numbers so rapid that the popu-
lation temporarily escapes regulation by its enemies. Nonterritorial
species of vertebrates are normally controlled by predation and, when
that fails, by competition; 4.n both cases juvenile individuals are
most affected. In territorial vertebrates competition for suitable
territories determines the size of the breeding population. Populations
of vertebrate species (excepting man) are in general regulated by the
production of adult individuals being a decreasing function of popula-
tion density.
INTRODUCTION
Changes in size of animal populations usually follow seasonal
and other variations in the environment. An important question 18:
Are these changes determined solely by the environment, or does the
density of the population itself affect these changes?
This study
aimed at determining the relation between the rate at which a popula-
tion grows or declines and the population density. Two approaches
were used: (1) an examination of mathematical models of populations,
and (2) an analysis of the records of many dirferent animal populations.
In the second approach, data were obtained from the literature and
statistically analyzed to test whether the growth rate of each popu-
lation was or was not a function of population density.
The change in numbers of a population with respect to time,
an/at, equals rn, where N is the number in the population and r is the
rate of change per unit population: r = an/nat. For conciseness r
will hereafter be called a population's "growth rate," even though it
can measure a decrease as well as an increase in population size. The
growth rate equals the conventional birth rate (number horn per unit
population per time) minus the death rate (number died per unit popu-
lation per time), and the growth rate will be positive, negative, or
zero depending on the relative values of its two components.
If r is a positive constant, the population will grow exponen-
tially; if it is a negative constant, the population will decline
exponentially; and If It is zero, the size will not change. Since
populations never do any of these for an indefinitely long period, the
value of r must vary from time to time. If its value is determined
solely by the environment, r is inäependent of the population density.
Otherwise r 1s some function of the population density. If r 18 a
decreasing function of density, declining as the density increases as
in Fig. 1A, and if the environment is relatively stable for a suffic-
iently long time, the population vill stabilize at the density where
r 18 zero. If r 18 an increasing function of density, as in Fig. 1B,
the population will increase to infinity or decline to extinction,
again assuming that the environment does not change.
This research was completed while I was working in the Radiation
Ecology Section, Health Physics Division, Oak Ridge National Laboratory;
members of this section helped me to clarify many of the ideas expressed
in this paper. Most of the statistical calculations were performed by
the Mathematics Panel of the Oak Ridge National Laboratory.
RATES IN MATHEMATICAL MODELS
Mathematical models for animal populations present the popula-
tion size, N, as a function of time, or the derivative of this with
respect to time, an/at. The growth rate of the population, r, equals
this derivative divided by N. The relation between r and N in these
1
equations can be determined by differentiating f with respect to N.
If the resulting derivative is negative, r is a decreasing function
op N as in Fig. 1A; if it is positive, r'is an increasing function as
in Fig. 1B; 18 It is zero, r 18 independent of N. The only models
examined here are those which have been found to describe the changes
in actual populations.
The logistic equation, widely used in population studies, 18
discussed by Allee et al. (1949, Ch. 21), who give numerous examples
of this equation fitted to the growth of actual populations. The
derivative, dr/an, of the logistic equation is negative, showing that
the growth rate decreases with increasing N, as in Fig. 1A. Several
persons (Gause and Witt, 1935; Smith, 1963b) have modified the logistic
for one reason or another, but in these modified forms r remains a
decreasing function of N.
Neyman, Park, and Scott (1958) constructed mathematical models
for Tribolium populations besed on extensive experiments in rearing
these beetles; one model is for the numbers of a species living alone
and another consists of two equations for two species of Tribolium
living together. Nicholson and Bailey (1935) developed equations for
.
-
.
--..
are.
..
-
....
-
-
-
the numbers of a host and its parasite; the first part of the oscillating
curve predicted by these equations was followed by seven generations
of a host-parasite population (De Bach and Smith, 1941). In all of
these equations the derivative of the growth rate with respect to
numbers is negative. Utida (19570) presented different equations
which predicted the course of his oscillating host-parasite populations.
The derivative for the parasite population is negative; that for the
host population is positive or negative depending upon conditions other
than the density of the host.
In summary, in mathematical models which have been shown to
parallel the changes of actual populations, the derivative of r with
respect to N is usually negative, 1.e., the growth rate is a decreasing
function of the density.
RATES IN ANIMAL POPULATIONS
Few persons have appraised the effect of population density on
population growth rates. Errington (1945, 1954) plotteà per cent gain
in bobwhite and muskrat (scientific names are listed in Appendix B)
populations &s a function of density and obtained curves showing smaller
rates of increase with higher densities. Smith (1961, 1963a) found
that in populations of thrips the growth rate over a month was nega-
tively correlated with the population density at the start of the
month. He also (1963b) showed that cultured daphnia populations
possessed growth rates that were non-linear, decreasing functions of
density.
For this part of the study, I obtained from the literature the
records of many different animal populations, calculated the growth
rates from the recorded aumbers, and tested the hypothesis that these
rates were independent of the population densities.
Sources of data
The data were obtained from censuses of animal populations that
met the following requirements: (1) actual counts of a population
inhabiting a definite space so that density was directly proportional
to the number, or a reliable index of the density of the population;
(2) counts or estimates made periodically so that the growth rates
were based on constant intervals (at); (3) a sufficiently long series
of such counts to enable a meaningful test despite random variations.
About half of the populations reported in Table I were censused.,
For the remainder an index of density was available. This index for
the fish populations was calculated from the annual commercial catch
corrected for variations in fishing intensity. Fur returns were used
as indexes of density for some northern mammals ; Keith (1962) gave
reasons for believing that these were valid indexes of density when
the returns came from constant geographic areas where fur prices had
not failed. Hunting kill statistics were not considered to be a
reliable index of density because of variations in hunting pressure
with periods of war, changing economic conditions, and changing recre-
ational habits. Exceptions to this were some bag records made on
British game preserves which had been managed intensively for decades;
Keith (1962) believed that these were reliable indexes of density.
Fur returns from the Hudson Bay Company and game bag records for some
British preserves supplied the longest series of records for this
study.
I found records of all populations, representing 71 species,
that met the requirements of this study.
Methods of analysis
A population at time t has a size it. Its growth rate propor-
tional to population size at this time 18 ft Expressing this rate in
terms of the meas ured change in numbers, N., over a definite period,
At,
M
Hypothetically, rt 18 a function of the environmental conditions and
the density of the population at this time.
Iet: 2 x a constant
Et = the suitability of the environment at time t
(climate, predators, food, etc., but not numbers
of the same species, all combined into one value).
Dop the coefficient of regression of rt on Et i
N = the number in the population at time .t; the density
will be directly proportional to N. 1f the popula-
tion inhabits a definite area or volume.
bp = the coefficient of regression of rt on N
= a random variable (the difference between an actual
rt measured as AN./Nat and an ideal 17, a function
only of Et and Nand the parameters a, bg, and Dp).
It = a + but + bnNE + 6 •
(2)
EL 18 unknown, but the following possibilities exist:
1.) E18 constant. This could be true in short term studies and
in cultures of animals 'Where physical conditions are regulated and food
18 constantly renewed. It is obviously not the case in most natural
situations. If E: 18 constant, (a + b ) 18 a constant and equation.".,
ecomes
·
Ft * &' + bnNt + 8 .
(2a)
2) Et fluctuates randomly so that there is no correlation be-
tween successive values of Ex. If the successive values of E. are
independent, but can be considered as part of 6 and equation (2)
becomes
ft = a + bytt +6
(20)
3) For many species periods of abundance alternate with periods
of scarcity (e.g., rainy and dry seasons). In such cases we can assume
that E has one value, En, during periods of high numbers and another,
Es, during periods of scarcity. Let a + b = en and a + b F. - &g,
then equation (2) becomes two equations :
rt = n + bxN+ + 6, when populations are high;
(3)
r+ = a + bylt + 6, when populations are low.
This idea could be applied to more than two population levels, but that
18 not attempted in this paper.

܂
ܢܐ
The question of whether r18 or 18 not a function of population
density was answered by the following argument and procedure. In a
population inhabiting a definite space and censusod at regular inter-
vals, an equals Nt+1 miilus N. The density 18 directly proportiona.).
to N, and at can be calied 1. The record of population censuses pro-
vides a measure of rt (equation 1) for each N., except the last; the
number of paired values of rt, and N. 18 one less than the number of
censuses of the population. If the environment fits either possibility
(1) or (2) described above, equations (1) and (2a) or (2b) can be com
bined into the regression equation
= a (or a') + BxN+ + € :
(4)
equat
Ch tih
Sene coeIILC
coeppich
If the environment has changed as hypothesized above in possibility (3),
the same procedure can be followed except for calculating two regression ...
equations (equation 3) with the same coefficient, Bro
The null hypothesis to be tested 1s that bp = 0. If the null
hypothesis is accepteå, rt 18 considered to be independent of N. If
the null hypothesis is rejected, rt 1.8 a decreasing function of N. (Fig.
1A) 1f by 18 negative, or an increasing function (18. 2B) 18 by 18
positive.
Linear regression methodh were used to estimate by for each
species listed in Table I. When two or more censuses were available
for one species, a single weighted mean coefficient (weighted by the
inverse of the variances of the separate censuses) was calculated for
the species. To test the null hypothesis, the distribution of Student's
t was used as a test of significance.
12
In the above discussion, rt was assumed to be a linear function
of population size. Smith (1963b) has shown that in Daphnia magna
the population growth rate 18 a non-linear, decreasing function of
density (Fig. 10). Ricker (1954) hypothesized a series of "reproduction
curves" from which it can be calculated; the curve which he considered
· best fits the data from a number of fish and invertebrate populations
has the non-linear, decreasing relation between rt and density shown
in Fig. ID. It can be seen from Figs. ic and ID that these curves can
be approximated by straight lines which will indicate correctly that
It is a decreasing function of density. Therefore for the purposes
of this study, rt can be assumed to be a linear function of density.
Before performing the regression analyses, it was necessary to
prove that the values of Nt for each population are serially correlated
or are non-random, because a series of random numbers analyzed by the,
method used for the populations will produce a b significantly dif-
ferent from zero. The reason for this result is that if N.+ 18
independent of Nu, for each Nt the expected value of N+1 18 its mean
value, Nt+2Then 8+ = (NH+1 - N_)/N4 = N4+1/4 - 1. Since N4+1 18
a constant, this is the equation of a hyperbola, and a regression of
rt on No will necessarily result in a value of by different from zero.
If on the other hand N 47 18 correlated with N, by will be different
from zero only if AN/N. changes approximately linearly with changing
Each series of population counts was tested for non-randomness
by the method of runs. Any in which the probability of randomness was
not less than 5% was eliminated.
To enable comparisons of the relative sizes of the coefficients
(bn) among the different species, the coefficients must be adjusted
· for several factors, because the absolute value of by depends upon the
length of the time interval between censuses, the number in the popu-
lation, and some relation between the animals' size and habits and
the effect of density.
Adjusting by to the generation time of the species gave values
of the same relative size for species censused at different intervals.
Generation time 18 here defined as the age at which a female usually
first reproduces. The generation times included in Table I were com-
piled from a variety of sources.
:
.
bx for generation time = bx for at x (generation time)/At (5)
Each population cens used was believed to inhabit a constant
area or volume so that the population density was proportional to its
numbers. In most cases the actual space inhabited was unknown, but
it was probably proportional to the mean size of the population, N;
1.e., area = kſ. Let De be the population density at time t and by
be the coefficient of regression of rt on Dt; then
De = N//area = Nz/KT
It = a + burnt + 6 = a + bp Pt + 6 = 8 + b(Nz/KN) + 6
by T = bg/l .
.
(6)
.

.24
This coefficient, by/k, 18 independent of the size of the population
on which the regression is based and 18 characteristic of the species.
An attempt to evaluate k did not add any meaning to the value of the
adjusted regression coefficient, so the coefficients reported in
Table I have been adjusted only for generation time and the mean size
of the population. This adjustment of by, the coefficient of regression
in equation (4), 18 summarized in a combination of equations (5) and (6):
,
·
by x (generation time)/at x Ñ = bx/k
(7)
Results and discussion of results
The results of the regression analyses are presented in Table I.
For each species the table contains information on the populations and
on the censuses or other measurements. The regression coefficient,
by/k, has been adjusted by equation (7) for generation time and the
size of the population censused. The probability of the result under
the null hypothesis (the regression coefficient equalling zero) 18 in
the last column. The populations eliminated because their records were
not significantly different from a series of random numbers are listed
at the end of the table.
The results in Table I were obtained by using equation (4),
based on the assumption that the environment either was constant or
varied in a random manner. Equations (3) and the associated explanation
shows how it is possible to allow for two levels of the environment,
and this equation was used for two species. The number of thrips
(Davidson and Andrewartha, 1948) increased each year to a high level
during the Australian spring and then dropped to a low level during
approximately ten months of drouth and low temperatures. The starling
population inhabited an area where the death of many trees beginning
about the sixteenth year of the series (Kendeigh, 1956) produced many
more nesting sites and a period of relative abundance of these birds.
For each of these records two regression equations were calculated
with the same coefficient of regression but with different constants
representing different levels of the environment. The regression
coefficients for the thrips calculated in this way 18 -0.278 with the
probability of this under the null hypothesis being less than 0.001;
that for the starling 18 -0.738 with the probability being less than
0.001. Thus these regression coefficients are very significantly
different from zero while those calculated from a single regression
are not so (see Table I). This procedure was used only for these two
species; for all the others the coefficients as calculated by a single
equation were significantly different from zero (at the 0.05 level or
better) or there was no direct evidence of environmental changes to
justify using equation (3).
Some of the populations for which the coefficient is not sig-
nificantly different from zero exhibited characteristics helping to
explain the lack of significance. The wax scale insect and the four
lepidopterous insects showed marked and occasionally very sudden changes
in population size; the causes of these changes are not known. The
•
16.
blowfly had a time lag between population density and the response to
density, resulting in oscillating numbers (Nicholson, 1954). The lake
trout population became almost extinct with a sudden and accelerating
decline in numbers (Eschmeyer, 1957).
species remaining after 7 were eliminated, 42 species had coefficients
thrips and starling, for which the use of equations (3) as described
above produced a coefficient significantly different from zero, are
counted here. Only one, the human population of the world, which has
been increasing at a greater rate with the increase in population
density, has a coefficient that is significantly positive. Furthermore,
of the 21 species for which the regression coefficient did not differ
significantly from zero, only one did not have a negative estimated
value (see : Table I). Population rates of change are usually, there-
fore, a decreasing function of population density; as can be seen from.
Tables I and II, this conclusion applies to animals which are taxonomi-
cally very different.
.
..
-
Changes between spring and fall
Spring and fall censuses over a number of years were made for
the four species listed in Table III, enabling a test of whether or
not the population rate of change is a function of density during a
period of increase, from spring to fall, and during a decrease, from
fall to spring. The procedure is the same as that described for
17
equation (4) except that, when considering spring to fall changes,
Nt becomes the number in spring and N+ the number the following fell;
there is a parallel procedure in studying fall to spring changes. In
adjusting the regression coefficient by equation (7), the generation
time for all four species 18 one year, At is six months, and Ñ is the
The results of the regression analyses are shown in Table III
and are the same for all four species. For the spring to fall changes,
the coefficient with respect to density is negative and significantly
different from zero; for the fall to spring changes, the coefficients
are not significantly different from zero: Possible explanations for
the seasonal differences are discussed in the following section.
CAUSES OF POPULATION CHANGES
.
The analyses described in the preceding sections have shown that
in most of the species tested the population growth rate is a decreas-
ing function of density. Different ways of saying the same are (1)
that as a population increases, its growth rate decreases, or if it
1.8 declining the negative rate increases toward zero, so that a curve
depicting its rise or fall will flatten as it approaches a certain
level; or (2) thet as a population becomes more dense the probability
of a further increase becomes less and that of a decrease becomes
greater, and vice versa. The net result is that population numbers
tend to remain stable, rarely increasing to excessive numbers and
: rarely decreasing to extinction.
It is true that many animal populations
in constant environments do fluctuate in size, but fluctuation within
limits 18 a degree of stabilization. Many fluctuations or oscillations
of animal populations result from a delay or time lag in the response
of growth rate to density (Frank, 1960; Wangersky and Cunningham, 1957).
My purpose now is to discuss the causes of population change,
the processes that affect the size of a population, to determine which
are responsible for the growth rate being a decreasing function of
density.
I am avoiding the use of the term "density-dependent factor"
which has been used frequently in connection with this subject but with
different meanings by different ecologists; see Macfadyen (1963, p.
-
-
-
-
-
-
-
-
-
.
-
-
152-157) for a review of the literature concerning this term. The
phrase "increasing (or cecreasing) function of density" 18 explicit
and is used here for any change whose rate increases (or decreases)
with increasing population density, and for the processes causing such
a change. The term "density-independent" is used in the absence of
any relation between rate and density..
The growth rate equals the birth rate minus the death rate.
If the growth rate is a decreasing function of density and only the
birth rate varies, the birth rate must also be a decreasing function
of density. Ecologists often assume that each species has a potential
reproductive rate determined by its physiology and life history, and
• that environmental factors reduce the actual rate to something below
the potential. These factors reducing the birth rate at higher densities
must then be operating as increasing functions of density. If, on the
19
other hand, only variations in the death rate are producing the changes
in the growth rate, death rates are increasing with greater population
density and so must the effectiveness of the processes producing the
mortality. In summary, the growth rate 18 usually a decreasing function
of density, but the causes of population change operate as increasing
functions of donsity.
The statistical analyses of population changes do not give any
clue as to which processes are producing the changes, except that in
four species (Table III) changes that were functions of density occurred
only between spring and fall and therefore the processes must be con-
fined to that season. The most frequently suggested processes that
may operate as increasing functions of density are competition, pre-
dation, and disease.
Competition results when the supply of some resource is inade-
quate for the population, or if interference or struggle for a common
resource results in harm to individuals. Competition 18 obviously an
increasing function of density. Examples of scarcity of a resource,
usually fooå, limiting a population, lowering fertility, or increasing
mortality are many (Andrewartha and Birch; 1954, p. 368-376; Andrewartha
and Browning, 1961; Armstrong, 1964; Gibb, 1960; Klomp, 1964; Lack,
1954, Ch. 7 and 11; Slobodkin, 1954). Among vertebrates, competition
for food frequently results in higher mortality of juvenile individuals
rather than of adults; examples of this are known for fish (Beverton,
1962; Ivlev, 1961), for owls (Southern, 1959), for voles (Hoffman,
1958), and for deer. (Leopold, Sowls, and Spencer, 1947). .
20
20
Individuals of some species defend a territory against encroach-
ment by other individuals, and in these territorial species the amount
of suitable habitat may be a limiting resource. Examples of vertebrates
where the size of the breeding population appears to be limited by the
number of suitable territories are golden eagles ( Brown and Watson,
1964), red grouse (Jenkins, 1963), titmice (Kluyver and Tinbergen,
1953), chaffinches (Glas, 1960), song sparrows (Tompa, 1962), several
bird species nesting in the spruce-fir forest (Hensley and Cope, 1951;
Stewart and Aldrich, 1951), and muskrats (Errington, 1946). Dragonflies
(Odonata) offer one example of this phenomenon in insects (Moore, 1964).
The amount of suitable habitat for a territorial species may
limit the population in another way. Errington (1946, 1956) has shown
that in muskrats mortality from predation is much higher among Indi-
viduals that fail to establish themselves in a suitable territory than
..do not obtain territories In suitable habitats (Jenkins', 1963; Jenkins,
Watson, and Miller, 1963).
The information included in Table III 18 consistent with the
hypothesis that the populations of these four species are regulated
by competition, either by territoriality determining the size of the
breeding population or by competition for food and perhaps other re-
sources determining the survival of the juveniles through the summer
and early fall. The growth rate for these four species was a decreasing
function of density only during the period from spring to fall; it was
density-independent for the rest of the year when no reproduction was
o wie weiteren
occurring and when young animals were becoming mature. Of the species
listed in Table III, the bobwhite and muskrat are territorial, the
first only during the nesting season, and the pheasant and grouse are
territorial only in the sense that the polygamous males defend a dis-
play area during the breeding season.
Interference between individuals affects population growth, as
is illustrated by the following examples. Allee et al. (1949, p. 349-
352) summarize the detrimental effects of interference on the repro-
ductive rates of a few species of insects. The food intake of individual
fishes in a crowded situation decreased because of interference from
others, fright, or actual fighting (Ivlev, 1961, Ch. 5). Titmice
fought over food most often in midwinter when they spent the greatest
proportion of their time searching for food (Gibb, 1954). As a result
of stress from increased social contacts, the birth rate of rodents
and hares decreased, and the mortality of infant rodents increased
because of failure of lactation, with increasing population density
(Christian, 1959 and 1963).
Predation is a cause of mortality that may be an increasing
function of density under some conditions and density-independent or
even a decreasing function under others. Holling (1961), in a review
of the effects of predation on insect numbers, concluded that the
number of prey killed per predator generally increases with prey
density, until prey density becomes so great that either the predators
are satiated or the prey obtains protection from the large group; pre-
dation may therefore be an increasing function of density at low prey
22
densities and a decreasing function at high densities. Holling's
conclusions are supported by studies of predation on insects by birds
made by Gibb (1962), minbergen (1960), and Tinbergen and Klomp (1960).
The role of predation among vertebrates appears to be more
complex. Errington (1946, 1956) was convinced that in territorial
species those individuals that failed to establish good territories
were removed by predation, which therefore removed only the surplus,
but in nonterritorial species whose behavior allows crowding, predation
may determine the size of the population. Deer are nonterritorial
species and deer populations were at least originally controlled by
predators as all overpopulations of deer in the United States followed,
and none preceded, the disappearance of large predators or the initi-
ation of predator control (Leopold, Sowls, and Spencer, 1947).
The effects of predation on populations of small mammals were
studied by Craighead and Craighead (1956) on an area in Michigan for
two fall and winter seasons. Their publishec. results do not contain
an estimate of the mortality rate due to predation, but from some of
their data I have estimated this rate for four species of small mammals
preyed upon by nine species of hawks and owls; mammalian predators
appeared to be insignificant. Appendix A describes the method of
estinating the rate. Table IV summarizes the results. Large mouse
populations were present in the 1941-42 fall and winter, smaller num-
bers in the 19147-48 season. The rate of mortality due to predation on
the two species of mice was about the same for each year, or density-
independent. The rabbit and fox squirrel populations were, on the other
23
hand, lowest in 1941-42 but suffered the higher mortality rate from
predation this year; for these two species predation was a decreasing
function of density. The obvious reason for this is that the large
numbers of mice present in 1941-42 attracted tu the area many more
hawks and owls than normal, and this increased the probability of an
attack upon an individual rabbit or squirrel.
The results presented in Table III indicate that for these four
species predation is a density-independent cause of mortality. In the
period from fall to spring population changes result solely from mor-
tality, and predation is responsible for most of this. During this
period the rates of population decline were independent of population
density.
Young or juvenile individuals are more likely to be killed by
predators than are adults. Beverton (1962) concluded that populations
of plaice are regulated by varying mortality from starvation and pre-
dation of the larval fish. Murie (1944) reported that predation on
caribou and Dall sheep was concentrated on young animals.
Cannibalism is a special case of predation, resembling compe-
tition in that the process is intraspecific.
It appears to be an
increasing function of density in flour beetles, Tribolium (Allee et al.,
1949, p. 370-371; Neyman, Park, and Scott, 1958), and it is at least
an important cause of mortality in planarian worms (Armstrong, 1964).
Ricker (1954) discusses the hypothetical effects of cannibalism on
population size.
24
Mortality due to parasites or disease is an increasing function
of density for insects and does regulate population size according to
Macfadyen (1936, p. 263). For birds and mammals, however, lack (1954)
believes that disease is with few exceptions not a cause of mortality
increasing with density. The evidence for both of these conclusions
was obtained from a few cases, and the conclusions may not have general
validity. Disease often follows starvation, and then starvation should
be considered the primary cause of mortality.
There are two other possible mechanisms for regulating population
size, although few observations and measurements pertain to either.
The secretion of substances into the environment which inhibit the
activities of other organisms would have results similar to those of
competition; this subject is briefly reviewed by Rose (1960). Fiynne-
Edwards (1962) describes a number of behavior patterns which he sug-
gests can reduce reproductive rates in dense populations.
che
The processes regulating natural populations of herbivorous
insects and some vertebrates are beginning to be understood. Herbi-
vorous insects are normally regulated by their predators and parasites.
When favorable conditions for reproduction and survival exist, the
numbers of insects may increase so rapidly as to "escape" their pre-
dators and parasites and an outbreak occurs. The peak population is
subdued by the favorable period ending or by parasites and disease
overtaking the dense population. The hypothesis that fluctuating
insect populations are regulated only when at their lower levels 13
supported by the data in Table V. The five species of insects listed
here had populations which' fluctuated greatly. The statistical analyses
based on their entire series of population counts indicated that their
population changes were independent of density (Table I). When each
population record was divided into periods of scarcity and abundance,
however, and a test was made only for the periods of scarcity, the
regression coefficient of growth rate on density was negative and
significantly different from zero (at the 0.05 level or better) for
three of the five species. This is evidence that the low or "normal"
levels of these populations were being controlled by processes oper-
ating as increasing functions of density.
In vertebrates, the processes controlling population size are
different between non-territorial and territorial species. Non-territorial
species, including probably most fish, deer, hares, and most rodents,
are normally regulated by predation. When this fails, as when pre-
dators of deer are removed by man, competition for food becomes in-
portant. In deer and probably in fish, this results in starvation of
younger individuals. In hares and some rodents the reproductive rate
drops and the mortality of juveniles increases. The numbers of ter-
ritorial species are determined by competition for suitable territories .
Those that fail to obtain territories, usually immature individuals,
are not able to reproduce and are the first to be eliminated by pre-
dation or starvation. Because the processes regulating vertebrate
populations either lower the birth rate or increase the mortality of
young individuals, the general result of increased population density
in vertebrates 18 a lowering of the production of young adults.
Hairston, Smith, and Slobodkin (1960) inferred from some obvious
facts that populations of herbivores are generally controlled by pre-
dators and those of predators by competition. This generalization 18
supported only in part by my conclusions presented above. Although
herbivorous insects and some herbivorous mammals are nornally controlled
by predators or parasites, competition is important for territorial
herbivores like muskrat and red grouse and also in some rodents whose
reproductive rate falls with increased crowding. Competition does
appear to be the regulating process in predaceous birds and mammals,
largely through the mechanism of territoriality; by far the majority
of species known to be territorial are carnivorous birds and mammals.
The processes regulating natural populations of animals other
than herbivorous insects, birds, and mamma.ls are still mostly unknown.
The statistical analyses .summarized in Tables I and II indicate that
these processes, like those discussed in this section, must somehow
regulate by varying the population's growth rate so that it is a
decreasing function of density.
APPENDIX A
Data published in the book by Craighead and Craighead (1956)
were used to calculate the mortality rates due to avian predators
shown in Table IV. Because this calculation involved drawing data
from scattered parts of the book, and because the conclusion drawn
from Table IV loes not agree with their conclusion (p. 309) that the
predation observed tended to regulate the prey populations, the pro-
cedure I used is here outlined. Lable and page numbers cited below
refer to Craighead and Craighead (1956).
Let:
W, = grams of food consumed by each species of raptor during
fall and winter (from Table 90).
W, = average individual weight for each prey species (from
Table 100).
Bar = number of individuals of each prey species (8) consumed
during fall and winter by each species of raptor (r).
N = total number of all prey individuals consumed by each
species of raptor, - gener
for = frequency of each prey species in food of each species
of raptor, = n/N, (estimated in samples reported in .
Tables 25-28).
:
Wo sr". = & Nofors = No fare
N. can then be calculated since the other factors have been estimated.
Total number of each prey species consumed = & Near
The rate of mortality due to predation is this number divided by
the size of the prey population (p. 356-357).
APPENDIX B
Scientific names for species named in text and tables.
PELECY PODA. Clam Tivela stultorum.
ARACHNIDA. Herbivorous mite Eotetranychus sexmaculatus. Predatory
mite Typhlodromus occidentalis.
CRUSTACEA. Daphnia Daphnia magna (unless otherwise stated).
INSECTA. Thrips Thrips imaginis. Florida wax scale Ceroplastes
floridensis. Chaff scale Parlatoria pergandei. ordered white
moth Bupalus piniarius. Pine spinner moth Dendrolimus pini.
Pine hawk moth Hyloicus pinastri. Pine beauty moth Panolis
griseovariegata. Rice weevil Calandra oryzae. Bean weevil
Callosobruchus chinensis. Broad-horned flour beetle Gnathocerus
cornutus. Red flour beetle Tribolium casteneum. Confused flour
beetle Tribolium confusum. Granary beetle Trogoderma versicolor.
Parasitic wasp Heterospilus prosopidus. Sheep blowfly Lucilia
cuprina. Housefly Musca domestica. Sewage fly Spaniotoma
minima
OSTEICHTHYES. Atlantic salmon Salmo salar. Lake trout Salvelinus
namaycush. Goldeye Hiodon alosoides. Northern pike Esox lucius.
Yellow perch Perca flavescens. Walleye Stizostedion vitreum.
Freshwater drum Aplodinotus grunnlens. Plaice Pleuronectes
platessa
AVES. Fulmar Fulmarus glacialis. Gannet Morus bassanus. Heron Ardea
cinerea. Stork Ciconia ciconia. Golden eagle Aquila chrysaetos.

Appendix B, continued
Ruffed grouse Bonasa umbellus. Red grouse Lagopus scoticus.
Sharp-tailed grouse Pedioecetes phasianellus. Prairie chicken
Tympanuchus cupido. Bobwhite quail Colinus virginianus. Partridge
Perdix perdix. Ring-necked Pheasant Phasianus colchicus. Pied
flycatcher Muscicape hypoleuca. Coal tit Parus ater. Blue tit
larus caeruleus. Crested tit Parus cristatus. Great tit Parus
major. House wren Troglodytes aedon. Wood thrush Hylocichli
mustelina. Sterling Sturnus vulgaris. Red-eyed vireo Vireo
olivaceus. Ovenbird Selurus aurocapillus . Hooded warbler
Wilsonia citrina. Chaffinch Fringilla coelebs. Song Sparrow
Melospiza melodia. Indigo Bunting Passerina cyanea.
MAMMALIA. Wo.lverine Gulo luscus. Fisher Martes pennanti. Mink Mustela
vison. Weasel Mustela vulgaris. Arctic fox Alopex lagopus .
Coyote Canis latrans. Wolf Canis lupus. Gray fox Urocyon
cinereoargenteus. Colored fox Vulpes fulva. Lynx Lynx canadensis.
Fur seal Callorhinus ursinus. Fox squirrel Sciurus niger.
Levant vole Microtus guentheri. Meadow vole Microtus pennsylvanicus.
Mus krat Ondatra zibethicus. White-footed mouse Peromyscus leucopus .
Snowshoe hare Lepus americanus. European hare Lepus europaeus .
European rabbit Oryctolagus cuniculus . Cottontail rabbit
Sylvilagus floridanus. Caribou Rangifer arcticus. Reindeer
Rangifer tarandus. Pronghorn Antilocepra americana. Dall sheep
Ovis dalli.
-
-
- -
...
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o
.
.so
.
..
--
----
-
-
-
-
-
-
-
Fig. 1. Graphs of r, population growth rate (1 • aN/Nat) as a
function of population density.
A. r is a decreasing linear function of density.
.
r 18 an increasing linear function of density.
r is a decreasing nonlinear function of density, found in growing
daphnia populations (Smith, 1963b).
D. ris a decreasing nonlinear function of density, from the repro-
-
-
--
-
.
duction curves of Ricker (1954).
-
-
-
-
-
-
-
-
-
-
UNCLASSIFIED
ORNL DWG. 64-5930
B
AX


DENSITY O
DENSITY
T


F
O
DENSITY
DENSITY
Ī
TABLE I.
Populations subjected to regression analysis, the regression
coefficient (coefficient of r, on population density), and
results of tests of significance on the regression coefficient.
Column A: Nature of population and source of data. Scientific
names are in Appendix B.
B: Number of populations combined.
.
.
C: Average size of population(s).
D: Interval between counts or measurements
generation time (see text)
-
- -
E: Number of intervals (average number 1f two or more
populations have been combined).
F: Calculated value of (bp is the regression
coefficient of rt on popluation density, see text
for meaning of ky.
G:
Level of significance; the probability is less than
this figure that the null hypothesis (Ho: bp = 0)
is true.
* - see also further analysis in text.
B
C
D
E
F
G
5 day
Herbivore mite in culture.
(Huffaker, 1958)
2586
10 day
30
- 1.69
0.001
Predatory mite in culture.
(Huffaker, 1958)
27
5 day
20 day
30
- 3.00
0.02
Daphnia magna in cultures.
(Pratt, 1943)
4 day
A day
53
-1.27
0.02
Daphnia obtusa in cultures.
(Slobodkin, 1954)
284 10 days ag
- 1.54 0.001
Thrips in roses in garden.
(Davidson and Andre-
wartha, 1948)
month
1
352
20 hay
80
-0.24 0.20* ·
Table I, continued - 2
B
C
D
E
F
G
Florida wax scale on
citrus leaves
(Bodenheimer, 1958).
month
1
322
4 month
95
- 2.67
0.50
Chaff scale on citrus
leaves
(Bodenheimer, 1958)
1
282
month
4 month
87
- 1.71
0.001
Bordered white moth pupae
hibernating in forest floor.
(Varley, 1949)
year
year
2
127
- 0.24 0.30
.
Pine spinner 'moth larvae
hibernating in forest floor,
(Varley, 1949)
year
I'
672
year
-0.11
- 0.11
0.50
Pine hawk moth pupae hiber-
nating in forest floor.
(Varley, 1949)
11
1
:
38
year
year
49
- 0.23
0.40
Pine beauty moth larvae
hibernating in forest floor.
(Varley, 1949)
year
43
year
59
-0.58 0.30
Rice weevil in culture.
(Birch, 1953)
4 week
6 week
533.
- 1.55
0.001
Bean weevil in culture.
(Utida, 19578)
1
382
382
gener-
&tion
- lila
0.001
Red flour beetle in culture.
• (Park and Frank, 1950) 1
30 day
45 day
157
68
- 1.06
0.01
.
Confused flour beetle in
culture.
(Park et al., 1941;
Park and Frank, 1950)
1361
30 day
45 day
43
. 4.50
0.001
Parasitic wasp in culture.
(utida, 19578)
gener-
ation
111
- 0.61
0.001
Sheep blowfly in culture.
( Nicholson, 1954)
2 day
15 day
32
- 3.61
0.10
Housefly in culture.
(Pimentel et al., 1963) I
7 day
4 week
82
-1.92
0.20
Table I, continued - 3
C
D
E
F
G
month
Sewage fly in sewage beds.
(Hoyd, 1943)
1
1703
43 day
79
-0.67
0.01
Atlantic salmon weight of
· catch
(Allee et al., 1949)
(56,300
year
year
Lake trout abundance index
based on fisheries catch
(Eschmeyer, 1957)
(2218000
pounds)
year
5 year
25
+ 1.74
0.10
Goldeye abundance index
based on fisheries catch.
(Smith and Krefting, 1954) 1
year
(66,659
pounds)
2 year
23
- 2.37
0.30
year
( 39,100
pounds)
5 year
23
- 2.07
0.05
Northern pike abundance index
based on fisheries catch.
(Smith & Krefting, 1954) 1
Yellow perch abundance index .
based on fisheries catch.
(Smith & Krefting, 1954) 1
year
(212,365
pounds)
3 year
23.
- 3.16
0.05
Walleye abundance index
based on fisheries catch.
(Pycha, 1961)
1
( 258,034
pounds)
year
5 year
28
- 2.17
0.02
Freshwater drum abundance index
based on fisheries catch.
(Smith & Krefting, 1954) 1
(17,108
year
3 year
23
-0.51
0.30
5 year
23,000
year
14
-0.36
0.20
Fulmar breeding birds in
Britain.
(Fisher, 1962) 1
Gannet estimated world popu-
lation.
(Fsher and Vevers, 1944) i
202,600
5 year
5 year
24
-0.06
0.20
year
Heron nesting birds.
(ack, 1953)
("ack, 1953)**8
3
423
-0.59
0.01
Stork nesting birds.
(Lack, 1954)
year
4. year's
24
- 0.10 0.80
Ruffed grouse census in
habitat.
(Hickey, 1955; Marshall,
1954)
on w
year
278
year
30
- 2.46
0.05
Table I, continued - 4
..
.
A
B
C
D
E
F
G
Red Grouse game bag records
(Mackenzie, 1992;
Middleton, 1934)
6
year
year
66
- 1.20
0.001
Sharp-tailed grouse census
in habitat
(Hickey, 1955)
year
year
34
-0.33
0.05
Prairie chicken counts of
displaying males.
(Shelford & Yeatter, 1955) 1:
year
year
17
- 0.35
0.10
Partridge game bag records ;
breeding birds.
(Middleton, 1934;
Severtzoff, 1934)
year
year
86
- 0.92
0.001
Pheasant introduced popu-
lation
( Lack, 1954)
難 ​無 ​數顯 ​數顯 ​數顯 ​數顯 ​顯
​year
year
5
-0.89
0.10
Pied flycatcher breeding
population
(Lack, 1954)
year
year
9
-0.86
0.05
Coal tit breeding population
(Kl.uyver, 1952)
year
year
- 1.32
0.01
Blue tit breeding population
(Kluyver, 1951)
year
year
3.1
-1.90
0.001
Crested tit breeding
population
(Kluyver, 1951)
1
year
year
31
-0.81 0.02
House wren breeding bird
census
(Kendeigh, 1944; Kendeigh
et al., 1948-63)
year
year
24
- 0.62
0.001
Wood thrush breeding bird
census
(Williams, 1947-50)
1
year
year
18
- 0.56
0.10
TABLE I, continued - 5
А
B
C
D
E
F
G
Starling breeding bird
census.
(Kendeigh, 1944; Kendeigh
et al., 1948-63)
year
year
23
- 0.21
0.30*
Red-eyed vireo breeding bird
census.
(Kendeigh, 1944; Kendeith
et al., 1948-63; Williams,
1947-56)
38
year
year
20
- 0.36
0.05
Indigo bunting breeding bird
census.
( Kendeigh, 194/4; Kendeigh
et al., 1948-63)
1
.
44
year
year
23
-0.51
0.02
Wolverine fur yields.
(Buckley, 1954)
1
311
2 year
41
-0.31
0.20
year
Fisher fur yields.
(Keith, 1962)
860
2 year
38
- 1.12
0.001
Mink fur yields.
(Keith, 1962)
10,570
year
year
43
-0.05
0.70
Weasel game bag returns .
(Middleton, 1934)
year
year
49
- 0.52
0.001
Arctic fox fur yields .
(Buckley, 1954)
4772
year
year
41
-1.15
0.01
Coyote fur yields.
(Keith, 1962)
18,600
year
2 year
34
-1.18
0.001
Wolf fur yields.
(Buckley, 1954)
325
year
2 year
41
-0.52
0.05
-
-
UV
Gray fox bounty records.
(Richmond, 1952)
8724
year
year
34
- 0.15
0.20
Colored fox fur yields.
(Keith, 1962)
year
year
7927
36
-0.17
0.20
Lynx fur yields.
(Buckley, 1954; Eton
& Nicholson, 1942b;
Keith, 1962)
1932
year
year
55
• 0.23
0.001
TABLE I, continued - 6
,
A
B
C
D
E
-
Fur seal census on breed:ing
grounds.
(Scheffer, 1955)
year
1
547,300 3year
19
-0.69
0.01
month
1
8
1.5 mo
82
-0.06
0.30
Levant vole census in
habitat.
(Bodenheimer, 1958)
. Muskrat census in habitat,
fur yields.
(Buckley, 1954; Elton &
Nicholson, 1942a;
Errington, 1954 and 1957;
Keith, 1962)
6
207,379
year
year
48
-0.41
0.001
Snowshoe hare fur yields.
(Lack, 1954; MacLulich,
1957)
2
112,682
year
year
37
-1.90
0.05
European hare game bag
records .
(Middleton, 1934)
2
209
year
year
- 0.52
0.001
year
5,071
8 month
66
-0.20
0.001
302
year
2 year
- 0.20
0.01
European rabbit game has
records.
(Middleton, 1934)
2
Reindeer introduced population.
(Scheffer, 1951)
· Pronghorn census in habitat.
(Shelford, 1954) tete z
Human population of the
world.
(various sources)
year
545
year
18
- 1.14 0.02
1
844x206
17 year 6
+0.11 0.05
NOTE: The following censuses were eliminated from the analysis because the
. series of counts were not significantly different from a series of
random numbers: Clams in ocean beach (Coe, 1957). Broad-horned
flour beetle in culture (Park et al., 1941). Granary beetle in
TABLE I, continued - 7
culture (Park et al., 1941). Lake trout fisheries catch (Fry,
1949). Walleye fisheries catch (Smith and Krefting, 1954).
Bobwhite quail census in habitat (Errington, 1957). Ruffed grouse
census in New Jersey habitat (Hickey, 1955). Great tit breeding
population (Kluyver, 1951.). Ovenbird breeding bird census
(Williams, 1947-50). Hooded warbler breeding bird census
(Williams, 1947-50). Arctic fox fur yields (Elton, 1942). Mink
fur yields (Buckley, 1954).
-
-
-
-
-
...
-
u
-..
:
.
.
.
.
.
.
TABLE II. Summary of results in Table I.
..
Colum A:
Total number of species.
B: Number of species eliminated because their popu-
lation record was not significantly different from
a random series.
C: Coefficient, by/k, not significantly different
from zero, probability of the null hypothesis
exceeding :05.
D: Coefficient positive and significantly different
from zero.
E: coefficient negative and significantly different
from zero.
A_
B
C_
E
.
Invertebrates other
than insects
1
.
i
Insects
m
Fish
7
-
3
-
w
Birds
a
Mammals
*
.
29
72.
3
2
7
13
42
Totals
"The human population of the world.
TABLE III. Results Based on Spring and Fall Censuses of Wild Populations.
Column A: Species and source of data.
B: Number of populations combined.
C: Average size of spring population(s).
D: Number of intervals (average number for the six
mus krat populations).
E: Calculated value of b/k.
F: Level of significance; the probability is less than this
· figure that the null hypothesis (H : D = 0) 18 true.

:
B
C
D
Spring to Fall
Fall to Sprin
FD
E
Bobwhite Quail,
1
145
17
-0.47 0.01
18
-0.01
.40
(Errington, 1957)
1
405 | 6
-0.50
0.05
5
+ 0.01
40
-
Phasianus
(Lack, 1954)
-
-
...-.
-
-
Ruffed Grouse,
Bonase
(Bump et al., 1947)
1
177
23
-0.44
0.05.12
-0.07
.20
-
-
+
+
Muskrat,
Ondatra.
(Errington, 19
283
16
-0.46 0.02
15
-0.01
.90
1957)
-
-
and owls during fall and winter, estimated from the data of
Craighead and Craighead (1956).
Meadow
Vole
Prey Species
White-foot Cottontail
Mouse
Rabbit
FOX
Squirrel
1941-42 . .
Population
Lost by predation
300
300
303,000
55,600
0.184
33,000
8,066
10
162
0.540
Predation rate
0.244
0.033
1947-48
Population
1,000
75,000
13,500
0.180
27,000
6,200
1,200
234
Lost by predation
Predation rate.
ll
0.230
0.195
0.011
=
-
-
+
-
=
-
-
-
-
·
TABLE V. Insect species from natural habitats analyzed during their
periods of scarcity. See Table I for more information.
Column A: Species.
B: Average size of population (during period of
scarcity).
.
C: Number of intervals.
D: Calculated value of b/k.
E: Level of significance.
_
Florida wax scale
54.7
ô
- 0.12
- 2.12
47.3
ô
w & Bņģle
.
--
- 0.42
15.9
ô
Bordered white moth
Pine spinner moth
Pine hawk moth
Pine beauty moth
**
8.7
- 1.32
0.20
.
4.9
- 0.89
0.01

.
2
A
..EY
END
-
sy
*
.
27
DATE FILMED
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el
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..
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.
.
.
.
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.
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"
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