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et al.Richard M. Sibly,
Birds, Fish, and Insects
On the Regulation of Populations of Mammals,
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List, and K. M. A. Chan for comments on the manuscript.
This study was supported by the National University of
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and the John Simon Guggenheim Memorial Foundation.
Supporting Online Material
Materials and Methods
25 April 2005; accepted 17 June 2005
On the Regulation of
Populations of Mammals,
Birds, Fish, and Insects
Richard M. Sibly,1* Daniel Barker,1Michael C. Denham,2
Jim Hone,3Mark Pagel1
A key unresolved question in population ecology concerns the relationship
between a population’s size and its growth rate. We estimated this rela-
tionship for 1780 time series of mammals, birds, fish, and insects. We found
that rates of population growth are high at low population densities but,
contrary to previous predictions, decline rapidly with increasing population
size and then flatten out, for all four taxa. This produces a strongly concave
relationship between a population’s growth rate and its size. These findings
have fundamental implications for our understanding of animals’ lives,
suggesting in particular that many animals in these taxa will be found living
at densities above the carrying capacity of their environments.
The way a population_s size changes through
time—its dynamics—depends on the way it
grows when smalland declines when big.More
specifically, the dynamics result from the
precise relationship between the population_s
size (N) and its per capita growth rate (pgr),
1). The simplest case is a straight-line rela-
tionship, such that pgr declines linearly with
increasing N (Fig. 1A, left). Linearity produces
the well-known logistic population growth
equation NðtÞ 0
parameter representing pgr when N 0 0, N0is
the size of the population at time 0 0, and K is
the population_s carrying capacity (1).
The relationship between pgr and N is
generally taken to be monotonic and de-
creasing and can be either concave or convex
(2). Convex relationships (Fig. 1B) imply that
pgr varies little until population size is near
carrying capacity, then drops rapidly. Con-
cavity (Fig. 1C) means that pgr is initially
relatively high, so small populations grow
quickly, but pgr then declines rapidly as
, where t is time (Fig.
ðKjN0Þejr0tþ N0, where r0is a
population size increases, later flattening out
so that the approach to carrying capacity is
relatively slow. In a variant possible in theory
and occasionally reported in nature, the slope
of the relationship between pgr and N be-
comes positive in small populations, such that
pgr actually increases with N over a narrow
range of population sizes, giving an Allee
The way in which pgr declines with
population size is conventionally modeled by
the theta-logistic equation, given by
pgr 0 r0E1 j ðN=KÞq^
where r0and K are as before, and q is a
parameter describing the curvature of the
relationship (2). In practice, population density
is sometimes used in place of population size,
and r0is best replaced by rm, representing pgr
when population size N is at a defined low
value, corresponding to a population of, for
example, one individual (5) (Fig. 2). Values of
q greater or less than 1 correspond to convex
and concave relationships, respectively (Figs. 1
and 2). Mechanistically, the value of q must
depend on the ways that animals interact at dif-
ferent densities (6).
There has been a persistent suggestion that
the shape of the pgr-density relationship
depends on a species_ life history (5, 7, 8).
The widely cited argument (9–14) is as follows.
Large, long-lived species generally live close to
the carrying capacity of their environments,
being limited mainly by resources, and are only
rarely subject to natural selection for increased
performance at low population density. As a
consequence, these species_ population growth
rates are relatively unaffected until populations
are nearing carrying capacity, producing the
convex curve of q 9 1 seen in Fig. 1. By con-
trast, species that spend most of their time at
densities much lower than carrying capacity are
selected for a high maximum rate of increase.
As a result, these species are affected even at
relatively low densities in their abilities to ac-
quire foods, and so the concave relationship of
q G 1 between pgr and N arises. There are a
number of cases of density dependence that
together have suggested that pgr-density rela-
tionships are convex for large mammals and
similar species but concave for species with life
histories like those of insects and some fish
(5, 7, 15).
The form of the pgr-density relationship has
implications beyond population dynamics, and
it is used to make predictions and to analyze
management options in areas such as conser-
vation (16), pest management (17), risk assess-
ment (18), and disease epidemiology (19). In
spite of this, there have been few attempts to
establish generalities about how pgr varies
with population size (5, 15, 20). Here we
analyze an extensive compilation of time se-
ries data from 4926 different populations in
the Global Population Dynamics Database
(GPDD) (21, 22). The GPDD is a collection of
time series of population counts or indices of
these, together with other taxonomic details
of more than 1400 species.
After exclusion of time series that were very
short, did not vary, or contained zeros, the
GPDD contained 3766 time series from 1084
species (table S1). We further excluded 469
series (12%) that showed a significantdecline in
size with time, because unknown factors may
have prevented population recovery and biased
the form of the estimated pgr-density relation-
ship, and 1% that showed positive density
dependence (i.e., pgr increased with density),
because these show no evidence of population
regulation. We examined the remaining 3269
series for evidence of Allee relationships, but
these were rare if present at all: There were
only 20 cases in which a quadratic regression of
pgr on N fitted better (P G 0.05) than a linear
regression, with a turning point of the required
1School of Animal and Microbial Sciences, University
of Reading, Whiteknights, Reading RG6 6AJ, UK.
2Statistical Sciences Europe, GlaxoSmithKline Re-
search and Development Limited, New Frontiers
Science Park, Third Avenue, Harlow, Essex CM19
5AW, UK.3Institute for Applied Ecology, University of
Canberra, Canberra ACT 2601, Australia.
*To whom correspondence should be addressed.
R E P O R T S
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on May 14, 2008
type and within the range of observed values of
N, and in only 5 was the pattern clearly non-
monotonic (22). We conclude that there is clear
evidence of Allee effects in only a small
minority (0.2%) of GPDD data sets. Absence
of Allee effects in bird studies has been noted
by Saether et al. (23).
WefittedEq.1 to each of the 3269 tractable
time series in the GPDD using a nonlinear
least-squares procedure and discarded cases
yielding relatively imprecise estimates of q
(22). The main taxonomic groups represented
were birds (150 species), mammals (79), bony
fish (64), and insects (381). Figure 3 shows the
frequencies of occurrence of fitted values of q
in each of the main taxonomic groups. In each
taxon, there are far more instances where q G 1
(concave) than where q 9 1 (convex). The av-
erage proportion of cases where q G 1 is 0.78.
This is higher than the proportion (0.62) found
in the only comparable study (5), but that was
based on only 13 species of birds. The pro-
portion of cases where q G 1 differs only a little
between the major taxonomic groups, though
there is a suggestion that the proportion is
higher in fish than in mammals, birds, and
insects (taking the average value of q for each
species and counting species, c3
significant) (Fig. 3).
It is not possible to apply explicit phyloge-
netic methods to these data because no phylog-
eny exists to describe them, but we repeated
analyses using genus means and then family
means as a way of controlling for the lack of
independence among species (counting genera,
P G 0.05). q was significantly (P G 0.05)
different from 1 in 613 of the 1780 time series
analyzed, being less than 1 in 581 cases and
greater than 1 in 32. The reason only 613
estimates were significantly different from 1 is
that some of the estimates are imprecise. Our
strategy for dealing with imprecision was to
remove very imprecise estimates (22), but this
retained quite a number that still had wide
confidence intervals. Among the small propor-
tion (0.22) of cases where q 9 1, there are no
obvious taxonomic or other patterns: All major
taxonomic groups are represented. Our results
suggest that in mammals, birds, fish, and
insects, population regulation is generally the
result of a concave relationship between a
population_s growth rate and its size.
The histograms of q suggest that q is
normally less than 1, but they do not directly
test the hypothesis that q would increase with
a species_ body size (7, 8, 15). However, there
is no suggestion of such a relationship in any
of the four taxonomic groups we analyzed.
In mammals, the reverse is the case: The
relationship is negative, not positive as pre-
dicted Eregression with one point per species:
r360 –0.32, P G 0.05 (fig. S2); with one
point per family or genus, the relationship is
still negative: r110 –0.253, nonsignificant, and
20 6.9, non-
20 8.2, P G 0.05; counting families, c3
r280 –0.291, nonsignificant, respectively^. Our
results, based on a much larger data set than
previous analyses, appear to rule out the pos-
sibility that the shape of the pgr-density rela-
tionship is strongly associated with taxonomy
or body size.
Fig. 1. (A to C) (Left)Therelationshipsbetweenpopulationgrowthrates(pgr) and size (N) with (right)
their associated population time series. The observed values on the left are calculated from the time
series, and the fitted curves are of the type of Eq. 1. The data come from three insect populations in
the GPDD with (A) q , 1 (Acyrthosiphon pisum, GPDD main ID 8383), (B) q 9 1 (Inachis io, ID 3276),
(C) q G 0 (Xylena vetusta, ID 6321). The form of pgr-N relationships are specific to the time and place
in which the data were collected (32).
Fig. 2. Illustration of the
curves generated by the
theta-logistic equation (Eq.
1) for different values of q.
N represents population size
or density. Each curve is
constrained to go through
(1, 0.1) and (100, 0); thus,
the minimum population
size is 1 and rm0 0.1 and
K 0 100. There is no partic-
ular significance in our
choice of N 0 1 for the
lower constraint; similar
families of curves are ob-
tained at other values of N,
provided that these are
nonzero and small in com-
parison with K (supporting online text).
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Values of q around zero can arise from
measurement error (24), and so it is important
to exclude that possibility here (25). If all
variation in population size arises from random
measurement error, it is straightforward to
derive the predicted relationship between pgr
and the logarithm of population density. The
relationship is linear with slope –1.0. We an-
alysed the slope of the relationship between pgr
and the logarithm of population density within
each of our four taxonomic groups and for
simulated data in which all variation in density
was random (Fig. 4). In three of our four groups,
the slopes differed markedly from –1.0 (t tests,
P ¡ 0.001), suggesting that measurement error
does not have a dominant role in these groups.
The simulated data have, as expected, a mean
very close to –1.0, and our insect populations
also fall close to –1.0, possibly suggesting that
measurement error has affected the estimates of
q in these populations. As a further check, we
used the GPDD grading of time series to
indicate perceived quality, which may in some
cases be affected by measurement error (21).
Restricting our analysis to the top two grades
out of five did not affect our conclusions.
An important implication of our finding, that
the relationship between pgr and density is
generally concave, is that many animals may
spend most of their time at or above carrying
capacity. To see this, consider the effects of
variations in population size induced by density-
independent environmental factors. Assume
that increases and decreases occur with similar
magnitudes and frequencies and that q G 1. For
q G 1, returns to carrying capacity are faster if
the population is below than if it is above
carrying capacity. The rate of return is given by
the absolute value of pgr, and the rate of return
is faster from a point below carrying capacity
than from a point an equivalent distance above
carrying capacity. This is seen in the dynamics
of the population of Xylena vetusta (Fig. 1C)
for which the estimated carrying capacity
(where pgr 0 0) is 512. Note that upward steps
are generally larger than downward. The result
is that populations spend more time above than
below carrying capacity. This process will
produce a tail extending to the right in the fre-
quency distribution of population size. In line
with this prediction, 88% of the 1849 GPDD
cases analyzed here are positively skewed (P G
0.001, mean skewness 1.08 T 0.024). Halley
and Inchausti (26) obtained a similar result.
Because bird and mammal populations may
generally be regulated by their food supplies
(27–29), our finding that most individuals live
in environments above carrying capacity sug-
gests they have less food than is needed for
population replacement. However, other fac-
tors, such as predation and social interactions
within the species, may in some circumstances
override the role of food.
Factors whose effects are not felt immedi-
ately may also be important in determining
population growth or decline (30), and we
considered carefully the possibility of including
time delays in our analysis. Adding two time
lags would have added a minimum of two extra
parameters to be estimated. Our conclusion was
that the additional complexity to the model was
not warranted, given the quality of data sets in
our analysis. However, we believe it would be
interesting to explore the possibility of includ-
ing time lags in future studies.
Our conclusion that the most common pgr-
density relationships are concave in birds,
mammals, fish, and insects should have wide
implications for understanding how the abun-
dance and dynamics of populations are con-
trolled and for our practical ability to make
predictions about how such species respond to
environmental change. For example, if a linear
relationship is assumed and values of rmand K
are estimated from other sources—for instance,
rmis sometimes estimated from life-history
data in optimal environments—then concavity
means that pgr is overestimated when the
population is below carrying capacity (Fig. 2).
This would have dangerous consequences in
wildlife and fisheries management, because
populations would recover from disturbances
more slowly than predicted. Pest control, by
contrast, would be more successful than ex-
pected. Knowledge of the shapes of the pgr-
density relationship is required in all areas of
population ecology to make projections as to
future abundance and population dynamics
(18, 28, 29, 31).
References and Notes
1. A. Tsoularis, J. Wallace, Math. Biosci. 179, 21 (2002).
2. P. Turchin, Complex Population Dynamics (Princeton
Univ. Press, Princeton, NJ, 2003).
3. P. A. Stephens, W. J. Sutherland, Trends Ecol. Evol.
14, 401 (1999).
4. F. Courchamp, T. Clutton-Brock, B. Grenfell, Trends
Ecol. Evol. 14, 405 (1999).
5. B.-E. Saether, S. Engen, E. Matthysen, Science 295,
6. T. Royama, Analytical Population Dynamics (Chapman
& Hall, London, 1992).
7. C. W. Fowler, Ecology 62, 602 (1981).
8. M. E. Gilpin, F. J. Ayala, Proc. Natl. Acad. Sci. U.S.A.
70, 3590 (1973).
9. A. R. E. Sinclair, Philos. Trans. R. Soc. London Ser. B
358, 1729 (2003).
10. P. Bayliss, D. Choquenot, Philos. Trans. R. Soc. London
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11. J. Lindstrom, H. Kokko, Ecol. Lett. 5, 338 (2002).
12. R. Lande, B.-E. Saether, S. Engen, Ecology 78, 1341 (1997).
Simulation Insects FishBirdsMammals
Fig. 4. Slopes of regressions of pgr versus loge
density, showing that the observed relationships
are not simply a result of measurement error.
Measurement error alone predicts a slope of –1
in regressions of pgr against logedensity, in
marked contrast to fitted slopes, except in the
case of the insects. For comparison, we also
show the effects of measurement error sim-
ulated with 3920 time series of length 30 and
processed as for the GPDD data sets. Bars show
one standard error of the mean. Frequency
distributions of these regression coefficients are
shown in fig. S3.
Fig. 3. Histograms of q for the four major taxonomic groups in the GPDD database: (A) mammals,
(B) birds, (C) fish, and (D) insects. A hybrid scale is used for q, linear between –1 and 2 and log10
elsewhere. This scale is used to give similar weights to each of the principal regions of interest in
Fig. 2. Where there existed within-species replication, we used the average value, so that each species
is here represented only once.
R E P O R T S
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on May 14, 2008
13. B.-E. Saether, Trends Ecol. Evol. 12, 143 (1997).
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B 357, 1153 (2002).
16. S. R. Beissinger, D. R. McCullough, Eds., Population
Viability Analysis (Univ. of Chicago Press, Chicago, 2002).
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Management (Blackwell Science, Cambridge, MA, 1994).
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Eds., Ecological Modeling in Risk Assessment (CRC/
Lewis, London, 2002).
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Humans: Dynamics and Control (Oxford Univ. Press,
20. P. Turchin, Oikos 84, 153 (1999).
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for Population Biology, Imperial College (1999), Global
Population Dynamics Database, available at www.sw.
22. Materials and methods are available as supporting
material on Science Online.
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24. T. M. Shenk, G. C. White, K. P. Burnham, Ecol.
Monogr. 68, 445 (1998).
25. To identify the effects of measurement error on q
estimation, we carried out computer simulations of
time series governed by Eq. 1 but subject to log-
normal environmental perturbations and with mea-
surement errors also being log-normally distributed.
Preliminary results suggested that over the range
of parameters of interest, q can be recovered with-
out appreciable bias, provided that measurement
error is less than half of environmental variation,
and that useful information is still obtainable when
measurement error and environmental variation are
26. J. Halley, P. Inchausti, Oikos 99, 518 (2002).
27. A. R. E. Sinclair, in Ecological Concepts, J. M. Cherrett,
Ed. (Blackwell Scientific, Oxford, 1989), pp. 197–241.
28. A. R. E. Sinclair, in Frontiers of Population Ecology, R. B.
Floyd, A. W. Sheppard, P. J. De Barro, Eds. (CSIRO,
Melbourne, 1996), pp. 127–154.
29. A. R. E. Sinclair, C. J. Krebs, Philos. Trans. R. Soc.
London Ser. B 357, 1221 (2002).
30. R. Lande, S. Engen, B.-E. Saether, Stochastic Popula-
tion Dynamics in Ecology and Conservation (Oxford
Univ. Press, Oxford, 2003).
31. S. Jennings, M. J. Kaiser, J. D. Reynolds, Marine
Fisheries Ecology (Blackwell Science, Oxford, 2001).
32. C. J. Krebs, Philos. Trans. R. Soc. London Ser. B 357,
for Population Biology at Silwood Park for generous and
efficient help in supplying the data and to B.-E. Saether
and A. Berryman for extensive constructive comments
on an earlier version of the manuscript. Supported by
NERC grant no. NER/B/S/2001/00867 (R.M.S. and M.P.).
Supporting Online Material
Materials and Methods
Figs. S1 to S3
7 February 2005; accepted 18 May 2005
Host Suppression and Stability in
a Parasitoid-Host System:
William Murdoch,1* Cheryl J. Briggs,2Susan Swarbrick1
We elucidate the mechanisms causing stability and severe resource suppression
in a consumer-resource system. The consumer, the parasitoid Aphytis, rapidly
controlled an experimentally induced outbreak of the resource, California red
scale, an agricultural pest, and imposed a low, stable pest equilibrium. The
results are well predicted by a mechanistic, independently parameterized model.
The key mechanisms are widespread in nature: an invulnerable adult stage in
the resource population and rapid consumer development. Stability in this
biologically nondiverse agricultural system is a property of the local interaction
between these two species, not of spatial processes or of the larger ecological
Although some consumer-resource (e.g.,
predator-prey) populations famously cycle in
abundance, most appear to be stable, even
when the predator strongly suppresses prey
abundance (1). Yet, any theory that includes
only a few basic predator properties—time
lags and limited killing capacity of individual
predators—generally predicts instability, i.e.,
large-amplitude oscillations or even predator-
driven extinction of the prey (2, 3). Model
stability is particularly difficult to achieve when
the predator can drive the prey to densities far
below the limits set by the prey_s own resources
Ethe Bparadox of enrichment[ (4)^, and almost
all theoretically stabilizing mechanisms achieve
stability only by causing the prey density to
increase close to that limit (1).
California red scale (Aonidiella aurantii),
an insect pest of citrus worldwide, is controlled
by the parasitoid Aphytis melinus (5). This
system exemplifies in extreme the features—
ecological simplicity, high productivity, and
severe suppression of the pest—that should
engender instability. (i) It is an almost pure
specialist consumer-resource interaction. Citrus
groves contain, in addition to red scale, only a
few, scarce, herbivore species. Under biolog-
ical control, red scale are attacked mainly by
Aphytis melinus; one other parasitoid and one
or two predator species are typically present
but scarce. (ii) Citrus provides a rich resource
for scale. deBach (6) showed that when
dichloro-diphenyl-trichloroethane (DDT) was
applied to citrus trees (which killed Aphytis but
not the resistant scale), scale outbreak density
reached several hundred times higher than
controlled populations and was not brought
back under control for more than 3 years
(presumably, when Aphytis was able to re-
invade the tree). Yet, in our study area, red
scale under control have persisted for 940
years (80 scale generations) with little tempo-
ral variation, at densities G1% of the limit set
by the citrus plant.
Over two decades, we and our colleagues
have tested and ruled out many mechanisms
by which Aphytis might achieve this remark-
able control with stability, including parasitoid
aggregation to, or independent of, local host
density (7), as well as density-dependence in
the parasitoid sex-ratio (8). Stability also does
not depend on spatial processes, including
metapopulation dynamics. Dynamics were not
altered when a spatial refuge from parasitism
was removed, or when populations in indi-
vidual trees were isolated from the larger
population in the grove (9): Control and sta-
bilizing mechanisms act locally within a single
tree. Feasible remaining mechanisms explored
in models involve life-history details, e.g., a
long adult host stage invulnerable to parasitism
(10). In previous studies, we could not detect
temporal density-dependence in parasitism, host-
feeding, or predation (11), a difficult task within
the narrow range of densities of a stable system
near equilibrium (12). A density-perturbation ex-
periment mightuncover both density-dependence
and the mechanisms causing return to equi-
librium. Density manipulations at the appro-
priate spatial scale typically are logistically
daunting, but in the Aphytis–red scale sys-
tem, the appropriate spatial scale is the in-
dividual tree (9).
We created experimental red scale out-
breaks (13). We caged individual trees and
increased scale recruitment over a period
somewhat longer than it takes scale to develop
from birth to adult (this development period
defines the time unit, t). We followed the
dynamics of these outbreak populations, to-
gether with caged and uncaged control pop-
ulations, over three to five scale development
times. Three separate experiments gave the
same result. We present only the third ex-
periment, which had four outbreak trees.
Control of the outbreak and stability—
return to equilibrium density—occurred rap-
idly (Fig. 1). Scale density began to decline
even before crawler additions stopped and
before one scale development period had
passed, and most suppression occurred by t 0 2;
i.e., within 2 months after we added scale. By
1Department of Ecology, Evolution and Marine Biolo-
gy, University of California, Santa Barbara, CA 93106,
USA.2Department of Integrative Biology, University
of California, Berkeley, CA 94720–3140, USA.
*To whom correspondence should be addressed.
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on May 14, 2008