Ecological Entomology (1998) 23, 216–222
CRITICAL APPRAISAL
Critical appraisals allow the analytical review of existing knowledge on current topics of significance in ecological entomology.
They should assess the worth or quality of the work in the field and suggest areas for investigation.
Cycles in insect populations: delayed density
dependence or exogenous driving variables?
M A R K D . H U N T E R 1 and P E T E R W . P R I C E 2
U.S.A. and
2Biological
1Institute
of Ecology, University of Georgia,
Sciences, Northern Arizona University, U.S.A.
Summary. Delayed density dependence, and the cycles in insect populations that it
can generate, are often investigated using time-series analysis. Recently, several
authors have raised concerns about the validity of using time-series analysis to detect
density dependence. One particular concern is the suggestion that exogenous driving
variables, such as cyclic weather patterns, can lead to the spurious detection of density
dependence in natural populations.
Using non-biological data (the electricity bills of one of the authors), we show how
easy it is to be misled by the results of time-series analysis. We then present 16 years’
data on the gall-forming sawfly, Euura lasiolepis (Hymenoptera: Tenthredinidae), and
show that cycles in weather, specifically winter precipitation, lead to the spurious
detection of density dependence in time-series analysis. We conclude that time-series
analysis cannot stand alone as a method for inferring the action of density dependence,
and urge further investigation of the effects of apparent cycles in abiotic forces on
insect populations.
Key words. Density dependence, Euura lasiolepis, insect population dynamics,
population cycles, time-series analysis.
Introduction
Cycles in insect populations are usually attributed to delayed
density–dependent interactions between insects and their food,
competitors, or natural enemies (Turchin, 1990; Royama, 1992;
Berryman, 1994). A second potential source of population
cycles that has received less attention is periodic fluctuations
in abiotic factors (Burroughs, 1992; Swetnam & Lynch, 1993).
If weather patterns are both cyclic and major determinants of
insect population change, then weather can generate insect
population cycles (Williams & Liebhold, 1995, 1997). Timeseries analysis (Box & Jenkins, 1976) is a technique that is
commonly used to indicate the time-lag on which negative
feedback processes are apparently acting. If significant lags
are detected in population data using time-series analysis,
Correspondence: Mark D. Hunter, Institute of Ecology, University
of Georgia, Athens, GA 30602–2202, U.S.A.
E-mail: [email protected]
216
delayed density dependence is often assumed to be operating
(Turchin, 1990; Berryman & Turchin, 1997). However, timeseries analysis cannot distinguish delayed density dependence
from the action of an exogenous driving variable, such as
weather, if weather patterns, themselves, are cyclic (Williams
& Liebhold, 1995, 1997). This inherent weakness of timeseries analysis adds to the general difficulty of demonstrating
the presence of density dependence in natural populations
(Pollard et al., 1987; Pollard, 1988; Stiling, 1988; Wolda &
Dennis, 1993; Dennis & Taper, 1994).
In this article, we wish to emphasize the dangers of inferring
the operation of density-dependent factors on insect populations
from time-series analysis alone. First, we subject a nonbiological data set (the cost of power consumption in the home
of one of the authors) to time-series analysis and show
cyclic changes in power costs. In the absence of background
information on the data, it might be concluded from the analysis
that delayed density dependence was generating the cycle.
In reality, of course, the cycle reflects seasonal changes in
temperature and concomitant uses of power: temperature is
© 1998 Blackwell Science Ltd
Cycles in insect populations
Table 1. Time-series analysis of monthly domestic electricity bills
for a house in Athens, Georgia, U.S.A. from February 1995 to June
1996. Data represent the r2 values of the Pearson product-moment
correlation coefficients for regressions between the per capita rate of
change of electricity costs with previous electricity costs at time t–1,
t–2, t–3 and t–4. d.f. 5 degrees of freedom in the analysis; NS 5 nonsignificant correlations.
Time
r-squared
d.f.
Significance
t-1
t-2
t-3
t-4
0.379
0.562
0.519
0.477
15
14
13
12
NS
P 5 0.02
P 5 0.05
NS
acting as an exogenous driving variable, and the apparent
detection of density dependence is spurious. Second, we subject
an insect population time-series of a similar number of timesteps to the same type of analysis. Again, apparent negative
feedback is detected in the series. Using a combination of lifetable data and field experiments, we show that changes in the
population density of this insect are also more readily explained
by the action of an exogenous driving variable than by the
action of any density dependence in the system.
Methods
Time-series 1: power costs
Monthly electricity bills from a home in Athens, Georgia,
U.S.A. from February 1995 to June 1996 were subjected to
time-series analysis (17 months of data, Table 1). We calculated
a pseudo per capita rate of increase, r, for electricity costs for
each pair of months in the time-series, using the equation rt 5
ln (Nt/Nt–1), where Nt is the power cost at time t. We then
correlated the ranges of r for each pair of months in the timeseries with power costs at time t–1, t–2, and t–3 to assess the
time-lag on which any apparent negative feedback process
might be operating. Although we recognize the inherent
autocorrelation that follows such analyses, and the statistical
difficulties that ensue (Williams & Liebhold, 1995, 1997), we
do not have sufficiently long time-series to apply maximum
likelihood techniques (Dennis & Taper, 1994) with any
significant increase in rigour. We follow previous authors (e.g.
Turchin, 1990) in using P-values from t–2 regressions (and
those with longer lags) to estimate the significance of those
relationships, and stress that the significance of t–1 relationships
should be viewed with caution. We then calculated a two-lag
model by stepwise regression (Turchin, 1990) to estimate the
significance of the t–2 and t–3 lags that resulted from timeseries analysis (below). Although time-series of 30–40 steps
are considered ideal for detecting factors that commonly
influence insect populations (Royama, 1992), shorter timeseries are commonly used to infer the action of delayed density
dependence (Turchin, 1990; Berryman, 1994) because longerterm data are rare.
© 1998 Blackwell Science Ltd, Ecological Entomology, 23, 216–222
217
Time-series 2: a gall-forming sawfly
Abundance of the univoltine gall-forming sawfly, Euura
lasiolepis (Hymenoptera: Tenthredinidae), was estimated each
year for 16 years from fifteen willow clones near Flagstaff,
Arizona, U.S.A. Details of the life history of E. lasiolepis on
its willow host, Salix lasiolepis, are well documented (Price
& Craig, 1984; Price, 1994). Briefly, adult sawflies generally
emerge in the spring, but in variable synchrony with the rapid
development of young willow shoots. Female sawflies oviposit
in the shoots and initiate gall formation. Larvae develop in the
gall, pupating the following spring. Individual willow genotypes
(genets) spread by layering, and the resulting clonal shrub is
composed of many stems (ramets) growing from the soil (Craig
et al., 1986).
The densities of E. lasiolepis were estimated between 1981
and 1996 from fifteen willow clones (Table 2). Willow clones
were situated on Museum of Northern Arizona property in
Flagstaff, Arizona, 2300 m above sea level. Twelve of the
clones were growing along a temporary stream bed that only
flows during snow-melt in some years. We characterize these
as dry-site clones. Three additional clones were associated
with high water supply (a spring or regular source of runoff),
and we characterize these as wet-site clones. The rarity of wet
sites in this region precluded use of more in the sampling
regimen. Dry-site clones were sampled from 1981 to 1996,
and wet-site clones from 1983 to 1996. All clones were within
1 km of each other. Densities of E. lasiolepis were estimated
annually after leaf fall in winter by throwing a tape over a
willow clone, and counting the number of galls on 100 shoots
touched by the tape. Ten replicates per clone were pooled to
give a single estimate of gall density per 1000 shoots per clone
per year.
Sawfly densities were, on average, 13-fold higher on wetsite clones than on dry-site clones (Table 2). As with the data
on electricity costs, we used time-series analysis to establish
the dominant time-lags in the action of apparent negative
feedback processes on sawfly populations on each clone by
regressing the per capita rate of change of sawflies with sawfly
density at times t–1, t–2, and t–3 (above). We also subjected
local precipitation data to time-series analysis (Table 2).
Precipitation per month from 1981 to 1996 was provided in
weather records from Flagstaff airport, 11 km from the study
site, and pooled into a single estimate of precipitation for
October–May for each year. Six years’ precipitation data from
a private weather station 1.5 km from the study site demonstrate
that the airport precipitation records accurately reflect
precipitation at the study site (Y 5 3.84 1 0.86X, n 5 6, r2 5
0.97, P , 0.01; Price & Clancy, 1986).
Sawfly survivorship was estimated over a 10-year period
(1981–90). Galls were collected from each clone in the spring
of each year, after pupation but before the emergence of adults.
From 1981 to 1986, galls were dissected and sawflies scored
as alive, dead or parasitized. From 1987 to 1990, galls were
reared to emergence in the laboratory, and scored as before.
Before dissection or rearing, galls were scored visually for
predation by mountain chickadees (Parus gambeli). Overall,
sawfly mortality was partitioned among parasitism by
218
Mark D. Hunter and Peter W. Price
Table 2. Densities of the gall-forming sawfly, Euura lasiolepis, on fifteen clones of the willow, Salix lasiolepis, precipitation (mm) from October
to May at the sawfly sampling sites, and monthly electricity bills (U.S. Dollars) of one of the authors. Twelve clones are at dry sites, and three at
wet sites. N refers to the number of values in the time series. The lag times t–1, t–2 and t–3 refer to the results of time-series analysis, and represent
the lag in which apparent negative feedback processes are operating on the series. Asterisks represent the significance of the Pearson productmoment correlation coefficient of regressions between per capita rates of change, r, and density at times t–1, t–2 and t–3 (*P , 0.05, **P , 0.01,
***P , 0.001, NS 5 not significant).
Clone
N
Mean
Min
Max
t–1
t–2
t–3
MNA1
MNA2
MNA3
MNA4
MNA5
MNA6
MNA7
NP4
NP5
NP7
NP8
NP9
Dry-site mean
BD1
CS1
CS2
Wet-site mean
Precipitation
Electricity costs
16
16
16
16
16
16
16
16
16
16
16
16
16
15
14
14
14
16
17
31.56
609.50
186.63
11.38
14.44
77.75
121.19
52.88
101.88
85.38
236.56
131.94
107.00
1552.73
1162.00
1141.50
1447.75
391.41
38.77
0.00
27.00
13.00
0.00
0.00
4.00
0.00
1.00
4.00
4.00
15.00
21.00
7.00
36.00
0.00
144.00
60.00
40.89
24.43
162.00
1287.00
590.00
55.00
42.00
291.00
433.00
181.00
258.00
265.00
783.00
374.00
330.00
3914.00
4052.00
3720.00
3825.00
831.09
69.19
*
NS
*
*
*
NS
*
NS
NS
*
NS
*
NS
NS
NS
NS
NS
*
NS
NS
*
*
*
NS
**
**
**
**
*
**
*
**
NS
***
***
**
NS
*
NS
NS
NS
NS
NS
NS
NS
NS
NS
NS
NS
NS
NS
**
*
*
**
*
*
Pteromalus sp. (Hymenoptera: Pteromalidae) with very minor
contributions from other parasitoid species, predation by
chickadees, and host-plant factors (Price & Craig, 1984; Price
& Clancy, 1986). One hundred galls were sampled per clone
per year where possible, but sample size varied with availability
(mean sample size 5 85.51 6 4.27). The percentage mortalities
caused by parasitoids or chickadees at time t were correlated
against sawfly population densities on each clone at times t,
t–1, and t–2 to test for direct or delayed density-dependent
mortality. Per cent mortalities among clones were also
correlated with sawfly density in each year to test for spatial
density-dependent mortality.
Results and discussion
Time-series 1: power costs
Time-series analysis of electricity costs detected apparent
negative feedback operating on time-lags of t–2 and t–3
(Table 1), combining to explain about 40% of the variance in
domestic electricity bills. This apparent density dependence is,
of course, spurious. High summer electricity costs (and low
gas costs) in Athens, Georgia, reflect an increased use of airconditioning (and decreased use of heating) associated with
high summer temperatures in the south-eastern United States.
As an amusing aside, gas costs at t–1 can be added to the
model, and are significantly and negatively associated with
electricity costs at time t (r2 5 0.48, P 5 0.028, Fig. 1). If we
were to approach these data without prior knowledge of the
mechanism, we might conclude from time-series analysis that
Fig. 1. Seasonal changes in the domestic bills (U.S. Dollars per
month) for electricity and gas for a house in Athens, Georgia, U.S.A.
Electricity costs exhibit both t–2 and t–3 lags in apparent negative
feedback processes, and a negative relationship with gas costs. The
apparent negative feedback and interaction are spurious, and seasonal
changes are driven by the exogenous driving variable, temperature.
there was density dependence acting on electricity costs and
evidence of a negative interaction between electricity and gas
(Fig. 1)! Rather, both costs are driven by the exogenous driving
variable of seasonal fluctuations in temperature.
Time-series 2: a gall-forming sawfly
Populations per clone of E. lasiolepis on dry-site clones
exhibited either t–1, t–2, or both t–1 and t–2 lags in apparent
© 1998 Blackwell Science Ltd, Ecological Entomology, 23, 216–222
Cycles in insect populations
219
Table 3. Per cent parasitism by Pteromalus sp. and predation by chickadees on Euura lasiolepis galls on fifteen clones of the willow Salix
lasiolepis. N 5 number of years in which both sawfly densities and mortality factors were measured. Data represent the Pearson product-moment
correlation coefficients for regressions between per cent mortality in year t and sawfly density in years t, t–1 and t–2, *P , 0.05, **P , 0.01,
NS 5 not significant. ID 5 clones for which there were insufficient data for regression. For example, parasitism was observed in only 2 years on
clone NP4, and sawfly mortality from chickadees was zero on most clones in most years.
% parasitism against sawfly density
Clone
N
time t
MNA1
MNA2
MNA3
MNA4
MNA5
MNA6
MNA7
NP4
NP5
NP7
NP8
NP9
Dry-site mean
BD1
CS1
CS2
Wet-site mean
10
10
10
10
8
8
8
8
8
8
10
10
10
9
8
8
8
0.757
0.052
–0.111
0.738
0.502
0.358
0.124
ID
0.390
–0.156
0.565
–0.160
0.311
–0.514
0.529
–0.051
–0.211
time t–1
**
NS
NS
**
NS
NS
NS
NS
NS
NS
NS
NS
NS
NS
NS
NS
–0.100
–0.265
–0.566
0.193
0.035
0.263
–0.351
ID
–0.145
–0.443
–0.472
–0.334
–0.734
0.014
0.288
–0.472
–0.288
% predation against sawfly density
time t–2
NS
NS
NS
NS
NS
NS
NS
NS
NS
NS
NS
*
NS
NS
NS
NS
–0.462
0.035
–0.062
0.358
–0.403
0.398
–0.770
ID
0.056
–0.623
–0.169
0.375
–0.441
–0.080
0.202
–0.665
–0.066
density-dependent processes. Gall populations on wet-site
clones all exhibited t–3 lags, with two clones exhibiting t–2
lags as well. Precipitation data exhibited t–1 and t–3 lags,
evidence for significant autocorrelation in precipitation
(Table 2).
There was evidence of density-dependent sawfly mortality
from the parasitoid Pteromalus sp. on two willow clones
(Table 3). Per cent parasitism was either unrelated (twelve
clones) or negatively related (clone MNA7) to gall density on
the remaining clones. When sawfly densities were averaged on
dry- and wet-site clones, parasitism was either inversely
density-dependent (dry-site clones) or unrelated to sawfly
density (wet-site clones) (Table 3). There was no evidence of
spatial density-dependent mortality from parasitoids (Table 4).
Although the density-dependent parasitism detected on two
clones could generate t–1 lags in time-series analysis, its
absence from most of the clones on which t–1 lags were
detected makes it an unconvincing candidate for generating
negative feedback. We conclude that parasitism cannot be
responsible for the apparent negative feedback detected in
time-series analyses of sawfly densities, and this agrees with
previous sampling and experimental work that found negligible
or inversely density-dependent effects of parasitoids on
E. lasiolepis (Price, 1988, 1994).
There was no evidence of temporal density-dependent
mortality on sawflies caused by chickadees at dry-clone sites
(Table 3). Galls on clones at dry sites suffered negligible
mortality from chickadees in most years. Sawflies on one of
the wet-site clones (CS2) showed some evidence of densitydependent mortality from chickadees, but the trend was based
on high levels of predation in 2 years of extremely high gall
abundance. The data indicate that chickadees ignore gall-
© 1998 Blackwell Science Ltd, Ecological Entomology, 23, 216–222
NS
NS
NS
NS
NS
NS
*
NS
NS
NS
NS
NS
NS
NS
NS
NS
time t
time t–1
ID
0.376 NS
0.390 NS
ID
ID
ID
ID
ID
ID
ID
ID
ID
ID
0.0003 NS
0.562 NS
0.753 *
0.455 NS
ID
–0.230
–0.094
ID
ID
ID
ID
ID
ID
ID
ID
ID
ID
–0.122
0.109
–0.058
0.007
time t–2
NS
NS
NS
NS
NS
NS
ID
–0.445
–0.395
ID
ID
ID
ID
ID
ID
ID
ID
ID
ID
0.323
–0.483
–0.130
–0.027
NS
NS
NS
NS
NS
NS
Table 4. Per cent parasitism by Pteromalus sp. and predation by
chickadees on Euura lasiolepis galls on clones of the willow Salix
lasiolepis. N 5 number of clones from which both sawfly densities
and mortality factors were measured in a given year. Data represent
the Pearson product-moment correlation coefficients for regressions
between per cent mortality and sawfly density among clones within
years, ***P , 0.001, NS 5 not significant. ID 5 years for which there
were insufficient data for regression: levels of predation by chickadees
were zero on most clones in most years and 1987 was the only year
when galls on most clones received some level of attack by chickadees.
Year
N
Parasitism
81
82
83
84
85
86
87
88
89
90
6
7
15
15
15
15
15
15
15
15
0.094
0.051
–0.327
0.017
–0.271
0.174
–0.371
0.391
0.074
0.040
Predation
NS
NS
NS
NS
NS
NS
NS
NS
NS
NS
ID
ID
ID
ID
ID
ID
0.763***
ID
ID
ID
forming sawflies most of the time, but switch to this food
source in years and on clones of very high sawfly abundance.
This is further supported by one year (out of 10 possible years)
when there was spatial density-dependent mortality caused by
chickadees: in 1987, the percentage sawfly mortality from
chickadees increased with gall density on a clone. Sawflies
were particularly abundant in 1987 (Fig. 2). However, the
absence of delayed temporal density-dependent mortality from
chickadees, on either wet-site or dry-site clones, leads us to
220
Mark D. Hunter and Peter W. Price
Fig. 2. Periodic fluctuations in precipitation and densities of the sawfly
Euura lasiolepis on (a) twelve dry-site and (b) three wet-site willow
clones. Sawfly populations track changes in precipitation on delay.
conclude that chickadees are not responsible for the apparent
time-lagged negative feedback detected in time-series analysis
of sawfly densities.
In contrast, patterns of precipitation appear to be capable of
generating changes in sawfly abundance that would produce
the apparent t–1, t–2 and t–3 lags in the action of negative
feedback processes. First, time-series analysis of precipitation
data suggests that there is autocorrelation in precipitation with
lags of t–1 and t–3 (Table 2). These match two of the lags
exhibited by the sawfly population data. Second, precipitation
at time t is positively correlated with gall density at time t 1 1
(Fig. 3) and t 1 2 (Fig. 4, dry-site clones only). In other words,
high levels of precipitation appear to favour high sawfly
populations for 2 subsequent years. This supports previous
experimental work in which application of water to willow
clones resulted in improved sawfly performance in subsequent
generations (Price & Clancy, 1986). The combination of
periodic precipitation, exhibited by apparent t–1 and t–3
lags in time-series analysis, and the response of gall-forming
sawflies to the quantity of precipitation 2 years previously, can
combine to generate all of the apparent delayed density-
Fig. 3. Effects of winter precipitation (October–May) at time t on
Euura lasiolepis populations at time t 1 1 on (a) twelve dry-site and
(b) three wet-site willow clones. Data are from 1981 to 1996 on wetsite clones and from 1983 to 1996 on dry-site clones.
dependent lags observed in time-series analysis of the sawfly
data (t–1, t–2 and t–3). The overall relationship between
precipitation and sawfly density is illustrated in Fig. 2. The
densities of E. lasiolepis closely follow changes in precipitation,
but on delay, in a manner remarkably similar to the output of
simple predator–prey models (Varley et al., 1973).
The mechanism by which precipitation influences willow
quality for E. lasiolepis has been established previously (Price
& Clancy, 1986). Both the number of willow shoots per stem
and the length of willow shoots decrease following dry years
and increase following wet years. Female sawflies markedly
avoid oviposition and gall initiation on water-stressed willows
because both egg and larval mortality are considerably higher
on such plants (Price & Clancy, 1986; Preszler & Price, 1988;
Craig et al., 1989). The combination of low resource availability
following drought and increased sawfly mortality results in
low sawfly populations for 2 years following drought events
(Price & Clancy, 1986). We cannot yet explain why sawfly
populations exhibit t–1 and t–2 lags on dry-site clones, but
© 1998 Blackwell Science Ltd, Ecological Entomology, 23, 216–222
Cycles in insect populations
221
of using time-series analysis alone to infer that density
dependence is operating (Holyoak, 1994; Williams & Liebhold,
1995, 1997) appear to be justified. Combined with life-table
studies and experimental manipulation, time-series analysis
can be a powerful approach to understanding insect population
dynamics and the relative importance of resource availability,
abiotic factors, and depredation to population change (Hunter
et al., 1997). Standing alone, it generates hypotheses about
potential sources of population change that require rigorous
testing. Finally, we join with others who suggest that more
attention should be paid to the periodicity of abiotic factors
(Burroughs, 1992; Swetnam & Lynch, 1993) that can influence
insect population change.
Acknowledgements
We gratefully acknowledge support from the National Science
Foundation (grant DEB-9527522 to M.D.H. and grant DEB9318188 to P.W.P.) during the preparation of this manuscript.
We also thank Simon Leather for the invitation to contribute
this Critical Appraisal.
References
Fig. 4. Effects of winter precipitation (October–May) at time t on
Euura lasiolepis populations at time t 1 2 on (a) twelve dry-site and
(b) three wet-site willow clones. Data are from 1981 to 1996 on drysite clones and from 1983 to 1996 on wet-site clones. The regression
is significant for dry-site clones only.
t–2 and t–3 lags on wet-site clones, but suggest that shoot
growth responses of willow to precipitation may vary depending
upon previous water availability.
The data on costs of power (Fig. 1) were analysed over a
17-month period, whereas the data on changes in sawfly
abundance (Fig. 2) were analysed over a 16-year period.
Although we admit to a certain amount of irreverence in our
choice of example, the results of the two time-series analyses
are remarkably similar. In both cases, exogenous driving
variables (temperature for power, precipitation for sawflies)
drive cyclic changes over time that could easily be
misinterpreted as evidence for density dependence. Indeed,
similar evidence has been used to infer the action of density
dependence in previous studies of insect populations (Turchin,
1990; Berryman, 1994). Our purpose is not to discourage the
use of time-series analysis as a tool for population analysis,
nor to cast doubt on the conclusions of previous studies. Rather,
we wish to emphasize that recent concerns over the validity
© 1998 Blackwell Science Ltd, Ecological Entomology, 23, 216–222
Berryman, A.A. (1994) Population dynamics: forecasting and diagnosis
from time series. Individuals, Populations, and Patterns in Ecology
(ed. by S. R. Leather, A. D. Watt, N. J. Mills and K. F. A. Walters),
pp. 119–128. Intercept, Andover, U.K.
Berryman, A.A. & Turchin, P. (1997) Detection of delayed density
dependence: comment. Ecology, 78, 318–320.
Box, G.E.P. & Jenkins, G.M. (1976) Time Series Analysis: Forecasting
and Control. Holden-Day, San Francisco, California.
Burroughs, W.J. (1992) Weather Cycles: Real or Imaginary? Cambridge
University Press, Cambridge.
Craig, T.P., Itami, J.K. & Price, P.W. (1989) A strong relationship
between oviposition preference and larval performance in a shootgalling sawfly. Ecology, 70, 1691–1699.
Craig, T.P., Price, P.W. & Itami, J.K. (1986) Resource regulation by a
stem-galling sawfly on the Arroyo willow. Ecology, 67, 419–425.
Dennis, B. & Taper, M.L. (1994) Density dependence in time series
observations of natural populations: estimation and testing.
Ecological Monographs, 64, 205–224.
Holyoak, M. (1994) Identifying delayed density dependence in timeseries data. Oikos, 70, 296–304.
Hunter, M.D., Varley, G.C. & Gradwell, G.R. (1997) Estimating the
relative roles of top-down and bottom-up forces on insect herbivore
populations: a classic study re-visited. Proceedings of the National
Academy of Sciences, 94, 9176–9181.
Pollard, E. (1988) Temperature, rainfall, and butterfly numbers. Journal
of Applied Ecology, 25, 819–828.
Pollard, E., Lakhani, K.H. & Rothery, P. (1987) The detection of
density-dependence from a series of annual censuses. Ecology, 68,
2046–2055.
Preszler, R.W. & Price, P.W. (1988) Host quality and sawfly populations:
a new approach to life table analysis. Ecology, 69, 2012–2020.
Price, P.W. (1988) Inversely density-dependent parasitism: the role of
plant refuges for hosts. Journal of Animal Ecology, 57, 89–96.
Price, P.W. (1994) Evaluating the role of natural enemies in latent
and eruptive species: new approaches in life table construction.
222
Mark D. Hunter and Peter W. Price
Individuals, Populations, and Patterns in Ecology (ed. by S. R.
Leather, A. D. Watt, N. J. Mills and K. F. A. Walters), pp. 221–
232. Intercept, Andover, U.K.
Price, P.W. & Clancy, K.M. (1986) Multiple effects of precipitation
on Salix lasiolepis and populations of the stem-galling sawfly, Euura
lasiolepis. Ecological Research, 1, 1–14.
Price, P.W. & Craig, T.P. (1984) Life history, phenology, and
survivorship of a stem-galling sawfly, Euura lasiolepis
(Hymenoptera: Tenthredinidae), on the Arroyo willow, Salix
lasiolepis, in Northern Arizona. Annals of the Entomological Society
of America, 77, 712–719.
Royama, T. (1992) Analytical Population Dynamics. Chapman and
Hall, London.
Stiling, P. (1988) Density-dependent processes and key factors in insect
populations. Journal of Animal Ecology, 57, 581–594.
Swetnam, T.W. & Lynch, A.M. (1993) Multicentury, regional-scale
patterns of western spruce budworm outbreaks. Ecological
Monographs, 63, 399–424.
Turchin, P. (1990) Rarity of density dependence or population regulation
with lags? Nature, 344, 660–663.
Varley, G.C., Gradwell, G.R. & Hassell, M.P. (1973) Insect Population
Ecology: an Analytical Approach. Blackwell Scientific
Publications, Oxford.
Williams, D.W. & Liebhold, A.M. (1995) Detection of delayed density
dependence: effects of autocorrelation in an exogenous factor.
Ecology, 76, 1005–1008.
Williams, D.W. & Liebhold, A.M. (1997) Detection of delayed density
dependence: reply. Ecology, 78, 320–322.
Wolda, H. & Dennis, B. (1993) Density dependence tests, are they?
Oecologia, 95, 581–591.
Accepted 20 December 1997
© 1998 Blackwell Science Ltd, Ecological Entomology, 23, 216–222