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Factors influencing detection of density dependence in British birds

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Oecologia (1996) 108:54-63 9 Springer-Verlag 1996
Marcel Holyoak 9 Stephen R. Baillie
Factors influencing detection of density dependence in British birds
II. Longevity and population variability
Received: 26 July 1995/Accepted: 31 March 1996
Abstract
If censuses are taken at less than generationintervals, the number of successive censuses in which agiven individual is recorded will depend on longevity.Repeatedly recording the same individuals could pro-duce under-estimates of population variability andinfluence detection of density dependence. We investi-gated this possibility in 60 time series of abundancesof British birds compiled from the Common BirdsCensus data and then used simple population modelsto illustrate the proposed mechanism. Species had aver-age lifespans of 2-10 years and were censused annu-ally. Density dependence was detected (at P < 0.05)much more frequently in bird species with long life-spans than in those with short lifespans; 75% of the12 longest-lived species showed density dependencecompared to 46% of all species. Population variabilitymeasured in annual censuses (termed "annual vari-ability") was lower in bird species with longer lifespans.We used discrete time models based on difference equa-tions to demonstrate how longevity influences popula-tion variability and detection of density dependence inseries of annual censuses. A model in which onlyfirst-year birds experienced density dependence wasrejected because annual variability was greater anddetection of density dependence was less likely whenlongevity was greater, the opposite of the observedeffects of longevity in birds. A model in which all ageclasses experienced density dependence gave time serieswith lower annual variability and in which densitydependence was detected more frequently when long-
M. Holyoak (~)1NERC Centre for Population Biology,Imperial College at Silwood Park, Ascot, UKS. R. BaillieBritish Trust for Ornithology, Thetford, Norfolk, UK
Present address:
~Department of Entomology,University of California, Davis, CA 95616, USAfax: 916 752 1537
evity was greater, which is the pattern observed inBritish birds. Analysis of data from this model showedthat the amount of density dependence actually pre-sent caused only small changes in annual variability,whereas
detection
of density dependence from simu-lated series was strongly influenced by annual vari-ability. The high annual variability of series fromshort-lived bird species could mask any density depen-dence that was present. Correcting for trends leadus to detect density dependence in 75 % of the 12 longestlived bird species. There is no reason to believethat this rate is not also representative of short-livedspecies.
Key
words Common Birds Census 9 Discrete-timemodels 9 Longevity 9 Population variability 9 Time
series analysis
Introduction
To obtain a general measure of population variabilityit is necessary to use censuses carried out at generation(lifespan) intervals (Connell and Sousa 1983).Frequently, however, censuses are annual and the life-span is more than a year, so that the interval betweencensuses is less than a generation (Connell and Sousa1983; Pollard 1977; Vickery and Nudds 1984; Baillie1990; Crowley and Johnson 1992; Woiwod and Hanski1992). This causes estimates of population variabilityto be influenced by longevity, so that the variability ofspecies cannot be compared if lifespans differ (Connelland Sousa 1983). Longevity might also influence detec-tion of density dependence from time series of censusesconducted at less than generation intervals; this prob-lem has not previously been addressed. Population vari-ability and the conditional probability for detection ofdensity dependence are also correlated (Hanski andWoiwod 1993), so we considered both variability anddetection concurrently.
This paper builds on earlier work in which weshowed that temporal trends in abundance are partlyresponsible for hindering detection of density depen-dence in time series of British birds compiled from theCommon Birds Census data (Holyoak and Baillie1996). However, even when data were detrended, den-sity dependence was found in only 45 % of 60 bird timeseries; a frequency which is much lower than the equiv-alent rates of >80% from British aphids and moths(Woiwod and Hanski 1992). We believe that the lowfrequency with which we found density dependence inbird series is caused by the use of census data taken atless than generation intervals.We investigated the effect of longevity on detectionof density dependence and measurement of populationvariability in both the bird time series and simple dis-crete-time models based on difference equations. Weinvestigated (1) the correlation between mean lifespanand the frequency of detection of density dependence(at P < 0.05) and (2) the correlation between mean life-span and population variability measured from annualcensuses. The presence of the above two relationshipsin model data does not allow us to infer whether theamount of density dependence present is influencingpopulation variability. Alternatively, population vari-ability may influence only
detection,
and may not reflectthe actual
presence
of density dependence. We analysedmodel data to strengthen our inference that it is mainlydetection of density dependence that is influenced bypopulation variability.There are several potential problems with measur-ing population variability (Connell and Sousa 1983;McArdle et al. 1990; McArdle and Gaston 1992, 1993;Link and Nichols 1994). Measures of variabilityare inflated by zero values (McArdle et al. 1990),however the CBC index values do not contain zeroabundances. The degree of spatial synchrony inpopulation fluctuations is important to the reliabilityof measures of temporal population variability(McArdle and Gaston 1993). In the CBC data setpopulation variability should largely reflect thetemporal variability of local subpopulations, becausemost of the species examined show a high degreeof regional synchrony in their population fluctuationsfor the area of Britain covered by the CBC plots(M. Holyoak and S. Baillie, unpublished work).The high degree of synchrony is confirmed by the obser-vation that changes in abundance of most specieswithin years were synchronous across plots (Taylor1965; Bailey 1967). In this analysis we assume thatin general species have a high degree of spatialsynchrony.This paper is laid out in the following sequence:1. We first explore the relationships between longevity,population variability and detection of density depen-dence in the bird data.2. Next we present two population models which arepotential descriptors of the CBC data.OECOLOGIA 108 (1996) 9 Springer-Verlag 553. We use these models to explore the same relation-ships between longevity, population variability anddetection of density dependence that we explored inthe bird data.4. Finally, we use the preferred model to clarifywhether population variability influences detection ofdensity dependence or whether it is the amount ofdensity dependence present that influences populationvariability.
Analyses of bird data
MethodsIn this section we investigate how longevity and detec-tion of density dependence, and longevity and popu-lation variability are related in the bird time series.Mean annual survival rate was used as an index oflongevity. Only survival rates of first-year birds wereused, because survival rate varies little with age in thespecies considered (Dobson 1990; Baillie andMcCulloch 1993). Survival rates, calculated using themethod of Lack (1943a, b) were obtained from Dobson(1990) and survival rates calculated using the methodof Brownie et al. (1978) were obtained from Baillieand McCulloch (1993). The bird time series and meth-ods used to test for density dependence are fullydescribed in the first of this pair of papers (Holyoakand Baillie 1996). Since temporal trends in abundancecan effect detection of density dependence (Vickery andNudds 1984; Pollard et al. 1987; Woiwod and Hanski1992), all bird time series were detrended prior totesting for density dependence using the method ofPollard et al. (1987), as described in Holyoak and Baillie(1996).To simplify explanation of the results of tests fordensity dependence we calculated an index of the
inci-dence of density dependence
(Hanski and Woiwod1993). This is simply the P-value from the test for den-sity dependence, which was logit transformed to lin-earise the distribution of P-values and multiplied by-1 so that larger values represent lower conditionalprobabilities of rejecting the null hypothesis of densityindependence. If the assumptions of the density depen-dence test are correct, density dependence is more likelyat higher values of the incidence of density dependence.Logit is the natural logarithm of the odds ratio,
P/(1-P),
where P is the conditional probability fromPollard et al.'s test. Spearman's rank order correlationswere used to test for a relationship between incidenceof density dependence and longevity.We calculated two measures of population variabil-ity. Population variability measured from annualcensuses was termed
annual
population variability.
Per generation
variability was calculated using onlyabundances that were separated by generation inter-vals. This allowed us to check the extent to which
56 OECOLOGIA 108 (1996) 9 Springer-Verlag
annual variability under-estimated per generation vari-ability in species with different lifespans.Annual population variability was measured as thestandard deviation of In(annual abundance) fromdetrended bird time series. Per generation variabilitycalculations used only counts which were separated bya lifespan in detrended bird series. Lifespan was cal-culated from annual survival rate as the period until95% of individuals were dead (Schoener 1985) androunding to the nearest year; solving ~M/~s < 0.05,where (~ is mean annual survival rate and
MLS
is themean lifespan in years. Per generation variability wascalculated as the mean standard deviation from all pos-sible sets of three consecutive values of In(annual abun-dance) separated by lifespan intervals, in the CBCseries. Because of the apparent non-linearity of plotsof annual variability or per generation variabilityagainst longevity (survival rate) we used polynomialregression to investigate this relationship. Spearman'srank order correlations were used to investigate therelationship between annual variability and incidenceof density dependence. To check the reliability of analy-ses of the bird series which used annual abundances,we plotted per generation variability against annualvariability and analysed this relationship using linearregression.ResultsDensity dependence was more frequently detected atP < 0.05 in the species that had longer lifespans(Fig. 1). Spearman's rank order correlations betweenthe incidence of density dependence and longevity inthe bird series were 0.344 (P < 0.01) from all 60 seriesand 0.451 (P < 0.01) from just the longest 37 series withonly one series per species (as listed in the Appendixto Holyoak and Baillie, 1996). Figure 2 shows the effecton detection rate of including only species with sur-vival rates of at least those shown on the x-axis; thehigher the minimum survival rate, the greater theproportion of series in which density dependence wasdetected. Density dependence was detected from allthree of the longest-lived species, 75% of the 12 mostlong-lived species, and 46% of all species (45% of allseries).For most species (most clearly those with annualsurvival rates < 0.6) population variability declinedwith increasing longevity (Fig. 3). In three out of fourcases survival rate was non-linearly related to bothannual and per generation population variability; sec-ond order polynomials were significant for both mea-sures of population variability and for either all 60 CBCseries or for 37 series, including only the longest seriesfor each species (Fig. 3A, C and D). In the fourth casethe second order term of the polynomial was non-significant, giving a linear relationship between annualvariability and survival rate (Fig. 3B). Clearly, in all ofthese cases population variability is higher in the
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Fig. 1A, B The relationship between incidence of density depen-dence and longevity (measured as annual survival rate) in detrendedseries from British birds, A is with all 60 series and B is with 37species and only the longest series for each species.
Points
on orabove the
dashed line
were significant at P _< 0.05. The incidence ofdensity dependence on the
horizontal axis
is the logit of the P-valuefrom the test of Pollard et al. (1987) for density dependence, mul-tiplied by - 1 to give greater values when conditional probabilityfor density dependence is greater
more short-lived species than in the longer livedspecies.Figure 4 shows that annual variability under-esti-mated per generation variability in all series. However,annual variability under-estimated per generation vari-ability more in short-lived species than in long-livedspecies. For example in a species with an annual vari-ability of 0.1, which occurred only in long-lived species(Fig. 3A), annual variability was on average 89 % of pergeneration variability (Fig. 4). Whereas, when annualvariability was 0.5, which is typical of short-livedspecies (Fig. 3A), this was on average only 43 % of pergeneration variability.
Some simple models of bird pp0pulation dynamics
In this section we introduce two population models thatwe used to test whether longevity can produce the
OECOLOGIA 108 (1996) 9 Springer-Verlag 57
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.1 .2 .3 .4 .5 .6 .7 .8survival rate
Fig. 2 The proportion of the 60 British bird time series, havingannual survival rates _> he magnitude shown on the horizontal axis,for which density dependence was detected in detrended time series.Density dependence was detected if P < 0.05 in the test of Pollardet al. (1987)effects reported from the bird time series on detectionof density dependence and population variability if cen-suses are taken at less than generation intervals.Although density dependence has been detected inmany bird species, it is not clear from the literaturewhen during the life-cycles of birds density dependenceis expressed. Sinclair (1989) found reports of densitydependence for 19 bird populations; density depen-dence was found in fertility/egg production (26% ofstudies), early juvenile mortality (32%), late juvenilemortality (74%) and adults (21%). Sinclair's surveyshows that density dependence may occur in all life-stages and in some cases in more than one life-stage,but tells us little about the frequency of density depen-dence in different life-stages because studies in whichdensity dependence was not found were not reported.For simplicity we considered only two life-stages in ourmodels, representing first-year birds (< 1 year old) andolder birds (_> 1 year old). We constructed populationmodels with density dependence acting either only onfirst year birds (Eqs, 1 and 2), or on both age classes(Eqs. 1, 3 and 4). We did not consider density depen-dence acting in only the older age class because of thelarge number of studies which reported density depen-dence in juveniles (Sinclair 1989). We also assumed thatany changes in the mean annual survival rate betweenfirst year and older birds were small by comparison tointer-specific differences (Dobson 1990).The population models are both based on a modifiedtbrm of the Ricker equation (Ricker 1954; Cook 1965),which we have altered to represent two life-stages. Ifthese models had deterministic (fixed) parameters, thenonce any transient behaviour was eliminated, theywould both produce a fixed abundance level (the equi-
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.1 .2 .3 .4 .5 .6 ,7 .8survival rate
Fig.
3A-D The relationship between population variability ofannual censuses and longevity (measured as annual survival rate)for detrended time series of British land-bird abundances. See textfor explanations of annual and per generation variability. A and Care for 60 time series from 37 species and B and D are with onlythe longest series for each species (Appendix 1). A and B show plotsof annual population variability against longevity, whereas C andD show per generation variability plotted against longevity. The
lines
are second-order polynomials in A, C and D and a linearregression in B. Equations for lines are: A, Y=0.77-1.83X+ 1.39X2, B, Y = 0.46 - 0.43X, C, Z = 2.83 7.88X+ 5.88X2 andD, Z = 2.90 - 7.89X+ 5.83X 2, where Y is annual variability, Z isper generation variability and X is survival rate. In the polynomialregressions in A, R2 = 0.26 for X F1.57 = 15.49, P < 0.01 and for
X 2 F~,57
= 4.41, P < 0.05. Equivalent figures were R 2 = 0.17,
FI,34 =
7.62, P < 0.05, F],34= 2.60, P >> 0.05 for B, R2 = 0.58,F~ ~7 = 64.5, P < 0.00001
-FI 57
= 16.0, P < 0.0002 for C andR ~- = 0.61,
FI,34 =
43.2, P < 0.i)002/C'1,34
=
9.27, P < 0.005 for D. Inall cases the significance of regressions was not altered if the out-lying point at the lowest survival rate is excludedlibrium abundance). This makes it necessary to addstochasticity to produce temporally variable abun-dances. Hanski and Woiwod (1993) showed that addingstochasticity through the density-dependent parameter(equivalent to o~ below) was unrealistic, at least for themoths and aphids which they considered. We thereforeonly consider density-independent ways of adding sto-chasticity to models: through the intrinsic growth rateor annual survival rate, or as a separate parameter.The power law of Taylor (1961) describes the rela-tionship between the variability of abundance data andthe mean abundance. We use it here to check the appro-priateness of models for describing bird data. Temporalvariance of abundance, V, and mean density x, are gen-erally related by the equation V=
ax b
(Taylor 1961),where a and b are constants. Log-transforming this
58OECOLOGIA 108 (1996) 9 Springer-Verlag2.5
>, 2
,,13
.#
1.5
r0
rO
.1 .2 .3 .4 .5 .6" annual variability
Fig. 4 The relationship between population variability measured atgeneration intervals and population variability measured annuallyin British birds. Generation intervals were calculated, using meanannual survival rates, as the time taken for 95% of individuals todie and rounding to the nearest year. Per generation variability onthe
vertical axis
is the mean standard deviation of In(annual abun-dance) from all sets of 3 years with a generation between years thatwere included from the CBC time series; this was done to elimi-nate sample size effects. The
line
is from a linear regression, repre-sented by Z = - 0.15 + 2.65 Y, where Z is per generation variabilityand Y is annual variability;
R 2
= 0.80, Fla7 = 236.3 and P<0.000001
equation gives ln(V) = ln(a) + bln(x), so that a and bcan be measured using linear regression of ln(V) againstln(x) using all annual abundances, where a is the inter-cept and b is the slope. We assessed the linearity of plotsfrom simulated data by eye and used linear regressionto measure b, for comparison with the value of 0.90from the CBC time series. Both models generatedapproximately linear Taylor's power law plots with b-values that encompass that of 0.90 observed for thebirds. These effects were only seen at constant values of[3 and r, both of which have strong effects on b; a sim-ilar result was obtained by Hanski and Woiwod (1993).values are normally distributed random values with amean of zero and variance (~2, which is related to themean abundance. Specifically, we assume c~ = z(1/c0Owhere z and 13 are constants (Hanski and Woiwod1993). Density dependence in Eq. 1 is determined bothby o~ and r. Numbers of adult birds are given by:
N2,t + 1 = Nl,t + i + (N2,t" (~) + 0t (2)
where 0, is a normal random variable with a mean ofzero and variance of v. All results were obtained numer-ically. Equation 2 could theoretically give negative pop-ulation sizes; however, this was not observed with theparameter values used in our simulations.Density dependence in all age classesIn this model all age classes were made to experiencedensity dependent survival, so that at the determinis-tic equilibrium, individuals in the age class of _>1 yearold have a probability t of survival. The same densitydependence (cz) acts in both age classes, which is likelyif the same mechanism acts on both adult and imma-ture birds, such as competition for food or nest sites.Population dynamics of first-year birds are againdescribed by Eq. 1 and survival of older birds isdescribed by Eq. 3 for
Nt >
1/c~ and Eq. 4 for
Nt
-< 1/a:
N2,t+ 1 = Nl, t + N2,t "t'el-c~N2'~ + r176
(3)Nz,t+ ~ = NLt +
N2,t't'e ~
(4)where cot is a normal random variable with a mean ofzero and variance of v. Density dependent mortality inolder birds is moderated by survival t; thus the actualsurvival rate of birds of > 1 year old is determined bothby the deterministic equilibrium abundance (1/~z), rand the value of t. As with Eqs. 1 and 2 all results wereobtained numerically by iterating Eqs. 1, 3 and 4 fordifferent parameter values. Both models were used togenerate time series of 31 generations in length, thesame as the longest CBC series.Density dependence only in first year birdsIn this model we make the intrinsic growth rate, r, sto-chastic by adding a normally distributed random vari-ate (et) to it, and make production of first-year birdsdensity-dependent. Older birds undergo density-inde-pendent mortality with a fixed annual survival rate,(~. Numbers of young birds recruited to year t + 1
(Nl,t +
I) are given by numbers of older birds in year t(N2,t) less the numbers lost through density dependentmortality, which depends on numbers in both ageclasses:
Nl,t +
1 =
N2,t er(1 - c~(NE,+ N2.,)) + a, (1)
where c~, the density dependent parameter is the inverseof the deterministic equilibrium abundance. The st-Analysis of model data
Methods
In this section:1. We test whether the two models can produce timeseries in which density dependence is more frequentlydetected when survival rates are high, and produce anegative relationship between longevity and annualvariability.2. We use model data to investigate the relationshipbetween the incidence of density dependence andannual variability.3. We test the extent to which annual variability ofmodel data under-estimated per generation variability.

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