e c o l o g i c a l m o d e l l i n g 2 1 9 ( 2 0 0 8 ) 170–177
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The effects of environmental perturbation andmeasurement error on estimates of the shape parameterin the theta-logistic model of population regulation
Daniel Barkera,∗, Richard M. Siblyb,c
a Sir Harold Mitchell Building, School of Biology, University of St. Andrews, St. Andrews, Fife KY16 9TH, UKb School of Biological Sciences, University of Reading, Reading RG6 6AS, UKc Centre for Integrated Population Ecology1, Denmark
a r t i c l e i n f o
Article history:
Received 27 August 2007
Received in revised form
30 July 2008
Accepted 21 August 2008
Published on line 14 October 2008
Keywords:
Density dependence
Theta-logistic
Generalised logistic
Observation error
Environmental stochasticity
Bias
Simulation
a b s t r a c t
The theta-logistic is a widely used generalisation of the logistic model of regulated biological
processes which is used in particular to model population regulation. Then the parameter
theta gives the shape of the relationship between per-capita population growth rate and
population size. Estimation of theta from population counts is however subject to bias, par-
ticularly when there are measurement errors. Here we identify factors disposing towards
accurate estimation of theta by simulation of populations regulated according to the theta-
logistic model. Factors investigated were measurement error, environmental perturbation
and length of time series. Large measurement errors bias estimates of theta towards zero.
Where estimated theta is close to zero, the estimated annual return rate may help resolve
whether this is due to bias. Environmental perturbations help yield unbiased estimates
of theta. Where environmental perturbations are large, estimates of theta are likely to be
reliable even when measurement errors are also large. By contrast where the environment
is relatively constant, unbiased estimates of theta can only be obtained if populations are
counted precisely. Our results have practical conclusions for the design of long-term popula-
tion surveys. Estimation of the precision of population counts would be valuable, and could
Return rate be achieved in practice by repeating counts in at least some years. Increasing the length
of time series beyond ten or 20 years yields only small benefits. If populations are mea-
sured with appropriate accuracy, given the level of environmental perturbation, unbiased
estimates can be obtained from relatively short censuses. These conclusions are optimistic
ta.
fundamental attribute of a population because it integrates
for estimation of the
1. Introduction
At the core of the study of population ecology is the relation-
ship between a population’s density and its growth rate (Siblyand Hone, 2002; Sibly et al., 2003a). Population growth rate, pgrhereafter, is defined as the per capita growth rate of the pop-
∗ Corresponding author.E-mail address: [email protected] (D. Barker).
ulation, i.e. as (dN/dt)/N, where N is the number of organismsin the population, or density, and t is time. pgr is the most
age-specific birth and death rates and so reveals whether thepopulation will increase or decrease, and it shows how fastthese increases or decreases will be. The form of the relation-
hip between N and pgr is important because it determinesow a population is regulated and how it will respond to dis-urbances.
In practice pgr is often estimated from annual populationounts as
grt = loge(Nt+1/Nt) = loge(Nt+1) − loge(Nt). (1)
A simple model of the relationship between N and pgr isinear:
gr ∝ 1 − (N/K) (2)
here K is a constant known as the carrying capacity of theopulation. This is equivalent to the logistic model of density-ependent population regulation, written
N/dt ∝ N(1 − N/K). (3)
According to this model a population follows an S-shapedurve as it increases from a low value to its limit, familiar fromany textbooks (e.g. Begon et al., 2006). From any starting-
oint, the population will return to carrying capacity overime. In the approximate region of carrying capacity, the returnrom a density above carrying capacity is a mirror-image of theeturn from a point equally far below.
Although the logistic model imposes linearity on the rela-ionship between pgr and density, there is little a priori reasono assume this to be correct in actual populations. The theta-ogistic model is a widely used generalisation of the logistic
odel that uses an additional parameter to model possibleurvature in the relationship between pgr and density, thus:
gr = r0[1 − (N/K)�] (4)
here the constant of proportionality r0 is a theoretical quan-ity representing what per-capita growth rate ‘would be’ at= 0, and � represents the curvature in the relationship
Nelder, 1961; Sæther et al., 2002; Tsoularis and Wallace, 2002;ibly et al., 2005; Thornley et al., 2007; Sakanoue, 2007).
In the theta-logistic model, when � > 1, the graph of pgr vsis convex when viewed from above; when � < 1, the graph
s concave when viewed from above. Different values of �
epresent different, traditional models. For example, � = 1 rep-esents the logistic model (Eq. (3)), and � → 0 approximatesGompertz or logarithmic model, in which pgr ∝ −loge(N/K)
Richards, 1959; Nelder, 1961; Sibly et al., 2005). However, theheta-logistic is not confined to representing these specific
odels: � may take any value except precisely zero (Richards,959; Sibly et al., 2005).
If � < 1, N rises more rapidly from below K than it decreasesrom above K. If � > 1, N falls more rapidly from above K thant increases from below K. Thus the value of � has importantmplications, for example in conservation and pest manage-
ent. It has been suggested that � < 1 may be typical of speciesith small, short-lived individuals, such as many insects, and
> 1 may be typical of species with large, long-lived individu-ls, such as some mammals (Gilpin and Ayala, 1973; Fowler,981). However, on the basis of estimates of � for 1780 pop-lations of animals, these hypotheses were rejected by Sibly
9 ( 2 0 0 8 ) 170–177 171
et al. (2005). Sibly et al. (2005) found that � is generally <1 inmammals, birds, fish and insects, and that � decreases ratherthan increases with body size among mammals.
All such conclusions are limited by the methods and dataon which they are based (Reynolds and Freckleton, 2005;Freckleton et al., 2006; de Vladar and Pen, 2007; Doncaster,2008). Measurement error causes bias in estimates of �. Inthe extreme case where all variation in observed N is due tomeasurement error, the fitted value of � is approximately zeroirrespective of its true value (Shenk et al., 1998). In this case itwould be useful to be able to discriminate those cases where �
really is zero from those in which it only appears so as a resultof measurement error. We approach this question by lookingat the estimate of annual return rate obtained by fitting thetheta-logistic model. The annual return rate of a populationis given by the slope of the relationship between pgr and loge
density at carrying capacity (May et al., 1974; Sibly et al., 2003b,2007), i.e.
return rate = −[d pgr/d(logeN)]K (5)
which for the theta-logistic curve is equivalent to
return rate = r0� (6)
as can be shown by differentiation of Eq. (4). If measurementerror causes us to estimate � ≈ 0, we also expect our estimatedannual return rate to match expectations in conditions of highmeasurement error. Measurement error alone produces a fit-ted return rate close to 1 (Sibly et al., 2005). Although thereare other ways to estimate return rate (see e.g. Sibly et al.,2007) in the current study we are not interested in return rateper se, but rather its potential value in distinguishing correctfrom incorrect estimates of � ≈ 0, and so here use parametersalready estimated by fitting the theta-logistic.
Since measurement error causes bias in estimates of �, it isimportant to know which factors affect this bias, and how biasmay be alleviated. Perhaps bias could be reduced by obtain-ing more information, for example by collecting data for moreyears. Environmental perturbations are also likely to affectbias, but it is not clear a priori how. Environmental perturba-tions might increase bias in a similar manner to measurementerror, or they might make more information available for anal-ysis by moving population size away from carrying capacity,providing more opportunity to observe the way in which it sub-sequently returns towards carrying capacity. To establish theeffects of these possible sources of bias, we here use simula-tion to investigate the effects of measurement error, length oftime series and environmental perturbation on the accuracyof estimation of the value of �.
2. Methods
By environmental perturbation we mean the deviation ofthe true value of N from the value expected from the
t+1
theta-logistic model and the true value of Nt. Our ‘environ-mental perturbation’ represents any effects excluded fromthe theta-logistic model, for example migration. Environmen-tal perturbations are here presumed to occur after density
172 e c o l o g i c a l m o d e l l i n g
Table 1 – Parameters of the simulation. For eachcombination of parameters, ten population time serieswere simulated
dependence has acted. By measurement error we mean thedeviation of the observed value of Nt from the true value ofNt. Since variation in observed pgr is approximately normallydistributed (Hone, 1999), we assumed normal distributions forboth environmental perturbations and measurement errors inloge N, corresponding to lognormal perturbations and errors inN (cf. Niwa, 2007). We measured N in units of carrying capacity,K, and simulated population time series in which environ-mental perturbations and measurement errors in loge N weredrawn at random from normal distributions with mean = 0 andvarious standard deviations, as shown in Table 1. For eachseries, the initial, true value of loge N was set to equal theenvironmental perturbation for that year. Thus if initial per-turbation were zero, for example, then the initial value of loge
N = loge K = 0. A range of values of parameters r0 and � fromEq. (4) was investigated; these are given together with thedurations of the time series in Table 1. We varied r0� inde-pendently of � for two reasons. Firstly, r0� is a measure of theannual return rate of the population (Eq. (5)), and it is concep-tually interesting to distinguish populations with the sameshape parameter, �, but different return rates. Secondly esti-mates of r0 and � are highly negatively correlated when � → 0
(Sibly et al., 2005). A negative value of � indicates a relation-ship between pgr and N even more concave (when viewed fromabove) than at � → 0, and also implies a negative value of r0.This is unproblematic because r0 has no biological meaning, in
Fig. 1 – Overview of the effects of environmental perturbation ansimple linear regression for 1440 simulated time series, with (a)perturbation S.D. = 0.2, and (c) environmental perturbation S.D. =S.D. to environmental perturbation S.D.
2 1 9 ( 2 0 0 8 ) 170–177
contrast to pgr at N = 1 used in an alternative parameterisationof the theta-logistic model (Sæther et al., 2000).
Altogether we investigated the effects of five factorsas shown in Table 1, using an orthogonal design with 10replicates. Simulations were performed in Minitab 14.20(Minitab, Inc.) using a macro script (simpops.mac, available athttp://biology.st-andrews.ac.uk/cegg/software.html). For eachsimulated time series pgr was estimated from Eq. (1), and r0,�, and K were estimated in S-Plus 7.0 (Insightful Corp.) using ascript based on genlogistic.ssc (Sibly et al., 2005).
3. Results
Parameters of the theta-logistic, measurement errors, envi-ronmental perturbations, series length and estimates of � andr0�, for each of the 21600 simulated series, are given in theSupplementary Material.
A summary of the effects of measurement error and envi-ronmental perturbation on estimates of � is presented in Fig. 1.Clearly measurement error erodes the signal in the data, bias-ing fitted values of � towards zero. In contrast, the effects ofenvironmental perturbations are almost entirely beneficial.Large environmental perturbations tend to cause the fittedvalue of � to more closely resemble its true value. This isbecause environmental perturbation moves population sizeaway from carrying capacity, and � is estimated from theway the population subsequently returns towards carryingcapacity. The larger the perturbations, the more informa-tion is available for the estimation of �. Thus the maximumacceptable ratio of measurement error S.D. to environmentalperturbation S.D. varies, from approximately 0.5 if pertur-bation S.D. = 0.1 (Fig. 1a), to approximately 1 if perturbationS.D. = 0.4 (Fig. 1c). However, very high environmental perturba-tion (S.D. = 0.4) appears to cause fitted values of � to be slightlytoo high (Fig. 2m–r).
The value of annual return rate, r0�, has more complexeffects on the estimate of �, involving interactions (Fig. 2). Pro-vided r0� and environmental perturbations are not both low,and environmental perturbations exceed measurement error,
d measurement error on the estimate of �. Each line is aenvironmental perturbation S.D. = 0.1, (b) environmental0.4. ‘Error ratio’ indicates the ratio of measurement error
Fig. 2 – Effects of true annual return rate r0� on the estimate of �, at different levels of environmental perturbation and measurement error. Each line is a simple linearregression for 240 simulated time series, with environmental perturbation S.D. = 0.1 and (a) r0� = 0.1, (b)r0� = 0.25, (c) r0� = 0.5, (d) r0� = 1, (e) r0� = 1.5, (f) r0� = 2; environmentalperturbation S.D. = 0.2 and (g) r0� = 0.1, (h) r0� = 0.25, (i) r0� = 0.5, (j) r0� = 1, (k) r0� = 1.5, (l) r0� = 2; environmental perturbation S.D. = 0.4 and (m) r0� = 0.1, (n) r0� = 0.25, (o)r0� = 0.5, (p) r0� = 1, (q) r0� = 1.5, (r) r0� = 2. ‘Error ratio’ indicates the ratio of measurement error S.D. to environmental perturbation S.D.
174 e c o l o g i c a l m o d e l l i n g 2 1 9 ( 2 0 0 8 ) 170–177
Fig. 3 – Effects of time series length on the estimate of �, at different levels of environmental perturbation and measurementerror. Each line is a simple linear regression for 480 simulated time series, with environmental perturbation S.D. = 0.1 and (a)length = 10, (b) length = 20, (c) length = 40 years; environmental perturbation S.D. = 0.2 and (d) length = 10, (e) length = 20, (f)
(g) lenme
length = 40 years; environmental perturbation S.D. = 0.4 andratio’ indicates the ratio of measurement error S.D. to enviro
the estimate of � is not unduly affected by r0�. However, low r0�
(≤0.5, and particularly ≤0.25) combined with low environmen-tal perturbation S.D. (≤0.2) can lead to catastrophic effects onestimation of � (Fig. 2a–c, g–h). Where r0� is high (e.g. r0� = 2)and environmental perturbation S.D. is also high (≥0.2), thesituation is much improved: estimates of � are good, even ifmeasurement error is twice as strong as environmental per-turbation (Fig. 2l,r).
Length of time series has little effect on the quality of our
estimate of �, apart from a slight tendency for estimation to beimproved with time series that are at least moderately long (20or 40 years as opposed to 10 years; Fig. 3). This suggests that avery long time series is not necessary to estimate � correctly.
ngth = 10, (h) length = 20, and (i) length = 40 years. ‘Errorntal perturbation S.D.
Where � is estimated as close to zero, for example wherethe genlogistic.ssc script provides an estimate of � = 0.001, thismight be a result of measurement error, in which case the esti-mate of return rate r0� is of interest since measurement erroralone produces an estimate of return rate ≈1. In our currentsimulations, which are not of ‘pure measurement error’ butalso include environmental perturbations, high levels of mea-surement error bias our fitted return rate to a value slightlyabove 1 (Fig. 4). Still, where estimates of � ≈ 0 are not cor-
rect, they do tend to be associated with an estimated returnrate close to 1 (Fig. 5). There are two components contribut-ing to this observation. Firstly, measurement error biases notonly our estimate of �, but also our estimate of r0� (Fig. 4).
g 2 1 9 ( 2 0 0 8 ) 170–177 175
Srdamrt
4
AcoobtmtistBp6tdcadsbep
ruamoot
Fig. 5 – Potential of annual return rate to help distinguishspurious and correct estimates of � ≈ 0. Means of estimatedr0� (±2 S.E.) are shown for (a) series where fitted � = true
Fsem
e c o l o g i c a l m o d e l l i n
econdly, if measurement error is high compared to envi-onmental perturbations, a true value of r0� ≤ 1 can make itifficult to estimate � correctly (Fig. 2). For whichever reason,n estimated return rate close to 1 suggests an estimate of � ≈ 0ay be spurious. But where true � → 0 and the true annual
eturn rate is close to 1, this approach cannot, of course, dis-inguish correct from incorrect estimates of � ≈ 0.
. Discussion
s with any simulation, strict interpretation is limited to theonditions of the simulation but the results provide hintsf broader generality. We have assumed that measurementsf population size may be incorrect, but are not subject toias; that the population truly is density dependent; thathe population is regulated according to a theta-logistic
odel; that the underlying parameters do not change duringhe course of the survey; and that the impact of density isnstantaneous. Whether these assumptions are realistic is aeparate area of research (Shenk et al., 1998, and referencesherein; Freckleton et al., 2006; Doncaster, 2008). However,rook and Bradshaw (2006) found density dependence to beervasive in their study of 1198 time series (one per species;39 invertebrates, 529 vertebrates and 30 plants) and theheta-logistic encompasses all the classic models of densityependence as special cases (see Section 1). Although oururrent study is in one sense a best-case scenario in that ourssumptions are met, the levels of parameters investigatedo include the range that is at all likely in reality, and includeome extreme values of parameters which are far fromest-case scenarios. One would hope that a measurementrror twice as strong as environmental perturbation is rare inublished population time series, for example.
Our most striking finding is the beneficial effects of envi-onmental perturbations, which when large helped yieldnbiased estimates of � even when measurement errors arelso large (Fig. 1). The causes of environmental perturbations
ay be diverse, including migration, anthropogenic effects,
r indeed any genuine effect on density which is not partf the theta-logistic model. While large environmental per-urbations are beneficial, small perturbations provide little
ig. 4 – Overview of the effects of environmental perturbation animple linear regression for 1440 simulated time series, with (a)rror perturbation S.D. = 0.2, and (c) environmental error perturbaeasurement error S.D. to environmental perturbation S.D.
� = 0.001 (N = 179); and (b) series where fitted � = 0.001 andtrue � /= 0.001 (N = 370).
possibility to observe and correctly characterise the natureof the return to carrying capacity from above and below. Forexample, a population which begins at carrying capacity andstays there is not helpful for estimating �. In studies in whichthe environment is relatively constant, unbiased estimates of� can only be obtained if populations are counted precisely.Increasing the length of time series beyond 10 or 20 yearsyields only small benefits (Fig. 3).
In a situation of high environmental perturbation, mea-surement error is less serious, so it is not then necessaryto have such precise estimates of population size to obtaina relatively unbiased estimate of �. Conversely, in relativelyconstant environments, great care must be taken to mea-sure population size precisely, otherwise the estimate of �
will be strongly biased towards zero. In these latter situations,expending greater effort on measuring population size pre-cisely will bring greater benefit than extending the time serieslength. This is encouraging for those who wish to estimate �.
It may often be difficult to commit to measurement of a longtime series, due to uncertainties of staffing and funding, butit will often be practical to measure a short time series withsufficient accuracy.
d measurement error on the estimate of r0�. Each line is aenvironmental perturbation S.D. = 0.1, (b) environmentaltion S.D. = 0.4. ‘Error ratio’ indicates the ratio of
i n g
r
176 e c o l o g i c a l m o d e l l
Low annual return rates (r0� ≤ 0.25) can cause problemsestimating � (Fig. 2). Estimation of return rates can be help-ful where estimated � is close to zero, since it may helpresolve whether the cause is measurement error. Of the 1780population time series from the Global Population DynamicsDatabase, GPDD (NERC Centre for Population Biology, 1999)that passed the ‘filters’ of Sibly et al. (2005), 86 had an estimateof � = 0.001 (the closest value to zero allowed by the genlogis-tic.ssc script); of these, five series had a return rate, obtainedfrom linear regression of pgr on loge N, within the intervalshown in Fig. 5b (i.e. 0.96–1.03). These were series for Lynxrufus (GPDD ID 216), Vulpes (IDs 1318 and 1322), Ochlodes venata(ID 3861) and Rhospalosiphum padi (ID 8410). The estimate of� = 0.001 for these five series may result from measurementerror; alternatively, these series may genuinely have � ≈ 0 andreturn rate ≈1. For such cases a more sophisticated mod-elling approach, explicitly incorporating measurement error(e.g. Newman et al., 2006; Dennis et al., 2006; Wang, 2007), mayprove valuable.
How will estimates be affected by predation, parasitismand harvesting? Without quantitative specification of effectsthis cannot be answered, but in the simplest population modelwith no age structure, pgr = birth rate–death rate. If predation,parasitism or harvesting increase the death rate by �, say, thisshifts the pgr/density curve vertically downwards by �. Thenew curve can be derived from Eq. (4) by replacing r0 by r0−�,and K by K[1−(�/r0)]1/� . Thus both r0 and K are reduced. Sinceenvironmental perturbations were simulated in units of loge
K they now appear larger than before and this helps in theestimation of �. However reduction in r0 reduces the value ofr0 �, and below r0� ≤ 0.25 there can be problems in estimationof � as described in the preceding paragraph. Where harvest-ing exists it may be appropriate to include harvesting in themodel of population regulation (cf. Legovic, 2008). Harvest datathemselves can provide a reliable index of population size (e.g.Cattadori et al., 2003).
The relevance of our conclusions to models other than thetheta-logistic requires further study. However, we hypothesisethat variation in true population size caused by environmen-tal stochasticity, or failing that a high ratio of environmentalstochasticity to measurement error, may generally be moreimportant than a long time series for accurate parameter esti-mation. The extent to which this holds if measurement errorand process error are explicitly accounted for in the model(e.g. Wang, 2007) may be an interesting question for futureresearch.
5. Practical conclusions for populationsurveys
Our results have practical conclusions for the design of long-term population surveys. It would be extremely helpful toknow the size of measurement errors so that the ratio ofmeasurement error to environmental perturbation could beestimated. This could be achieved in practice if the size of
measurement errors were estimated by repeating populationcounts in at least some years. If measurement error and envi-ronmental perturbation in loge N are normally distributed, aswe have assumed, then their sum is also normally distributed.
2 1 9 ( 2 0 0 8 ) 170–177
If one knew the extent of measurement error, for examplefrom repeated measurements, then one could estimate thevariance of environmental perturbations by subtracting thevariance of measurement error from the variance of total vari-ation. Our results could then be used to indicate the quality ofestimated �.
Several ‘rules of thumb’ emerge. If environmental perturba-tion is high (e.g. perturbation in loge N has a standard deviationof 0.4) and measurement error does not exceed environmentalperturbation, estimates of � are likely to be of high quality foreven a short time series (10 years). With environmental per-turbation lower than this, lower measurement error is morebeneficial for the estimation of � than is a longer time series.Finally, where � appears to be zero, examination of annualreturn rate can reveal whether this is due to measurementerror or not.
Overall our conclusions are optimistic for estimation of �.If the theta-logistic is an appropriate model, then estimationof � is improved the larger the environmental perturbations,and unbiased estimates can be obtained from relatively shortcensuses.
Acknowledgements
We thank Mark Pagel for advice, and Len Thomas for help-ful comments on an earlier version of the manuscript. D.B.acknowledges the support of a Research Councils UK Aca-demic Fellowship.
Appendix A. Supplementary data
Supplementary data associated with this article can be found,in the online version, at doi:10.1016/j.ecolmodel.2008.08.008.
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