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Detecting Cycles and Delayed Density Dependence: a Comment on Hunter and Price (1998)

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Detecting Cycles and Delayed Density Dependence: a Comment on Hunter and Price (1998) Ecological Entomology (2000) 25, 119±121 NEW PERSPECTIVES New Perspectives is intended to allow the communication of comments, viewpoints, and speculative interpretation of issues in ecology pertinent to entomology.Comments, viewpoints, or suggestions arising from published papers intended to fuel discussion and debate are also welcome.Contributions should be as concise as possible, normally not exceeding 2000 words.Formal research reports will not be acceptable, but summarised novel data, suitably supported by statistics, may be allowed. Detecting cycles and delayed density dependence: a comment on Hunter and Price (1998) 1 2 PETER TURCHIN andALAN A.BERRYMAN 1Department of Ecology and Evolutionary Biology, University of Connecticut and 2Department of Entomology, Washington State University, Pullman, U.S.A. Key words. Density dependence, population cycles, time-series analysis. Controversy is a sign of health in a scienti®c discipline and, will be established statistically when longer time series for measured by this standard, population ecology must be a very these variables become available.On the other hand, the healthy science indeed, because the controversy over density apparent oscillatory tendency may be spurious, and will dependence and its detection shows no signs of abating (e.g. disappear with more data.We conclude that whatever patterns Turchin, 1995, 1999; Williams & Liebhold, 1995; Berryman & may be revealed after additional years of observations, at this Turchin, 1997).A recent addition to this debate is the article by point in time there is no need to explain population cycles in Hunter and Price (1998) in the Critical Appraisal section of saw¯ies, because their very existence has not been demon- Ecological Entomology.Hunter and Price focus on the issue of strated statistically. detecting population cycles and delayed density dependence in The second problem is Hunter and Price's insistence that we time-series data.On the basis of their analysis of two data sets, fail to consider the possibility that regular population one ecological and one nonecological, they conclude that the oscillations may be driven by exogenous factors that are approaches advocated by us and colleagues can lead to themselves cyclic (the very ®rst sentence in Hunter and Price spurious results and inappropriate conclusions. states: `Cycles in insect populations are usually attributed to There are three basic problems with the Hunter and Price delayed density-dependent interactions between insects and critique.The ®rst problem involves the de®nition of population their food, competitors, or natural enemies').Nothing could be cycle, for it is not clear whether Hunter and Price have the farther from the truth.In fact, the ®rst hypotheses for cyclic same de®nition in mind that we do.The most common population dynamics involved exogenous causation, such as de®nition of a population cycle, and one to which we adhere, is periodic climate and sunspot activity: see, for example, that the dynamics of the population are characterised by reviews by Finerty (1980) and Martinat (1987), and discussion statistically signi®cant periodicities.For this reason, a more by Turchin and Taylor (1992) and Royama (1992).This is not proper term may be regular oscillations (see Turchin, 1999). simply a theoretical idea, but has been investigated empirically Hunter and Price claim that data on precipitation and saw¯y in speci®c data sets by the present authors and colleagues.For abundance exhibit cyclic dynamics.Indeed, the degree of example, Berryman (1973) concluded that apparent cycles in ¯uctuation is impressive, but the conventional methods of ®r engraver beetle populations were driven by periodic time-series analysis (e.g. Box & Jenkins, 1976; Royama, 1992) oscillations in the abundance of its food supply, and Turchin fail to provide evidence for statistically signi®cant periodicity et al.(1991) considered, but rejected, the possibility that three in either data set (Fig.1).The autocorrelation functions (ACF) weather variables could have generated population cycles of for both populations exhibit very weak oscillatory tendencies, the southern pine beetle.In the same vein, Ellner and Turchin but autocorrelations at the apparent period of 6 (precipitation) (1995) discussed how models lacking seasonality may generate and 8 (saw¯y numbers) years are extremely weak and do not spurious results for monthly data. even approach statistical signi®cance.Perhaps the periodicity The third problem with the Hunter±Price analysis is their failure to use the appropriate methods when analysing time- Correspondence: Peter Turchin, Department of Ecology and series data.Time-series observations are generally not inde- Evolutionary Biology, University of Connecticut, Storrs, CT 06269- pendent of each other, and this needs to be taken into account by 3043, U.S.A. E-mail: [email protected] the statistical model.To give an example, suppose that the # 2000 Blackwell Science Ltd 119 120 Peter Turchin and Alan A. Berryman Fig. 1. (a) Fluctuations of saw¯y density on dry-site clones (after Hunter & Price, 1998; ®g.2a). (b) Autocorrelation function estimated for the saw¯y data (lines indicate 95% con®dence intervals for each autocorrelation).(c) Precipitation at the ®eld site.(d) Autocorrelation function for precipitation. observed data were generated by the ®rst order process analysing dynamic systems that do not operate on a per-capita Xt =a+bXt ±1+ et.It is likely that there will be a strong basis (after all, raindrops do not reproduce).Thus, in order to correlation between Xt and Xt ±2, but this correlation re¯ects not investigate second-order effects in the rainfall data, the the structural properties of the model but rather the fact that Xt is appropriate approach is to calculate the partial autocorrelation correlated with Xt ±1, which is, in turn, correlated with Xt ±2.By function.Second, it turns out that the analysis of the Hunter± contrast, the expected value of the partial correlation between Price precipitation data reveals no statistically signi®cant Xt and Xt ±2 is zero (Royama, 1992).Apparently, Hunter and partial correlations at any lag (the partial autocorrelations at Price were unaware of this point.To show that saw¯y dynamics the ®rst two lags are 0.15 and ± 0.39; to be signi®cant at 0.05 are driven by delayed density-dependent processes, they level, a correlation coef®cient would need to be at least 0.52 in regressed the realised per capita rate of population change, magnitude).In other words, not only is evidence for statistical rt = ln Nt /Nt ±1, directly on lagged population density, Nt ±2, periodicity absent in both data sets, but there is also no without ®rst removing the effect of Nt ±1.Hence, they did not evidence for second-order dynamics. follow the procedure described by Turchin (1990), who ®rst Finally, it is well known that one cannot use standard tests to regressed rt on Nt ±1and only then tested whether adding Nt ±2to evaluate the presence of density dependence (for a technical the regression would signi®cantly reduce the unexplained but very clear explanation of this problem, see Dennis & variance.Analysing the saw¯y data with the correct approach, Taper, 1994).This problem is particularly acute for short data we ®nd that the term involving Nt ±2 does not result in a series.Thus, contrary to the statement of Hunter and Price that signi®cant improvement of the regression at the 0.05 level `we do not have suf®ciently long time-series to apply (F1,11 = 4.81, 0.05 < P < 0.1). maximum likelihood techniques (Dennis & Taper, 1994)', it What about evidence for delayed density-dependence in the is precisely for short time series that one should use the precipitation data? First, using the per-capita rate of change rt parametric bootstrap method of Dennis and Taper. as the dependent variable is clearly inappropriate when Furthermore, showing that there is a strong correlation # 2000 Blackwell Science Ltd, Ecological Entomology, 25, 119±121 Detecting cycles and delayed density dependence 121 between population numbers and some exogenous variable is tion behavior and survival from 1964 to 1971. Canadian no grounds for claiming that signs of density dependence are Entomologist, 105, 1465±1488. spurious.For a very clear and forceful statement of this, we Berryman, A.A. & Turchin, P. (1997) Detection of delayed density refer the reader to Royama (1992). dependence: comment. Ecology, 78, 318±320. All statistical methods can be misapplied, and perhaps the Box, G.E.P. & Jenkins, G.M. (1976) Time Series Analysis: Forecasting Hunter and Price paper is useful in showing how time-series and Control. Holden Day, Oakland, California. Dennis, B.& Taper, B.(1994) Density dependence in time series analysis can be misused by looking for endogenous effects observations of natural populations: estimation and testing. when it makes no sense to look for them.Hunter and Price, Ecological Monographs, 64, 205±224. however, fail to show any speci®c instances where we or others Ellner, S.& Turchin, P.(1995) Chaos in a noisy world: new methods have ignored evidence for exogenous drivers and analysed the and evidence from time series analysis. American Naturalist, 145, data as though they were generated by endogenous second- 343±375. order mechanisms (in fact, the contrary is the case; see examples Finerty, J.P. (1980) The Population Ecology of Cycles in Small given above).Perhaps the population oscillations in Mammals: Mathematical Theory and Biological Fact.Yale Fennoscandian voles or larch budmoth in Switzerland will University Press, New Haven, Connecticut. eventually be explained by exogenous environmental in¯u- Hunter, M.D. & Price, P.W. (1998) Cycles in insect populations: ences, but so far nobody has been able to advance a convincing delayed density dependence or exogenous driving variables? proposal as to what these exogenous factors might be. Ecological Entomology, 23, 216±222. Finally, we completely agree with the general message of Martinat, P.J.
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