Why Do Populations Cycle? A Synthesis Of Statistical And Mechanistic Modeling Approaches

Ecology, Sept, 1999 by Bruce E. Kendall, Cheryl J. Briggs, William W. Murdoch, Peter Turchin, Stephen P. Ellner, Edward McCauley, Roger M. Nisbet, Simon N. Wood

INTRODUCTION

The population densities of many species can fluctuate nearly periodically over time, with periods that cannot be explained simply by seasonal variation [ILLUSTRATION FOR FIGURE 1 OMITTED]. These regular, large-amplitude oscillations invite explanation and indeed these data sets and others like them have fascinated generations of ecologists. How can we uncover the mechanisms that drive cycles in population density?

A common approach to studying population cycles has been to perform short-term experimental and observational studies that look directly at population processes that might cause the cycles. This approach has been used intensively, for example, in studies of Canadian hares (Krebs 1996) and British grouse (Watson et al. 1984, Hudson et al. 1992, Moss et al. 1996). This work is invaluable for building up a list of biologically plausible causes of cycles in a population, as was summarized for many species of outbreaking forest insects in Berryman (1988). Often the empirical evidence suggests that some aspect of fecundity or mortality varies with the population's density, and each such result is put forward as a potential governing factor. Nevertheless, these hypotheses are rarely tested rigorously, and the number of competing hypotheses tends to grow through time. A decisive test can be difficult. Although some kind of density dependence is required for population cycles to arise, not all density-dependent interactions will do the job; it is almost impossible to determine the effect of the interaction from verbal models of the population. Only mathematical population models can show which factors are even capable of generating cyclic dynamics; without such models the experimental approach will not be able to solve the question of population cycles.

At the same time, ecologists have been accumulating empirical data of another sort: long time series of population abundances. There are a few dozen "classic" time series, such as those in Fig. 1, that are widely known, but recent work has produced a deluge of new series. For example, a project collecting published time series of 10 or more years has passed the 5000 mark and is still going strong (J. Prendergast, personal communication). 700 of these series are longer than 25 yr, and 30% of those have periodic oscillations (Kendall et al. 1998). These data allow us to go beyond merely characterizing a population as "cyclic," and quantify aspects of the cycle such as the period, amplitude, and maximum growth rate.

How can we employ mathematical models to best use both the above-detailed information on population processes and any long-term population time series, and reach stronger conclusions about the causes of population cycles? We believe that the time is right to bring together two complementary quantitative approaches - time-series statistics and mechanistic population modeling. Four major features distinguish the two approaches: (1) the goals of the analysis; (2) the way in which the time series is treated; (3) the kinds of models that are developed; and (4) the methods used for judging the models' explanatory power. These approaches are now highly developed, and each has been applied individually to the problem of population cycles, but they have rarely been used in concert. Our purposes in this paper are to demonstrate that much is to be gained by bringing these traditions together, to outline a series of steps to achieve that end, and to show a successful case study.

The primary aims of time-series statisticians are to describe the data and extrapolate the time series into the future. This approach analyzes the time series directly to produce descriptors of the dynamics such as period, amplitude, and Lyapunov exponent. The models used are usually non-mechanistic - they either explicitly include statistical features of the data, such as the autocorrelation structure or the spectrum of periodic tendencies, or they use flexible "nonparametric" functions, such as splines or neural networks, to relate future population size to current and past abundances. There are well-understood quantitative assessments of model fit, such as the mean squared prediction error, which are used both for parameter estimation and to compare the fit of different models to a particular data set.

An example of a statistical model that is commonly used with population cycles is the linear autoregressive (AR) model (Royama 1992). This is used both to quantitatively characterize the period and strength of the cycle, and to look for correlations with other oscillatory variables. A well-known application of the latter is the search for the putative correlation between Canadian lynx and hare populations and the sunspot cycle (Moran 1949, Keith 1963, Sinclair et al. 1993, Ranta et al. 1997, Sinclair and Gosline 1997).

Nonlinear models of varying complexity have also been used, both to describe

time series (Lindstrom et al. 1995) and to estimate whether the fluctuations in various populations are chaotic (Hassell et al. 1976, Turchin 1993, Ellner and Turchin 1995).