Detecting chaotic dynamics of insect populations from long‐term survey data

Abstract

1. Estimates of the Lyapunov exponent, a statistic that measures the sensitive dependence of the dynamic behaviour of a system on its initial conditions, are used to characterize several sets of insect time series.2. A new method is described to overcome the difficulty of defining the dynamics of an observed, noisy, short ecological time series. This method provides two test statistics for the estimated Lyapunov exponent.3. This method is applied to forty-six time series comprising six aphid species from five sites and four moth species from six sites. There are few positive Lyapunov exponents and none is sufficiently large to characterize its time series as chaotic.4. Two methods to estimate the Lyapunov exponent are compared; that based on logarithmically transformed counts yields less variable estimates for highly variable insect data than that based on untransformed counts.

Population dynamics of two marine polychaetes: the relative role of density dependence, predation, and winter conditions

“…Second, polynomial models using log-transformed population densities were used, as suggested by Perry et al (1993) and Zhou et al (1997). The second-order model, using a second-degree (quadratic) polynomial, is written as: R t =b 0 +b 1 X t +b 11 X 2 t +b 2 X t 1 + b 22 X 2 t 1 +b 12 X t X t 1 + t .…”

“…Studies based on singlespecies models with time delays indeed pointed to the existence of chaotic dynamics in natural populations (Turchin and Taylor, 1992;Turchin, 1993Turchin, , 1996. However, other studies could not confirm this result (Perry et al, 1993;Zhou et al, 1997), and the debate over whether chaotic dynamics are relevant to ecological reality continues.…”

Temperature change and complex dynamics

“…1 are deleted, respectively. This response-surface technique has been used increasingly to study short ecological time series (Schaer 1985;Wolf et al 1985;Casdagli 1989;Turchin and Millstein 1993;Perry et al 1993;Ellner and Turchin 1995;Falck et al 1995;Zhou et al 1997). The equilibrium population density in this model is a complex quadratic function of temperature and other parameters, the number of which depend on the appropriate lag selected.…”

“…There are several methods to de®ne the dynamic behaviour of a model, such as simulation of dynamics, estimation of the Lyapunov exponent and stability analysis (Hassell and May 1973;May 1974May , 1975May , 1976Turchin 1991;Turchin and Taylor 1992;Turchin and Millstein 1993;Bascompte and Sole 1994;Ellner and Turchin 1995;Zhou et al 1997). For noisy time series, especially from ®eld data, the estimation of a Lyapunov exponent may be the best approach to distinguish chaotic dynamics from purely random stochastic¯uctuations.…”

“…Possible examples of chaotic dynamics in biological populations have been detected (Turchin and Taylor 1992;, but the methodology is still under development for short timeseries (Schaer 1985;Wolf et al 1985;Casdagli 1989;Turchin and Millstein 1993;Zhou et al 1997). The apparent rarity (May 1974;Hassell et al 1976) of complex population dynamics in nature could be due to overly simple models and estimation methods used, or to the underestimation of direct density dependence (Perry 1994), or to the lack of models with delayed density dependence (Turchin and Taylor 1992).…”