Complex Dynamics in Ecological Time Series

Abstract

Although the possibility of complex dynamical behaviors–limit cycles, quasiperodic oscillations, and aperiodic chaos–has been recognized theoretically, most ecologists are skeptical of their importance in nature. In this paper we develop a methodology for reconstructing endogenous (or deterministic) dynamics from ecological time series. Our method consists of fitting a response surface to the yearly population change as a function of lagged population densities. Using the version of the model that includes two lags, we fitted time—series data for 14 insect and 22 vertebrate populations. The 14 insect populations were classified as: unregulated (1 case), exponentially stable (three cases), damped oscillations (six cases), limit cycles (one case), quasiperiodic oscillations (two cases), and chaos (one case). The vertebrate examples exhibited a similar spectrum of dynamics, although there were no cases of chaos. We tested the results of the response—surface methodology by calculating autocorrelation functions for each time series. Autocorrelation patterns were in agreement with our findings of periodic behaviors (damped oscillations, limit cycles, and quasiperiodicity). On the basis of these results, we conclude that the complete spectrum of dynamical behaviors, ranging from exponential stability to chaos, is likely to be found among natural populations.

Will Sympatric Speciation Fail due to Stochastic Competitive Exclusion?

“…More precisely, we assume these random deviations are additive on a log scale, a common assumption in population modeling and ecological time series analysis (e.g., Royama 1981;Turchin and Taylor 1992;Bjørnstad et al 1995;Ives et al 1999). Populations with identical traits can be expected to respond to environmental fluctuations in a similar way; that is, we expect a high-correlation r between their respective deviations from the deterministic growth rate.…”

Effect of resource subsidies on predator–prey population dynamics: a mathematical model

“…Sustained temporal fluctuations in natural population densities are common [59], and predatorprey systems in particular are often noted for their oscillations [34,39,61]. Murdoch [41] suggested that such interactions can be stabilized when predators are able to switch from a preferred prey that is rare to alternatives that are more common [10].…”

Stochastic analogues of deterministic single-species population models

“…Population fluctuations can arise from variation in the environment over time (Walther et al, 2002;Saether, 1997), from intrinsic 'demographic' stochasticity arising from variation in the number of offspring produced per individual (Bartlett, 1960;Royama, 1992), from errors in observations (Valpine & Hastings, 2002) and from deterministic non-linear dynamics, such as cycles and chaos (May, 1976;Turchin, 2003;Berryman, 1999). The combination of these effects, coupled with the fact that ecological time-series are often short, makes the construction of predictive mathematical models notoriously difficult (Turchin & Taylor, 1992). Indeed, the central requirement in developing mathematical models of population ecology is that stochastic and deterministic factors can be weighted appropriately.…”