Estimating chaos and complex dynamics in an insect population

A defining hypothesis of theoretical ecology during the past century has be en that population fluctuations might largely be explained by relatively lo w-dimensional, nonlinear ecological interactions, provided such interaction s could be correctly identified and modeled. The realization in recent deca des that such nonlinear interactions might result in chaos and other exotic dynamic behaviors has been exciting but tantalizing, in that attributing t he fluctuations of a particular real population to the complex dynamics of a particular mathematical model has proved to be an elusive goal. We experi mentally tested a model-predicted sequence of transitions (bifurcations) in the dynamic behavior of a population from stable equilibria to quasiperiod ic and periodic cycles to chaos to three-cycles using cultures of the flour beetle Tribolium. The predictions arose from a system of difference equati ons (the LPA model) describing the nonlinear life-stage interactions, predo minantly cannibalism. We built a stochastic version of the model incorporat ing demographic variability and obtained conditional least-squares estimate s for the model parameters. We generated 2000 "bootstrapped data sets" with a time-series bootstrap technique, and for each set we reestimated the mod el parameters. The resulting 2000 bootstrapped parameter vectors were used to obtain confidence intervals for the model parameters and estimated distr ibutions of the Liapunov exponents for the deterministic portion (the skele ton) of the model as well as fur the full stochastic model. Frequency distr ibutions of estimated dynamic behaviors of the skeleton at each experimenta l treatment were produced. For one treatment, over 83% of the bootstrapped parameter estimates corresponded to chaotic attractors, and the remainder o f the estimates yielded high-period cycles. The low-dimensional skeleton ac counted for at least 90% of the variability in the population abundances an d accurately described the responses of populations to experimental demogra phic manipulations, including treatments for which the predicted dynamic be havior was chaos. Demographic stochasticity described the remaining noise q uite well. We conclude that the fluctuations of experimental hour battle po pulations are explained largely by known nonlinear forces involving canniba listic-stage interactions. Claims of dynamic behavior such as periodic cycl es or chaos must be accompanied by a consideration of the reliability of th e estimated parameters and a realization that the population fluctuations a re a blend of deterministic forces and stochastic events.