Estimating Chaos and Complex Dynamics in an Insect Population

Abstract

A defining hypothesis of theoretical ecology during the past century has been that population fluctuations might largely be explained by relatively low-dimensional, nonlinear ecological interactions, provided such interactions could be correctly identified and modeled. The realization in recent decades that such nonlinear interactions might result in chaos and other exotic dynamic behaviors has been exciting but tantalizing, in that attributing the fluctuations of a particular real population to the complex dynamics of a particular mathematical model has proved to be an elusive goal. We experimentally tested a modelpredicted sequence of transitions (bifurcations) in the dynamic behavior of a population from stable equilibria to quasiperiodic and periodic cycles to chaos to three-cycles using cultures of the flour beetle Tribolium. The predictions arose from a system of difference equations (the LPA model) describing the nonlinear life-stage interactions, predominantly cannibalism. We built a stochastic version of the model incorporating demographic variability and obtained conditional least-squares estimates for the model parameters. We generated 2000 ''bootstrapped data sets'' with a time-series bootstrap technique, and for each set we reestimated the model parameters. The resulting 2000 bootstrapped parameter vectors were used to obtain confidence intervals for the model parameters and estimated distributions of the Liapunov exponents for the deterministic portion (the skeleton) of the model as well as for the full stochastic model. Frequency distributions of estimated dynamic behaviors of the skeleton at each experimental treatment were produced. For one treatment, over 83% of the bootstrapped parameter estimates corresponded to chaotic attractors, and the remainder of the estimates yielded high-period cycles. The low-dimensional skeleton accounted for at least 90% of the variability in the population abundances and accurately described the responses of populations to experimental demographic manipulations, including treatments for which the predicted dynamic behavior was chaos. Demographic stochasticity described the remaining noise quite well. We conclude that the fluctuations of experimental flour beetle populations are explained largely by known nonlinear forces involving cannibalistic-stage interactions. Claims of dynamic behavior such as periodic cycles or chaos must be accompanied by a consideration of the reliability of the estimated parameters and a realization that the population fluctuations are a blend of deterministic forces and stochastic events.

Stable bifurcations in semelparous Leslie models

“…For these numbers, ρ 2 ranges from 0.17414 to −37.994 and, as a result, the bifurcating positive equilibria are stable for β 23 = 0 and unstable for β 23 = 1. This model prediction is confirmed by the experimental outcomes reported in [3,12,16]. It is also interesting to note that for β 23 = 1, the beetle cultures displayed synchronous oscillations in which the three life stages were non-overlapping.…”

“…The well-known LPA model is the basic model used in extensive experimental studies of nonlinear dynamics involving the species Tribolium (flour beetles) [4,12]. The LPA model is not semelparous, but in many of the key experiments, the protocol manipulated the adult death rate to nearly 100% and therefore gave the insects, in effect, a semelparous life history [3,12,16]. Therefore, one might reasonably take the semelparous LPA model as an approximate model for these experiments.…”

“…In the experimental studies of nonlinear dynamics in Tribolium [3,12,16], numerical estimates of the parameters were made from data and the inter-class competition parameter β 23 was experimentally manipulated. Specifically, β 31 = 0.01731, β 33 = 0.01310, s 1 = 0.8 and β 23 ranged from 0 to 1.…”

Cycle Chains and the LPA Model

“…However, the constraint m a ¼ 1 does not permit an application to the chaos experiment and study described in Refs. [4,8,10] where m a ¼ 0:96: Indeed, in that application the chaotic attractor is not synchronous.…”

“…In several long term experiments concerned with complex population dynamics (reported, for example, in Refs. [4,7,8,10]) the adult death rate was manipulated to equal 96% and hence m a ¼ 0:96: Motivated by this fact, we consider the LPA model (1) with m a ¼ 1: Biologically, this means no adults survive longer than one unit of time (two weeks in the case of applications to flour beetles). When m a ¼ 1 the inherent net reproductive number is…”