Nonlinear Stochastic Population Dynamics: The Flour Beetle Tribolium as an Effective Tool of Discovery

Prior to the seminal work of R.M. May in the 1970s, the prevailing paradigm viewed the unpredictable fluctuations in population time-series data as random effects due to environmental variability and/or measurement errors. In the absence of environmental variability, according to this view, population numbers would either equilibrate or settle into regular periodic oscillations. May's (1974) suggestion that simple deterministic rules might explain the complex fluctuations observed in animal abundances led to an intense search for chaos in extant population data. The results of the search were suggestive, but equivocal, and May's hypothesis remained the subject of lively debate (Zimmer, 1999; Perry et al., 2000). We took a different approach. Our interdisciplinary research team composed of statisticians, mathematicians, and biologists gathered together in the early 1990s to document experimentally the occurrence of nonlinear dynamic phenomena in biological populations. We began with the idea that nonlinear theory yields testable hypotheses concerning changes in the dynamical behaviors of populations. For example, in the case of the quadratic map (sometimes called the "logistic" map), changes in the intrinsic growth rate lead to a sequence of dynamical behaviors from equilibria to periodic cycles, to aperiodic chaotic behavior. Our thought was that a sequence of changes in dynamical behavior, which is a common feature of nonlinear models, could be tested, in principle, under controlled laboratory conditions. This would provide a connection between theory and data that was missing from ecology. From the very beginning of our collaboration, fundamental questions greeted us at every turn as we looked at historical time-series data and at data collected in our laboratories. Over the years we struggled to combine deterministic concepts such as equilibria, cycles, saddle nodes, bifurcations, basins of attraction, multiple attractors, resonance, and chaos with observations. What would a stable equilibrium, let alone chaos, look like in a population? Could a saddle node be invoked as an explanation for different transient behaviors of time series among replicate populations? Is chaos even possible if we consider discrete-state population models? Is it useful to consider populations as discrete-state stochastic systems? In ecological theory, a central (and abiding) problem is to situate deterministic theory in the context of biological systems where important demographic events are probabilistic. Chance variation, in such fundamental biological processes as the number of offspring per adult and the chance of an individual surviving to adulthood, is a part of population dynamics. Probabilistic variation enters the overall research effort in the statistical methods associated with model identification, parameter estimation, and model validation. Chance events are also a component of the interpretation of population behavior; probabilistic variation is essential to the explanation of ecological time-series data. We expand on these points. First, a mathematical population model, built and tested as a serious scientific hypothesis, must be somehow connected to data. A probabilistic version of the model must be constructed to account for inevitable deviations of data from the predictions of the deterministic model. Demographic/environmental variability must be modeled in order to construct an appropriate estimating function for the model parameters (based on the likelihood or conditional sums of squares, for instance). Statistical diagnostic procedures should be used to evaluate the uncertainty component of the model. Second, chance events interact with deterministic forces to produce emergent dynamic behaviors. The deterministic skeleton fixes the geometry of state space, providing a stage for the transient dance of stochasticity. Chance events allow the system to visit (and re-visit) the various deterministic entities on the stage, including unstable invariant sets, which under strict deterministic theory would have little or no impact on population time series. Ecological time series can display a stochastic mix of many of the dynamic features of the skeleton, including multiple attractors, transients, unstable invariant sets (such as saddles and unstable manifolds) and lattice effects. Stochasticity enlarges the repertoire of time-series orbits; each population, even in a set of laboratory replicates, may display a unique sequence of population abundances. In this chapter, we expand upon the message that in order to understand population fluctuations, deterministic and stochastic forces must be viewed as an integral part of the ecological system. We begin by explaining our models, both animal and mathematical. We then discuss how we estimated model parameters and validated the model. Next, with the parameterized model in hand, we present an overview of some of the nonlinear phenomena and related topics that we have documented in our experimental system: chaotic dynamics, population outbreaks, saddle nodes, phase switching, lattice effects, the anatomy of chaos and, finally, mechanistic models of stochasticity.