R. F. Costantino

NONLINEAR DEMOGRAPHIC DYNAMICS: MATHEMATICAL MODELS, STATISTICAL METHODS, AND BIOLOGICAL EXPERIMENTS'

Our approach to testing nonlinear population theory is to connect rigorously mathematical models with data by means of statistical methods for nonlinear time series. We begin by deriving a biologically based demographic model. The mathematical analysis identifies boundaries in parameter space where stable equilibria bifurcate to periodic 2-cy- cles and aperiodic motion on invariant loops. The statistical analysis, based on a stochastic version of the demographic model, provides procedures for parameter estimation, hypothesis testing, and model evaluation. Experiments using the flour beetle Tribolium yield the time series data. A three-dimensional map of larval, pupal, and adult numbers forecasts four possible population behaviors: extinction, equilibria, periodicities, and aperiodic motion including chaos. This study documents the nonlinear prediction of periodic 2-cycles in laboratory cultures of Tribolium and represents a new interdisciplinary approach to un- derstanding nonlinear ecological dynamics.

PHASE SWITCHING IN POPULATION CYCLES

Oscillatory populations may exhibit a phase change in which, for example, a high–low periodic pattern switches to a low–high pattern. We propose that phase shifts correspond to stochastic jumps between basins of attraction in an appropriate phase space which associates the different phases of a periodic cycle with distinct attractors. This mechanism accounts for two-cycle phase shifts and the occurrence of asynchronous replicates in experimental cultures of Tribolium.

Anatomy of a chaotic attractor: Subtle model-predicted patterns revealed in population data

Mathematically, chaotic dynamics are not devoid of order but display episodes of near-cyclic temporal patterns. This is illustrated, in interesting ways, in the case of chaotic biological populations. Despite the individual nature of organisms and the noisy nature of biological time series, subtle temporal patterns have been detected. By using data drawn from chaotic insect populations, we show quantitatively that chaos manifests itself as a tapestry of identifiable and predictable patterns woven together by stochasticity. We show too that the mixture of patterns an experimentalist can expect to see depends on the scale of the system under study.

Explaining and predicting patterns in stochastic population systems

Lattice effects in ecological time–series are patterns that arise because of the inherent discreteness of animal numbers. In this paper, we suggest a systematic approach for predicting lattice effects. We also show that an explanation of all the patterns in a population time–series may require more than one deterministic model, especially when the dynamics are complex.

What we learned

This chapter summarizes the concepts and models associated with chaos and bifurcations. It reviews the modeling methodology, nonlinear phenomenon, and dynamics of biological populations. Model assessment is important to distinguish between procedures to parameterize (calibrate) a model and those to validate a model. Part of the success of the deterministic LPA model can be attributed to the models design around the dominant mechanism known to drive the dynamics of the species. Another aspect of the mechanistic modeling approach that is considered is the inclusion of stochasticity. The connection of the deterministic model to real data is accomplished by modeling the expected deviations of data from model predictions in terms of probabilistic assumptions concerning environmental and demographic sources of noise. One conclusion drawn from the studies is that the deterministic LPA model provides a remarkably accurate description of the flour beetle cultures under various laboratory conditions. It made accurate a priori predictions of the long-term dynamics of beetle populations when demographic parameters of the populations are changed by direct manipulation. One of the features of nonlinearity is that responses are not (necessarily) proportional to disturbances; they might be very unexpected and nonintuitive. Animal numbers or densities come from a finite set of discrete numbers called lattice. The LPA model provides an accurate explanation of patterns. The chapter provides insights into nonlinear population dynamics and a modest step toward the hardening of ecological science.

Patterns in chaos

This chapter highlights one of the chaotic treatments in the route-to-chaos experiment. A study of complex dynamics—in particular, chaotic dynamics—requires a sufficiently long time series of data. In this experiment, chaotic treatment is considered in more detail by examining some temporal patterns predicted by the chaotic attractor. Sensitivity to initial conditions is the subject of a follow-up experiment designed to test hallmark property of chaos. The model-predicted consequences of small perturbations to population numbers near the most sensitive region of the chaotic attractor—the hot spot—are dramatically borne out by the experimental data. In addition to corroborating the influence of the chaotic attractor on the beetle populations, the hot-spot experiment also serves as an interesting demonstration of controlling chaos. Temporal patterns in chaos are complicated. The investigation of temporal patterns in the chaos treatment leads to another insights into modeling the dynamics of biological populations. Recurrent temporal pattern that is observed in the experimental data accounts for the lattice effect, a pattern arising from the fact that animals come in whole numbers and data necessarily lie on a discrete lattice of points in state space.