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 identities boundaries in parameter space where stable equilibria bifurcate to periodic 2-cycles 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 Hour 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 understanding nonlinear ecological dynamics.