Nonlinear Demographic Dynamics: Mathematical Models, Statistical Methods, and Biological Experiments

Abstract

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—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 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 understanding nonlinear ecological dynamics.

Dynamics of experimental populations of native and introduced blowflies (Diptera: Calliphoridae): mathematical modelling and the transition from asymptotic equilibrium to bounded oscillations

“…These results are even more significant in the light of recent findings by Cavalieri & Koçak (1995) who demonstrated the transition from steady cycles to continuous chaos by adjusting birth and death rates in populations of the corn borer, Ostrinia nubalis. Similar results were reported by Costantino et al (1995) and Dennis et al (1995) for Tribolium. The findings by Costantino et al (1995), Dennis et al (1995), and Cavalieri and Koçak (1995) …”

Multiple Attractors, Saddles, and Population Dynamics in Periodic Habitats

“…The LPA model has been validated by means of a number of experiments, and has been used successfully to predict transitions between equilibria, periodic cy-cles, invariant loops, and chaos, as well as saddle phenomena and phase switching in population cycles (see Costantino et al, 1995Costantino et al, , 1997Costantino et al, , 1998Cushing et al, 1996Cushing et al, , 1998Dennis et al, 1995Dennis et al, , 1997Desharnais et al, 1997;Beniot et al, 1998;Henson et al, 1998). Henson and Cushing (1997) and Costantino et al (1998) modified the autonomous LPA model (1) to account for the periodic flour volume.…”

“…To explain this phenomenon, the discrete LPA Tribolium model of Dennis et al (1995Dennis et al ( , 1997 and Costantino et al (1995Costantino et al ( , 1997 was modified by Henson and Cushing (1997) and Costantino et al (1998) to account for the periodic flour volume. Cannibalism between life cycle stages is the nonlinear mechanism driving flour beetle dynamics in these experiments (Park et al, 1970).…”

Modelling competition and hybridization between native cutthroat trout and nonnative rainbow and hybrid trout

“…These stochastic components provide a framework for parameter estimation and model comparison using one-step-ahead predictions and maximum likelihood estimation (MLE), a method that has been well developed for discrete-time population models [10,11,12,33,76]. The general model is given by…”