### 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 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.

Original language | English (US) |
---|---|

Pages (from-to) | 261-281 |

Number of pages | 21 |

Journal | Ecological Monographs |

Volume | 65 |

Issue number | 3 |

State | Published - 1995 |

### Fingerprint

### Keywords

- Bifurcation analysis
- Chaos
- Cycles
- Equilibrium
- Hypothesis testing evaluation
- Nonlinear demographic dynamics
- Parameter estimation
- Stability
- Tribolium

### ASJC Scopus subject areas

- Ecology

### Cite this

*Ecological Monographs*,

*65*(3), 261-281.

**Nonlinear demographic dynamics : Mathematical models, statistical methods, and biological experiments.** / Dennis, Brian; Desharnais, Robert A.; Cushing, Jim M; Costantino, Robert F.

Research output: Contribution to journal › Article

*Ecological Monographs*, vol. 65, no. 3, pp. 261-281.

}

TY - JOUR

T1 - Nonlinear demographic dynamics

T2 - Mathematical models, statistical methods, and biological experiments

AU - Dennis, Brian

AU - Desharnais, Robert A.

AU - Cushing, Jim M

AU - Costantino, Robert F

PY - 1995

Y1 - 1995

N2 - 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.

AB - 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.

KW - Bifurcation analysis

KW - Chaos

KW - Cycles

KW - Equilibrium

KW - Hypothesis testing evaluation

KW - Nonlinear demographic dynamics

KW - Parameter estimation

KW - Stability

KW - Tribolium

UR - http://www.scopus.com/inward/record.url?scp=0028976114&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0028976114&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0028976114

VL - 65

SP - 261

EP - 281

JO - Ecological Monographs

JF - Ecological Monographs

SN - 0012-9615

IS - 3

ER -