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The dynamics of density dependent population models

J Guckenheimer, G Oster, A Ipaktchi

    Journal of Mathematical Biology
    |May 23, 1977
    PubMed
    Summary
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    This study analyzes complex population dynamics using dynamical systems theory. It reveals chaotic behavior and "strange attractors" in density-dependent models, offering new analytical approaches.

    Area of Science:

    • Mathematical Biology
    • Dynamical Systems Theory
    • Population Ecology

    Background:

    • Density-dependent population models exhibit complex dynamics, often explored via numerical simulations.
    • Understanding the mathematical underpinnings of this complexity is crucial for ecological forecasting.

    Purpose of the Study:

    • To theoretically analyze the mechanisms driving complexity in density-dependent population models.
    • To apply modern dynamical systems theory to understand population dynamics.
    • To investigate bifurcations and chaotic behavior in a two-age-class Leslie model.

    Main Methods:

    • Theoretical analysis using dynamical systems theory.
    • Investigation of one-dimensional difference equations for chaotic behavior.
    • Application to a density-dependent Leslie model with two age classes.

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  • Analysis of bifurcations and the demonstration of strange attractors.

    • Main Results:

    • Demonstrated the existence of a strange attractor in a two-age-class Leslie model, representing a complex, non-equilibrium, non-cyclic attracting set.
    • Observed essentially random behavior near the strange attractor, indicative of chaos.
    • Identified a novel approach for the statistical analysis of dynamics within the chaotic regime.
    • Extended findings to higher-dimensional and continuous population models.

    Conclusions:

    • Dynamical systems theory provides a powerful framework for understanding complex population dynamics.
    • Strange attractors and chaotic behavior are inherent features in certain density-dependent models.
    • The study offers new theoretical tools for analyzing chaotic population dynamics across various model types.