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Equilibria in structured populations.

J M Cushing

    Journal of Mathematical Biology
    |January 1, 1985
    PubMed
    Summary
    This summary is machine-generated.

    This study analyzes population density equilibrium using bifurcation theory. It reveals a continuum of stable population states arising from general nonlinear models, impacting population dynamics.

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    Area of Science:

    • Mathematical Biology
    • Population Dynamics
    • Bifurcation Theory

    Background:

    • Population density and its stable equilibrium are crucial in ecological modeling.
    • Internal population structure (e.g., age, size) complicates equilibrium analysis.
    • Nonlinear density-dependent processes are common in biological systems.

    Purpose of the Study:

    • To investigate the existence and stability of positive equilibrium states for structured populations.
    • To analyze these states as a bifurcation problem using a birth modulus parameter.
    • To explore conditions for global positivity and stability of equilibrium states.

    Main Methods:

    • Bifurcation analysis of nonlinear population model equations.
    • Utilizing an inherent birth modulus (n) as the bifurcation parameter.

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  • Examining density-dependent birth and growth processes.

    • Main Results:

    • A global, unbounded continuum of nontrivial equilibrium pairs (n, rho) bifurcates from a critical point.
    • These equilibrium pairs are locally positive, with conditions for global positivity established.
    • Local stability is contingent on the direction of bifurcation.
    • A method for constructing global bifurcation diagrams is presented for specific density-dependent models.

    Conclusions:

    • The study demonstrates a general mechanism for generating diverse stable population equilibria in structured populations.
    • Bifurcation theory provides a powerful framework for understanding population stability and dynamics.
    • The findings offer insights into uniqueness and nonuniqueness of positive equilibria, with applications to existing ecological models.