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Equilibria in structured populations

Journal of Mathematical Biology volume 23, pages 15–39 (1985)Cite this article

Abstract

The existence of a stable positive equilibrium state for the density ρ of a population which is internally structured by means of a single scalar such as age, size, etc. is studied as a bifurcation problem. Using an inherent birth modulus n as a bifurcation parameter it is shown for very general nonlinear model equations, in which vital birth and growth processes depend on population density, that a global unbounded continuum of nontrivial equilibrium pairs (n, ρ) bifurcates from the unique (normalized) critical point (1, 0). The pairs are locally positive and conditions are given under which the continuum is globally positive. Local stability is shown to depend on the direction of bifurcation. For the important case when density dependence is a nonlinear expression involving a linear functional of density (such as total population size) it is shown how a detailed global bifurcation diagram is easily constructed in applications from the graph of a certain real valued function obtained from an invariant on the continuum. Uniqueness and nonuniqueness of positive equilibrium states are studied. The results are illustrated by several applications to models appearing in the literature.

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Affiliations

  1. Department of Mathematics, Building 89, University of Arizona, 85721, Tucson, Arizona, USA

    J. M. Cushing

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  1. J. M. Cushing
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Additional information

This research was done while the author was on leave at the Lehrstuhl für Biomathematik, Universität Tübingen, Auf der Morgenstelle 10, 7400 Tübingen 1, Federal Republic of Germany

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Cushing, J.M. Equilibria in structured populations. J. Math. Biology 23, 15–39 (1985). https://doi.org/10.1007/BF00276556

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  • DOI: https://doi.org/10.1007/BF00276556

Keywords

  • Structured populations
  • Equilibria
  • Stability
  • Bifurcation