Equilibria in structured populations

Research output: Contribution to journalArticle

28 Citations (Scopus)

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.

Original languageEnglish (US)
Pages (from-to)15-39
Number of pages25
JournalJournal of Mathematical Biology
Volume23
Issue number1
DOIs
StatePublished - Dec 1985

Fingerprint

Structured Populations
Population Density
Continuum
population density
Bifurcation
Equilibrium State
nonlinear models
Parturition
Population
Density Dependence
population size
Nonlinear Dynamics
Global Bifurcation
Growth Process
Nonuniqueness
Linear Functional
Local Stability
Bifurcation Diagram
Population Size
Nonlinear Model

Keywords

  • Bifurcation
  • Equilibria
  • Stability
  • Structured populations

ASJC Scopus subject areas

  • Mathematics (miscellaneous)
  • Agricultural and Biological Sciences (miscellaneous)

Cite this

Equilibria in structured populations. / Cushing, Jim M.

In: Journal of Mathematical Biology, Vol. 23, No. 1, 12.1985, p. 15-39.

Research output: Contribution to journalArticle

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