### 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 language | English (US) |
---|---|

Pages (from-to) | 15-39 |

Number of pages | 25 |

Journal | Journal of Mathematical Biology |

Volume | 23 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1985 |

### Fingerprint

### 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.

Research output: Contribution to journal › Article

*Journal of Mathematical Biology*, vol. 23, no. 1, pp. 15-39. https://doi.org/10.1007/BF00276556

}

TY - JOUR

T1 - Equilibria in structured populations

AU - Cushing, Jim M

PY - 1985/12

Y1 - 1985/12

N2 - 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.

AB - 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.

KW - Bifurcation

KW - Equilibria

KW - Stability

KW - Structured populations

UR - http://www.scopus.com/inward/record.url?scp=0022345912&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0022345912&partnerID=8YFLogxK

U2 - 10.1007/BF00276556

DO - 10.1007/BF00276556

M3 - Article

C2 - 4078497

AN - SCOPUS:0022345912

VL - 23

SP - 15

EP - 39

JO - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

SN - 0303-6812

IS - 1

ER -