### Abstract

The existence of a stable positive equilibrium density for a community of k interacting structured species is studied as a bifurcation problem. Under the assumption that a subcommunity of k-1 species has a positive equilibrium and under only very mild restrictions on the density dependent vital growth rates, it is shown that a global continuum of equilibria for the full community bifurcates from the subcommunity equilibrium at a unique critical value of a certain inherent birth modulus for the kth species. Local stability is shown to depend upon the direction of bifurcation. The direction of bifurcation is studied in more detail for the case when vital per unity birth and death rates depend on population density through positive linear functionals of density and for the important case of two interacting species. Some examples involving competition, predation and epidemics are given.

Original language | English (US) |
---|---|

Pages (from-to) | 627-649 |

Number of pages | 23 |

Journal | Journal of Mathematical Biology |

Volume | 24 |

Issue number | 6 |

DOIs | |

State | Published - Feb 1987 |

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### Keywords

- Bifurcation
- Communities of structured populations
- Competition
- Equilibria
- Interacting species
- Predator-prey inter-actions
- Stability

### ASJC Scopus subject areas

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

### Cite this

**Equilibria in systems of interacting structured populations.** / Cushing, Jim M.

Research output: Contribution to journal › Article

*Journal of Mathematical Biology*, vol. 24, no. 6, pp. 627-649. https://doi.org/10.1007/BF00275507

}

TY - JOUR

T1 - Equilibria in systems of interacting structured populations

AU - Cushing, Jim M

PY - 1987/2

Y1 - 1987/2

N2 - The existence of a stable positive equilibrium density for a community of k interacting structured species is studied as a bifurcation problem. Under the assumption that a subcommunity of k-1 species has a positive equilibrium and under only very mild restrictions on the density dependent vital growth rates, it is shown that a global continuum of equilibria for the full community bifurcates from the subcommunity equilibrium at a unique critical value of a certain inherent birth modulus for the kth species. Local stability is shown to depend upon the direction of bifurcation. The direction of bifurcation is studied in more detail for the case when vital per unity birth and death rates depend on population density through positive linear functionals of density and for the important case of two interacting species. Some examples involving competition, predation and epidemics are given.

AB - The existence of a stable positive equilibrium density for a community of k interacting structured species is studied as a bifurcation problem. Under the assumption that a subcommunity of k-1 species has a positive equilibrium and under only very mild restrictions on the density dependent vital growth rates, it is shown that a global continuum of equilibria for the full community bifurcates from the subcommunity equilibrium at a unique critical value of a certain inherent birth modulus for the kth species. Local stability is shown to depend upon the direction of bifurcation. The direction of bifurcation is studied in more detail for the case when vital per unity birth and death rates depend on population density through positive linear functionals of density and for the important case of two interacting species. Some examples involving competition, predation and epidemics are given.

KW - Bifurcation

KW - Communities of structured populations

KW - Competition

KW - Equilibria

KW - Interacting species

KW - Predator-prey inter-actions

KW - Stability

UR - http://www.scopus.com/inward/record.url?scp=0023098864&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0023098864&partnerID=8YFLogxK

U2 - 10.1007/BF00275507

DO - 10.1007/BF00275507

M3 - Article

C2 - 3572260

AN - SCOPUS:0023098864

VL - 24

SP - 627

EP - 649

JO - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

SN - 0303-6812

IS - 6

ER -