Equilibria in systems of interacting structured populations

Research output: Contribution to journalArticle

8 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)627-649
Number of pages23
JournalJournal of Mathematical Biology
Volume24
Issue number6
DOIs
StatePublished - Feb 1987

Fingerprint

Structured Populations
Bifurcation
Birth Rate
Population Density
Population
birth rate
Parturition
Mortality
Linear Functionals
Local Stability
population density
Growth
predation
Critical value
Modulus
Continuum
Restriction
Direction compound
Dependent
Community

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.

In: Journal of Mathematical Biology, Vol. 24, No. 6, 02.1987, p. 627-649.

Research output: Contribution to journalArticle

@article{f04a31abe86b487eb1ee6c3757d5de6b,
title = "Equilibria in systems of interacting structured populations",
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.",
keywords = "Bifurcation, Communities of structured populations, Competition, Equilibria, Interacting species, Predator-prey inter-actions, Stability",
author = "Cushing, {Jim M}",
year = "1987",
month = "2",
doi = "10.1007/BF00275507",
language = "English (US)",
volume = "24",
pages = "627--649",
journal = "Journal of Mathematical Biology",
issn = "0303-6812",
publisher = "Springer Verlag",
number = "6",

}

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 -