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.
In: Journal of Mathematical Biology, Vol. 24, No. 6, 02.1987, p. 627-649.Research output: Contribution to journal › Article
}
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 -