A bifurcation theorem for nonlinear matrix models of population dynamics
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Cushing-Farrell-a-general-bifu ...
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Final Accepted Manuscript
Affiliation
Univ Arizona, Dept MathIssue Date
2019-12-08Keywords
Nonlinear difference equationsmatrix equations
population dynamics
equilibrium
bifurcation
stability
Metadata
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TAYLOR & FRANCIS LTDCitation
J. M. Cushing & Alex P. Farrell (2019) A bifurcation theorem for nonlinear matrix models of population dynamics, Journal of Difference Equations and Applications, DOI: 10.1080/10236198.2019.1699916Rights
© 2019 Informa UK Limited, trading as Taylor & Francis Group.Collection Information
This item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at repository@u.library.arizona.edu.Abstract
We prove a general theorem for nonlinear matrix models of the type used in structured population dynamics that describes the bifurcation that occurs when the extinction equilibrium destabilizes as a model parameter is varied. The existence of a bifurcating continuum of positive equilibria is established, and their local stability is related to the direction of bifurcation. Our theorem generalizes existing theorems found in the literature in two ways. First, it allows for a general appearance of the bifurcation parameter (existing theorems require the parameter to appear linearly). This significantly widens the applicability of the theorem to population models. Second, our theorem describes circumstances in which a backward bifurcation can produce stable positive equilibria (existing theorems allow for stability only when the bifurcation is forward). The signs of two diagnostic quantities determine the stability of the bifurcating equilibrium and the direction of bifurcation. We give examples that illustrate these features.Note
12 month embargo; published online: 8 December 2019ISSN
1023-6198Version
Final accepted manuscriptSponsors
NSFNational Science Foundation (NSF) [DMS-1407564]ae974a485f413a2113503eed53cd6c53
10.1080/10236198.2019.1699916