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    A bifurcation theorem for nonlinear matrix models of population dynamics

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    Final Accepted Manuscript
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    Author
    Cushing, J. M.
    Farrell, Alex P.
    Affiliation
    Univ Arizona, Dept Math
    Issue Date
    2019-12-08
    Keywords
    Nonlinear difference equations
    matrix equations
    population dynamics
    equilibrium
    bifurcation
    stability
    
    Metadata
    Show full item record
    Publisher
    TAYLOR & FRANCIS LTD
    Citation
    J. M. Cushing & Alex P. Farrell (2019) A bifurcation theorem for nonlinear matrix models of population dynamics, Journal of Difference Equations and Applications, DOI: 10.1080/10236198.2019.1699916
    Journal
    JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS
    Rights
    © 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 2019
    ISSN
    1023-6198
    DOI
    10.1080/10236198.2019.1699916
    Version
    Final accepted manuscript
    Sponsors
    NSFNational Science Foundation (NSF) [DMS-1407564]
    ae974a485f413a2113503eed53cd6c53
    10.1080/10236198.2019.1699916
    Scopus Count
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