A global bifurcation theorem for Darwinian matrix models. Issue 8 (2nd August 2016)

• Record Type: Journal Article
• Authors: Meissen, Emily P.; Salau, Kehinde R.; Cushing, Jim M.
• Is Part Of: Journal of difference equations and applications. Volume 22:Issue 8(2016)
• Page Start: 1114
• Page End: 1136
• Publication Date: 2016-08-02
• Subjects: Nonlinear matrix models -- evolutionary population dynamics -- bifurcation -- stability -- Allee effects; 37N25 -- 39A60 -- 92D15 -- 92D25; Difference equations -- Periodicals; 515.625
• DOI: 10.1080/10236198.2016.1177522
• Languages: English
• ISSNs: 1023-6198

Abstract

Abstract : Motivated by models from evolutionary population dynamics, we study a general class of nonlinear difference equations called matrix models. Under the assumption that the projection matrix is non-negative and irreducible, we prove a theorem that establishes the global existence of a continuum with positive equilibria that bifurcates from an extinction equilibrium at a value of a model parameter at which the extinction equilibrium destabilizes. We give criteria for the global shape of the continuum, including local direction of bifurcation and its relationship to the local stability of the bifurcating positive equilibria. We discuss a relationship between backward bifurcations and Allee effects. Illustrative examples are given.