title: A bifurcation theorem for evolutionary matrix models with multiple traits.
abstract: One fundamental question in biology is population extinction and persistence, i.e., stability/instability of the extinction equilibrium and of non-extinction equilibria. In the case of nonlinear matrix models for structured populations, a bifurcation theorem answers this question when the projection matrix is primitive by showing the existence of a continuum of positive equilibria that bifurcates from the extinction equilibrium as the inherent population growth rate passes through 1. This theorem also characterizes the stability properties of the bifurcating equilibria by relating them to the direction of bifurcation, which is forward (backward) if, near the bifurcation point, the positive equilibria exist for inherent growth rates greater (less) than 1. In this paper we consider an evolutionary game theoretic version of a general nonlinear matrix model that includes the dynamics of a vector of mean phenotypic traits subject to natural selection. We extend the fundamental bifurcation theorem to this evolutionary model. We apply the results to an evolutionary version of a Ricker model with an added Allee component. This application illustrates the theoretical results and, in addition, several other interesting dynamic phenomena, such as backward bifurcation induced strong Allee effects.
authors.affiliation: Department of Mathematics, University of Arizona, 617 N Santa Rita, Tucson, AZ, 85721, USA. cushing@math.arizona.edu.|Interdisciplinary Program in Applied Mathematics, University of Arizona, 617 N Santa Rita, Tucson, AZ, 85721, USA. cushing@math.arizona.edu.
authors.forename: J M
authors.identifier: 0000-0003-1844-661X
authors.initials: JM
authors.lastname: Cushing
authors.affiliation: Department of Mathematics, Faculty of Sciences, University of Porto and LIAAD-INESC, Rua Do Campo Alegre 687, 4169-007, Porto, Portugal.
authors.forename: F
authors.initials: F
authors.lastname: Martins
authors.affiliation: Department of Mathematics, Faculty of Sciences, University of Porto and LIAAD-INESC, Rua Do Campo Alegre 687, 4169-007, Porto, Portugal.
authors.forename: A A
authors.initials: AA
authors.lastname: Pinto
authors.affiliation: Interdisciplinary Program in Applied Mathematics, University of Arizona, 617 N Santa Rita, Tucson, AZ, 85721, USA.
authors.forename: Amy
authors.initials: A
authors.lastname: Veprauskas
mesh_terms: D005075:Biological Evolution; D005716:Game Theory; D008954:Models, Biological; D017711:Nonlinear Dynamics; D010641:Phenotype; D011158:Population Growth
keywords: Bifurcation; Equilibria; Evolutionary game theory; Nonlinear matrix models; Stability; Structured population dynamics