Journal of Mathematical Biology

, Volume 75, Issue 2, pp 491–520 | Cite as

A bifurcation theorem for evolutionary matrix models with multiple traits

  • J. M. CushingEmail author
  • F. Martins
  • A. A. Pinto
  • Amy Veprauskas


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.


Nonlinear matrix models Structured population dynamics Evolutionary game theory Bifurcation Equilibria Stability 

Mathematics Subject Classification

92D25 92D15 39A28 37G99 



J. M. Cushing and A. Veprauskas were supported by the U.S. National Science Foundation grant DMS 0917435. A. A. Pinto thanks the support of LIAAD—INESC TEC through program PEst, the Faculty of Sciences of University of Porto and Portuguese Foundation for Science and Technology (FCT—Fundação para a Ciência e a Tecnologia) through the Project “Dynamics, Optimization and Modelling, with with reference PTDC/MAT-NAN/6890/2014. This work is financed by the ERDF—European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation—COMPETE 2020 Programme, and by National Funds through the FCT—Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) within project «POCI-01-0145-FEDER-006961. It is also supported by the project NanoSTIMA: Macro-to-Nano Human Sensing: Towards Integrated Multimodal Health Monitoring and Analytics/NORTE-01-0145-FEDER-000016” is financed by the North Portugal Regional Operational Programme (NORTE 2020), under the PORTUGAL 2020 Partnership Agreement, and through the European Regional Development Fund (ERDF). F. Martins thanks the financial support of Portuguese Foundation for Science and Technology (FCT—Fundação para a Ciência e a Tecnologia) through a PhD. scholarship of the programme MAP-PDMA. (Reference: PD/BD/105726/2014). The authors are grateful for the comments of two anonymous reviewers and the handling editor, which were exceptionally helpful in improving the paper.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ArizonaTucsonUSA
  2. 2.Interdisciplinary Program in Applied MathematicsUniversity of ArizonaTucsonUSA
  3. 3.Department of Mathematics, Faculty of SciencesUniversity of Porto and LIAAD-INESCPortoPortugal

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