### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 1-30 |

Number of pages | 30 |

Journal | Journal of Mathematical Biology |

DOIs | |

State | Accepted/In press - Jan 6 2017 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Modeling and Simulation
- Agricultural and Biological Sciences (miscellaneous)
- Applied Mathematics

### Cite this

*Journal of Mathematical Biology*, 1-30. https://doi.org/10.1007/s00285-016-1091-4

**A bifurcation theorem for evolutionary matrix models with multiple traits.** / Cushing, Jim M; Martins, F.; Pinto, A. A.; Veprauskas, Amy.

Research output: Contribution to journal › Article

*Journal of Mathematical Biology*, pp. 1-30. https://doi.org/10.1007/s00285-016-1091-4

}

TY - JOUR

T1 - A bifurcation theorem for evolutionary matrix models with multiple traits

AU - Cushing, Jim M

AU - Martins, F.

AU - Pinto, A. A.

AU - Veprauskas, Amy

PY - 2017/1/6

Y1 - 2017/1/6

N2 - 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.

AB - 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.

KW - Bifurcation

KW - Equilibria

KW - Evolutionary game theory

KW - Nonlinear matrix models

KW - Stability

KW - Structured population dynamics

UR - http://www.scopus.com/inward/record.url?scp=85008471347&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85008471347&partnerID=8YFLogxK

U2 - 10.1007/s00285-016-1091-4

DO - 10.1007/s00285-016-1091-4

M3 - Article

C2 - 28062892

AN - SCOPUS:85008471347

SP - 1

EP - 30

JO - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

SN - 0303-6812

ER -