### Abstract

The classic Beverton-Holt (discrete logistic) difference equation, which arises in population dynamics, has a globally asymptotically stable equilibrium (for positive initial conditions) if its coefficients are constants. If the coefficients change in time, then the equation becomes nonautonomous and the asymptotic dynamics might not be as simple. One reason the coefficients can change in time is their evolution by natural selection. If the model coefficients are functions of a heritable phenotypic trait subject to natural selection then, by standard methods for modeling evolution, the model becomes a planar system of coupled difference equations, consisting of a Beverton-Holt type equation for the population dynamics and a difference equation for the dynamics of the mean phenotypic trait. We consider a case when the trait equation uncouples from the population dynamic equation and obtain criteria under which the evolutionary system has globally asymptotically stable equilibria or periodic solutions.

Original language | English (US) |
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Title of host publication | Springer Proceedings in Mathematics and Statistics |

Publisher | Springer New York LLC |

Pages | 127-141 |

Number of pages | 15 |

Volume | 102 |

ISBN (Print) | 9783662441398 |

DOIs | |

State | Published - 2014 |

Event | 19th International Conference on Difference Equations and Applications, ICDEA 2013 - Muscat, Oman Duration: May 26 2013 → May 30 2013 |

### Other

Other | 19th International Conference on Difference Equations and Applications, ICDEA 2013 |
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Country | Oman |

City | Muscat |

Period | 5/26/13 → 5/30/13 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Springer Proceedings in Mathematics and Statistics*(Vol. 102, pp. 127-141). Springer New York LLC. https://doi.org/10.1007/978-3-662-44140-4_7

**An Evolutionary Beverton-Holt Model.** / Cushing, Jim M.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Springer Proceedings in Mathematics and Statistics.*vol. 102, Springer New York LLC, pp. 127-141, 19th International Conference on Difference Equations and Applications, ICDEA 2013, Muscat, Oman, 5/26/13. https://doi.org/10.1007/978-3-662-44140-4_7

}

TY - GEN

T1 - An Evolutionary Beverton-Holt Model

AU - Cushing, Jim M

PY - 2014

Y1 - 2014

N2 - The classic Beverton-Holt (discrete logistic) difference equation, which arises in population dynamics, has a globally asymptotically stable equilibrium (for positive initial conditions) if its coefficients are constants. If the coefficients change in time, then the equation becomes nonautonomous and the asymptotic dynamics might not be as simple. One reason the coefficients can change in time is their evolution by natural selection. If the model coefficients are functions of a heritable phenotypic trait subject to natural selection then, by standard methods for modeling evolution, the model becomes a planar system of coupled difference equations, consisting of a Beverton-Holt type equation for the population dynamics and a difference equation for the dynamics of the mean phenotypic trait. We consider a case when the trait equation uncouples from the population dynamic equation and obtain criteria under which the evolutionary system has globally asymptotically stable equilibria or periodic solutions.

AB - The classic Beverton-Holt (discrete logistic) difference equation, which arises in population dynamics, has a globally asymptotically stable equilibrium (for positive initial conditions) if its coefficients are constants. If the coefficients change in time, then the equation becomes nonautonomous and the asymptotic dynamics might not be as simple. One reason the coefficients can change in time is their evolution by natural selection. If the model coefficients are functions of a heritable phenotypic trait subject to natural selection then, by standard methods for modeling evolution, the model becomes a planar system of coupled difference equations, consisting of a Beverton-Holt type equation for the population dynamics and a difference equation for the dynamics of the mean phenotypic trait. We consider a case when the trait equation uncouples from the population dynamic equation and obtain criteria under which the evolutionary system has globally asymptotically stable equilibria or periodic solutions.

UR - http://www.scopus.com/inward/record.url?scp=84906841959&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84906841959&partnerID=8YFLogxK

U2 - 10.1007/978-3-662-44140-4_7

DO - 10.1007/978-3-662-44140-4_7

M3 - Conference contribution

AN - SCOPUS:84906841959

SN - 9783662441398

VL - 102

SP - 127

EP - 141

BT - Springer Proceedings in Mathematics and Statistics

PB - Springer New York LLC

ER -