### Abstract

I discuss one dimensional maps as discrete time models of population dynamics from an extinction-versus-survival point of view by means of bifurcation theory. I extend this approach to a version of these population models that incorporates the dynamics of a single phenotypic trait subject to Darwinian evolution. This is done by proving a fundamental bifurcation theorem for the resulting two dimensional, discrete time model. This theorem describes the bifurcation that occurs when an extinction equilibrium destabilizes. Examples illustrate the application of the theorem. Included is a short summary of generalizations of this bifurcation theorem to the higher dimensional maps that arise when modeling the evolutionary dynamics of a structured population.

Original language | English (US) |
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Title of host publication | Applied Analysis in Biological and Physical Sciences - ICMBAA 2015 |

Publisher | Springer New York LLC |

Pages | 41-62 |

Number of pages | 22 |

Volume | 186 |

ISBN (Print) | 9788132236382 |

DOIs | |

State | Published - 2016 |

Event | International Conference on Recent Advances in Mathematical Biology, Analysis and Applications, ICMBAA 2015 - Aligarh, India Duration: May 25 2016 → May 29 2016 |

### Other

Other | International Conference on Recent Advances in Mathematical Biology, Analysis and Applications, ICMBAA 2015 |
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Country | India |

City | Aligarh |

Period | 5/25/16 → 5/29/16 |

### Keywords

- Allee effects
- Bifurcations
- Difference equations
- Discrete time dynamics
- Equilibria
- Evolutionary dynamics
- Population dynamics
- Stability

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Applied Analysis in Biological and Physical Sciences - ICMBAA 2015*(Vol. 186, pp. 41-62). Springer New York LLC. https://doi.org/10.1007/978-81-322-3640-5_3