### Abstract

For nonlinear scalar difference equations that arise in population dynamics the geometry of the graph obtained by plotting the population growth rate as a function of inherent fertility leads to information about the number of positive equilibria and about the local stability of positive equilibria. Specifically, equilibria on decreasing segments of this graph are always unstable. Equilibria on increasing segments are stable in two circumstances: when the equilibrium is sufficiently close either to 0 or to a critical point on the graph. These geometric criteria are shown to hold for a class of nonlinear Leslie models in which (age-specific) survival rates are population density independent and fertilities are dependent on a weighted total population size. Examples are given to show how this geometric method can be used to identity strong Allee and hysteresis effects in these models.

Original language | English (US) |
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Title of host publication | Difference Equations and Discrete Dynamical Systems with Applications - 24th ICDEA 2018 |

Editors | Martin Bohner, Stefan Siegmund, Roman Šimon Hilscher, Petr Stehlík |

Publisher | Springer |

Pages | 201-211 |

Number of pages | 11 |

ISBN (Print) | 9783030355012 |

DOIs | |

State | Published - Jan 1 2020 |

Event | 24th International Conference on Difference Equations and Applications, ICDEA 2018 - Dresden, Germany Duration: May 21 2018 → May 25 2018 |

### Publication series

Name | Springer Proceedings in Mathematics and Statistics |
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Volume | 312 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Conference

Conference | 24th International Conference on Difference Equations and Applications, ICDEA 2018 |
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Country | Germany |

City | Dresden |

Period | 5/21/18 → 5/25/18 |

### Fingerprint

### Keywords

- Allee effects
- Bifurcation
- Hysteresis
- Leslie matrix models
- Stability

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Difference Equations and Discrete Dynamical Systems with Applications - 24th ICDEA 2018*(pp. 201-211). (Springer Proceedings in Mathematics and Statistics; Vol. 312). Springer. https://doi.org/10.1007/978-3-030-35502-9_8