### Abstract

Persistence and local stability of the extinction state are studied for discrete-time population models x n = F(x_{n-1}), n ∈ N, with a map F on the cone X_{+} of an ordered normed vector space X. Since sexual reproduction is accounted for, the first order approximation of F at 0 is an order-preserving homogeneous map B on X_{+} that is not additive. The cone spectral radius of B acts as the threshold parameter that separates persistence from local stability of 0, the extinction state. An important ingredient of the persistence theory for the induced semiflow is a homogeneous order-preserving eigenfunctional θ: X+ → R+ of B that is associated with the cone spectral radius and interacts with an appropriate persistence function σ. Applications are presented for spatially distributed or rank-structured populations that reproduce sexually.

Original language | English (US) |
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Pages (from-to) | 447-470 |

Number of pages | 24 |

Journal | Discrete and Continuous Dynamical Systems - Series B |

Volume | 21 |

Issue number | 2 |

DOIs | |

State | Published - Mar 1 2016 |

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### Keywords

- Basic reproduction number
- Basic turnover number
- Cone derivative
- Cone spectral radius
- Eigenfunctional
- Homogeneous map
- Krein-Rutman
- Local stability
- Order derivative
- Order persistence
- Persistence

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics