### 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 |

### Fingerprint

### 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

### Cite this

**An extinction/persistence threshold for sexually rep roducing populations : The cone spectral radius.** / Jin, Wen; Thieme, Horst.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - An extinction/persistence threshold for sexually rep roducing populations

T2 - The cone spectral radius

AU - Jin, Wen

AU - Thieme, Horst

PY - 2016/3/1

Y1 - 2016/3/1

N2 - Persistence and local stability of the extinction state are studied for discrete-time population models x n = F(xn-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.

AB - Persistence and local stability of the extinction state are studied for discrete-time population models x n = F(xn-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.

KW - Basic reproduction number

KW - Basic turnover number

KW - Cone derivative

KW - Cone spectral radius

KW - Eigenfunctional

KW - Homogeneous map

KW - Krein-Rutman

KW - Local stability

KW - Order derivative

KW - Order persistence

KW - Persistence

UR - http://www.scopus.com/inward/record.url?scp=84954533519&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84954533519&partnerID=8YFLogxK

U2 - 10.3934/dcdsb.2016.21.447

DO - 10.3934/dcdsb.2016.21.447

M3 - Article

VL - 21

SP - 447

EP - 470

JO - Discrete and Continuous Dynamical Systems - Series B

JF - Discrete and Continuous Dynamical Systems - Series B

SN - 1531-3492

IS - 2

ER -