An extinction/persistence threshold for sexually rep roducing populations

The cone spectral radius

Wen Jin, Horst Thieme

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

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.

Original languageEnglish (US)
Pages (from-to)447-470
Number of pages24
JournalDiscrete and Continuous Dynamical Systems - Series B
Volume21
Issue number2
DOIs
StatePublished - Mar 1 2016

Fingerprint

Spectral Radius
Extinction
Persistence
Cones
Cone
Local Stability
Normed vector space
Vector spaces
Ordered Vector Space
Threshold Parameter
Semiflow
Structured Populations
Discrete-time Model
Population Model
First-order
Approximation

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

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abstract = "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.",
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AU - Thieme, Horst

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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.

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KW - Basic reproduction number

KW - Basic turnover number

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KW - Cone spectral radius

KW - Eigenfunctional

KW - Homogeneous map

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KW - Local stability

KW - Order derivative

KW - Order persistence

KW - Persistence

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