### Abstract

If the individual state space of a structured population is given by a metric space S , measures μ on the θ-algebra of Borel subsets T of S offer a modeling tool with a natural interpretation: μ (T) is the number of individuals with structural characteristics in the set T. A discrete-time population model is given by a population turnover map F on the cone of finite nonnegative Borel measures that maps the structural population distribution of a given year to the one of the next year. Under suitable assumptions, F has a first order approximation at the zero measure (the extinction fixed point), which is a positive linear operator on the ordered vector space of real measures and can be interpreted as a basic population turnover operator. For a semelparous population, it can be identified with the next generation operator. A spectral radius can be defined by the usual Gelfand formula.We investigate in how far it serves as a threshold parameter between population extinction and population persistence. The variation norm on the space of measures is too strong to give the basic turnover operator enough compactness that its spectral radius is an eigenvalue associated with a positive eigenmeasure. A suitable alternative is the flat norm (also known as (dual) bounded Lipschitz norm), which, as a trade-off, makes the basic turnover operator only continuous on the cone of nonnegative measures but not on the whole space of real measures.

Original language | English (US) |
---|---|

Pages (from-to) | 1168-1217 |

Number of pages | 50 |

Journal | Mathematical Biosciences and Engineering |

Volume | 17 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2020 |

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### Keywords

- Basic reproduction number
- Eigenvectors
- Extinction
- Feller property
- Fixed point
- Flat norm
- Measure kernels
- Spectral radius

### ASJC Scopus subject areas

- Modeling and Simulation
- Agricultural and Biological Sciences(all)
- Computational Mathematics
- Applied Mathematics

### Cite this

**Discrete-time population dynamics on the state space of measures.** / Thieme, Horst R.

Research output: Contribution to journal › Article

*Mathematical Biosciences and Engineering*, vol. 17, no. 2, pp. 1168-1217. https://doi.org/10.3934/mbe.2020061

}

TY - JOUR

T1 - Discrete-time population dynamics on the state space of measures

AU - Thieme, Horst R.

PY - 2020/1/1

Y1 - 2020/1/1

N2 - If the individual state space of a structured population is given by a metric space S , measures μ on the θ-algebra of Borel subsets T of S offer a modeling tool with a natural interpretation: μ (T) is the number of individuals with structural characteristics in the set T. A discrete-time population model is given by a population turnover map F on the cone of finite nonnegative Borel measures that maps the structural population distribution of a given year to the one of the next year. Under suitable assumptions, F has a first order approximation at the zero measure (the extinction fixed point), which is a positive linear operator on the ordered vector space of real measures and can be interpreted as a basic population turnover operator. For a semelparous population, it can be identified with the next generation operator. A spectral radius can be defined by the usual Gelfand formula.We investigate in how far it serves as a threshold parameter between population extinction and population persistence. The variation norm on the space of measures is too strong to give the basic turnover operator enough compactness that its spectral radius is an eigenvalue associated with a positive eigenmeasure. A suitable alternative is the flat norm (also known as (dual) bounded Lipschitz norm), which, as a trade-off, makes the basic turnover operator only continuous on the cone of nonnegative measures but not on the whole space of real measures.

AB - If the individual state space of a structured population is given by a metric space S , measures μ on the θ-algebra of Borel subsets T of S offer a modeling tool with a natural interpretation: μ (T) is the number of individuals with structural characteristics in the set T. A discrete-time population model is given by a population turnover map F on the cone of finite nonnegative Borel measures that maps the structural population distribution of a given year to the one of the next year. Under suitable assumptions, F has a first order approximation at the zero measure (the extinction fixed point), which is a positive linear operator on the ordered vector space of real measures and can be interpreted as a basic population turnover operator. For a semelparous population, it can be identified with the next generation operator. A spectral radius can be defined by the usual Gelfand formula.We investigate in how far it serves as a threshold parameter between population extinction and population persistence. The variation norm on the space of measures is too strong to give the basic turnover operator enough compactness that its spectral radius is an eigenvalue associated with a positive eigenmeasure. A suitable alternative is the flat norm (also known as (dual) bounded Lipschitz norm), which, as a trade-off, makes the basic turnover operator only continuous on the cone of nonnegative measures but not on the whole space of real measures.

KW - Basic reproduction number

KW - Eigenvectors

KW - Extinction

KW - Feller property

KW - Fixed point

KW - Flat norm

KW - Measure kernels

KW - Spectral radius

UR - http://www.scopus.com/inward/record.url?scp=85075666751&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85075666751&partnerID=8YFLogxK

U2 - 10.3934/mbe.2020061

DO - 10.3934/mbe.2020061

M3 - Article

AN - SCOPUS:85075666751

VL - 17

SP - 1168

EP - 1217

JO - Mathematical Biosciences and Engineering

JF - Mathematical Biosciences and Engineering

SN - 1547-1063

IS - 2

ER -