Discrete-time population dynamics on the state space of measures

Horst R. Thieme

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


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 languageEnglish (US)
Pages (from-to)1168-1217
Number of pages50
JournalMathematical Biosciences and Engineering
Issue number2
StatePublished - 2020


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

ASJC Scopus subject areas

  • Modeling and Simulation
  • General Agricultural and Biological Sciences
  • Computational Mathematics
  • Applied Mathematics


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