Discrete-time population dynamics on the state space of measures

• Published in: Mathematical biosciences and engineering : MBE Vol. 17; no. 2; pp. 1168 - 1217
• Main Author: Thieme, Horst R
• Format: Journal Article
• Language: English
• Published: United States AIMS Press 01-01-2020
• Subjects: basic reproduction number; eigenvectors; extinction; feller property; fixed point; flat norm; measure kernels; spectral radius; extinction; fixed point; eigenvectors; measure kernels; Feller property; flat norm; spectral radius; basic reproduction number
• ISSN: 1551-0018; 1551-0018
• DOI: 10.3934/mbe.2020061

If the individual state space of a structured population is given by a metric space
, measures
on the
-algebra of Borel subsets
of
offer a modeling tool with a natural interpretation:
(
) is the number of individuals with structural characteristics in the set
. A discrete-time population model is given by a
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,
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
. For a semelparous population, it can be identified with the
. 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.