Abstract:
If a structured population is considered and the individual state space is given
by a metric space S, measures $\mu$ on the $\sigma$-algebra of Borel subsets T of S offer
a modeling tool with a natural interpretation: $\mu(T)$ is the number of individuals
with structural characteristics in the set T. A discrete-time population model is
given by a 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. F can
be interpreted as a population turnover map. Under suitable assumptions, F has
a first order approximation at the zero measure (the extinction fixed point), which
is a positive linear operator. This first order approximation can be interpreted as a
basic population turnover operator. In the case of a semelparous population, it can
be identified with the next generation operator. A spectral radius can be defined
by the usual Gelfand formula and be interpreted as a basic population turnover
number. We investigate in how far it serves as a threshold parameter between population
extinction and population persistence [1]. 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 or
a positive eigenfunctional which can be used in standard persistence techniques. A
suitable alternative is the flat norm (also known as (dual) bounded Lipschitz norm)
[2], 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 finite real measures.
References
[1] W. Jin, H.R. Thieme, An extinction/persistence threshold for sexually reproduc-
ing populations: the cone spectral radius, Disc. Cont. Dyn. Systems B 21 (2016),
447-470
[2] P. Gwiazda, A. Marciniak-Czochra, H.R. Thieme, Measures under the
at norm
as ordered normed vector space, Positivity 22 (2018), 105-138, Correction: Positivity
22 (2018)