University of Heidelberg, **Applied Analysis Seminar**

Prof. Dr. Horst Thieme (Arizona State University)

Discrete-time population dynamics on measures

*2/104, Statistics’ Seminar-Room*

Discrete-time population dynamics on measures

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)

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