TY - GEN

T1 - Reproduction Number Versus Turnover Number in Structured Discrete-Time Population Models

AU - Thieme, Horst R.

N1 - Funding Information:
The author thanks Odo Diekmann and Roger Nussbaum for helpful comments and two anonymous referees for their constructive remarks. Special thanks goes to Senada Kalabusic for the extraordinary help in adapting the script to the style demands of the proceedings.
Publisher Copyright:
© 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.

PY - 2023

Y1 - 2023

N2 - The analysis of the discrete-time dynamics of structured iteroparous populations involves a basic yearly turnover operator B= A+ H with a structural transition operator A and a mating and fertility operator H. A and H map a normal complete cone X+ of an ordered normed vector space X into itself and are (positively) homogenous and continuous on X+, A is additive and H is order-preserving. Assume that r(A) < 1 for the spectral radius of A. Let HR1 with R1=∑j=0∞Aj be the next generation operator and T= r(B), the spectral radius of B, be the (basic) turnover number and R= r(HR1) be the (basic) reproduction number. We explore conditions for a turnover/reproduction trichotomy, namely one (and only one) of the following three possibilities to hold: (i) 1 < T≤ R, (ii) 1 = T= R, (iii) 1 > T≥ R. In some cases, one may also like to consider the lower reproduction number R⋄= lim λ→1+r(HRλ), Rλ=∑j=0∞λ-(n+1)An. R⋄ is also useful to study the case r(A) = 1 to explore conditions for the dichotomy 1 = T≥ R⋄ or 1 < T≤ R⋄≤ ∞.

AB - The analysis of the discrete-time dynamics of structured iteroparous populations involves a basic yearly turnover operator B= A+ H with a structural transition operator A and a mating and fertility operator H. A and H map a normal complete cone X+ of an ordered normed vector space X into itself and are (positively) homogenous and continuous on X+, A is additive and H is order-preserving. Assume that r(A) < 1 for the spectral radius of A. Let HR1 with R1=∑j=0∞Aj be the next generation operator and T= r(B), the spectral radius of B, be the (basic) turnover number and R= r(HR1) be the (basic) reproduction number. We explore conditions for a turnover/reproduction trichotomy, namely one (and only one) of the following three possibilities to hold: (i) 1 < T≤ R, (ii) 1 = T= R, (iii) 1 > T≥ R. In some cases, one may also like to consider the lower reproduction number R⋄= lim λ→1+r(HRλ), Rλ=∑j=0∞λ-(n+1)An. R⋄ is also useful to study the case r(A) = 1 to explore conditions for the dichotomy 1 = T≥ R⋄ or 1 < T≤ R⋄≤ ∞.

KW - Cones

KW - Continuity of the spectral radius

KW - Eigenvector

KW - Extinction

KW - Feller kernel

KW - Generation growth factor

KW - Homogeneous operators

KW - Integral projection models

KW - Integro-difference equations

KW - Mating function

KW - Net reproductive value

KW - Ordered vector spaces

KW - Pair-formation function

KW - Population growth factor

KW - Rank structure

KW - Resolvent

KW - Stability

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U2 - 10.1007/978-3-031-25225-9_23

DO - 10.1007/978-3-031-25225-9_23

M3 - Conference contribution

AN - SCOPUS:85152583983

SN - 9783031252242

T3 - Springer Proceedings in Mathematics and Statistics

SP - 495

EP - 539

BT - Advances in Discrete Dynamical Systems, Difference Equations and Applications - 26th ICDEA, 2021

A2 - Elaydi, Saber

A2 - Kulenović, Mustafa R.S.

A2 - Kalabušić, Senada

PB - Springer

T2 - 26th International Conference on Difference Equations and Applications, ICDEA 2021

Y2 - 26 July 2021 through 30 July 2021

ER -