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 -