We have the pleasure, thanks to the support of the ESSEC IDS dpt, Institut des Actuaires, LabEx MME-DII, the group BFA of SFdS, and all ARLES (Aging Risks and their Long-term impact on the Economy and Society research project) partners to invite:
Prof. Peter JAGERS
Department of Mathematical Sciences
Chalmers University of Technology and
University of Gothenburg, Sweden
Date and place.
: Thursday, March 25, at 12:30 pm (Paris) and 7:30 pm (Singapore)
To participate in the Working Group (via Zoom), please click here
Password/Code : WGRisk
“ Salvaging Galton's Unreasonable Thesis: All Populations Must Die Out ! "
Science knows a number of basic, paradigmatic themes. Extinction or persistence of populations (or for that sake of population-reproduction-like structures like nuclear chain reactions or epidemic spread) is one of them. It was first formulated during the 19th Century, in a demographic context, the frequent extinction of (noble) family names. For long the established, mathematical truth about extinction was Galton’s contention, that all families must die out, not reconciled with Malthus’s thesis of exponential growth of populations as a whole. The conflict between these viewpoints was resolved only in 1930 (after being known to Bienaymé at least in 1845, but seemingly forgotten) and only for branching style processes where individuals act independently.
Exponential growth can not prevail in a finite world where there is a ceiling for total population size, a carrying capacity in ecological terminology. And such a barrier cannot be strict; it should allow population size to oscillate around it. This leads to populations changing at discrete (but arbitrary and possibly random) time points, the conditional expected change, given the complete past population history, being negative, whenever population size exceeds what is suitably termed a weak carrying capacity. Further, it is natural to assume that there be a strictly positive risk of population decrease, given the preceding population history, at times when the population is smaller than the weak carrying capacity. Then, the population must die out. This sharpens the result in Jagers and Zuyev (J. Math. Biol. 2020). The reader should note that nothing specific is assumed about life and reproduction of individuals, nor about independence between different individuals, just the assumptions mentioned. In mathematical terms they are:
Let Z(n) denote the population size immediately after the n-th population size change, n≥1, Z(0) the starting number of population members, and ℱn the sigma-field of all events in the population up to and including those at the n-th population change. By assumption, there is a smallest K>0 (the weak carrying capacity) such that :
(1) Z(n) >K ⟹ E[Z(n+1)| ℱn] ≤ Z(n)
(2) ∃ 𝜀 > 0; Z(n) ≤K ⟹ 𝐏(Z(n+1) < Z(n) | ℱn) ≥ 𝜀,
and (3) Z(n)=0 ⟹ Z(n+1)=0.
The consequence then, is that P(Z(n) →0) =1, i.e. the population dies out almost surely.
- Galton, F. and Watson, H. W. (1875) On the probability of the extinction of families. J. Anthropol. Soc. London 4, 138–144.
- Jagers, P, and Zuyev, S. (2020) Populations in environments with a soft carrying capacity are eventually extinct. J. Math. Biol. 81(3), 845-851. DOI 10.1007/s00285-020-01527-5.
- Jagers, P. and Zuyev, S. (2021) A soft carrying capacity combined with a definite decrease risk leads to population extinction. Submitted.
Jeremy Heng, Olga Klopp and Marie Kratz
and Riada Djebbar (Singapore Actuarial Society - ERM)