A birth–death model of ageing: from individual-based dynamics to evolutive differential inclusions
Ageing’s sensitivity to natural selection has long been discussed because of its apparent negative effect on an individual’s fitness. Thanks to the recently described (Smurf) 2-phase model of ageing (Tricoire and Rera in PLoS ONE 10(11):e0141920, 2015) we propose a fresh angle for modeling the evolution of ageing. Indeed, by coupling a dramatic loss of fertility with a high-risk of impending death—amongst other multiple so-called hallmarks of ageing—the Smurf phenotype allowed us to consider ageing as a couple of sharp transitions. The birth–death model (later called bd-model) we describe here is a simple life-history trait model where each asexual and haploid individual is described by its fertility period \(x_b\) and survival period \(x_d\). We show that, thanks to the Lansing effect, the effect through which the “progeny of old parents do not live as long as those of young parents”, \(x_b\) and \(x_d\) converge during evolution to configurations \(x_b-x_d\approx 0\) in finite time. To do so, we built an individual-based stochastic model which describes the age and trait distribution dynamics of such a finite population. Then we rigorously derive the adaptive dynamics models, which describe the trait dynamics at the evolutionary time-scale. We extend the Trait Substitution Sequence with age structure to take into account the Lansing effect. Finally, we study the limiting behaviour of this jump process when mutations are small. We show that the limiting behaviour is described by a differential inclusion whose solutions \(x(t)=(x_b(t),x_d(t))\) reach the diagonal \(\lbrace x_b=x_d\rbrace \) in finite time and then remain on it. This differential inclusion is a natural way to extend the canonical equation of adaptive dynamics in order to take into account the lack of regularity of the invasion fitness function on the diagonal \(\lbrace x_b=x_d\rbrace \).
KeywordsIndividual-based dynamics Age-structure Ageing Adaptive dynamics Differential inclusions Life-history evolution
Mathematics Subject Classification60J25 60J80 92D15 92D25
We acknowledge partial support by the Chaire Modélisation Mathématique et Biodiversité of Veolia Environment—École Polytechnique—Museum National d’Histoire Naturelle—FX. We acknowledge partial support by CNRS and the ATIP/Avenir-Aviesan kick-starting group leaders program. This work was also supported by a public Grant as part of the investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH. Finally, we acknowledge the reviewers for their interesting and constructive suggestions.
- Clément F, Robin F, Yvinec R (2017) Analysis and calibration of a linear model for structured cell populations with unidirectional motion: application to the morphogenesis of ovarian follicles. arXiv preprint arXiv:1712.05372
- Dambroise E, Monnier L, Ruisheng L, Aguilaniu H, Joly JS, Tricoire H, Rera M (2016) Two phases of aging separated by the smurf transition as a public path to death. Sci Rep 6(23):523Google Scholar
- Darwin C (1871) The origin of species by means of natural selection, or, the preservation of favoured races in the struggle for life. John Murray, LondonGoogle Scholar
- Fabian D, Flatt T (2011) The evolution of ageing. Nat Educ Knowl 3(3):1–10Google Scholar
- Haldane JBS (1942) New paths in genetics. George allen & Unwin, LondonGoogle Scholar
- Markus L (1956) Asymptotically autonomous differential systems. In: Contributions to the theory of nonlinear oscillations. Annals of mathematics studies, vol 3, no 36. Princeton University Press, Princeton, pp 17–29Google Scholar
- Medawar PB (1952) An unsolved problem of biology: an inaugural lecture delivered at University College London. H. K. Lewis, LondonGoogle Scholar
- Metz J, Geritz S, Meszéna G, Jacobs F, van Heerwaarden J (1996) Adaptive dynamics: a geometrical study of the consequences of nearly faithful reproduction. In: Stochastic and spatial structures of dynamical systems. Elsevier Science, Burlington, pp 183–231Google Scholar
- Tran VC (2006) Modèles particulaires stochastiques pour des problèmes d’évolution adaptative et pour l’approximation de solutions statistiques. PhD thesis, Université de Nanterre-Paris XGoogle Scholar
- Weismann A (1881) The origin of the markings of caterpillars. Studies in the theory of descent (trans and ed: R Meldola). Sampson Low Marston Searle Rivington, London, pp 161–389Google Scholar
- Weismann A, Poulton E, Bagnall S, Schönland S, Shipley AE, Arthur Everett S (1891) Essays upon heredity and kindred biological problems, vol 1. Clarendon Press, OxfordGoogle Scholar