# A birth–death model of ageing: from individual-based dynamics to evolutive differential inclusions

## Abstract

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 \(\lbrace x_b=x_d\rbrace \).

## Keywords

Individual-based dynamics Age-structure Ageing Adaptive dynamics Differential inclusions Life-history evolution## Mathematics Subject Classification

60J25 60J80 92D15 92D25## Notes

### Acknowledgements

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.

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