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

Sylvie Méléard, Michael Rera, Tristan Roget

Ageing's sensitivity to natural selection has long been discussed because of
its apparent negative effect on individual's fitness. Thanks to the recently
described (Smurf) 2-phase model of ageing we were allowed to 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, $x_b$ and
$x_d$ converge during evolution to configurations $x_b-x_d\approx 0$. This
guarantees that a certain proportion of the population maintains the Lansing
effect which in turn, confers higher evolvability to individuals. \\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
$\lbrace x_b=x_d\rbrace$.