| Abstract: |
| Understanding the complex interplay between mutation and selection in
asexuals is a central issue of evolutionary biology. To study the adaptation
of a population under these two forces, in the presence of a phenotype
optimum, we propose two frameworks: (1) an integro-differential approach with
context-dependent mutation kernels $\partial_t p(t,m)=U\, (J_y \star p-p)
(t,m) +p(t,m) (m - \overline{m}(t)), $ with $(J_y \star p)(t,m) =
\int_\mathbb{R} J_y(m-y)p(t,y) \, dy$ and $\overline{m}(t)= \int_\mathbb{R}
y\,p(t,y) \, dy$. In this case, we follow the dynamics of the fitness
distribution $p(t,m)$; (2) a nonlocal nonlinear transport equation satisfied
by a moment generating function of the fitness distribution; the derivation
of this equation is based on microscopic arguments. We show that these two
equations are connected and we derive several properties of their solutions.
These properties have implications in evolutionary biology, regarding the
effect of the parameters (e.g., mutation rate, dimension of the phenotypic
space) on the trajectories of adaptation and on the stationary states. In
particular, we give simple sufficient conditions on the parameters for the
existence and non-existence of a concentration phenomenon at the optimal
fitness value $m=0$. We compare our results with empirical results given by
stochastic individual-based simulations of Wright-Fisher type models. This is
a joint work with Marie-Eve Gil (BioSP, I2M), Fran\c cois Hamel (I2M) and
Guillaume Martin (ISEM). |
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