  When studying the dynamics of trait distribution of populations in a
heterogeneous environment, classical models from quantitative genetics choose
to look at its system of moments, specifically the first two ones.
Additionally, in order to close the resulting system of equations, they often
assume that the trait distribution is Gaussian (see for instance Ronce and
Kirkpatrick 2001). The aim of this paper is to introduce a mathematical
framework that follows the whole trait distribution (without prior assumption)
to study evolutionary dynamics of sexually reproducing populations.
Specifically, it focuses on complex traits, whose inheritance can be encoded by
the infinitesimal model of segregation (Fisher 1919). We show that it allows us
to derive a regime in which our model gives the same dynamics as when assuming
a Gaussian trait distribution. To support that, we compare the stationary
problems of the system of moments derived from our model with the one given in
Ronce and Kirkpatrick 2001 and show that they are equivalent under this regime
and do not need to be otherwise. Moreover, under this regime of equivalence, we
show that a separation bewteen ecological and evolutionary time scales arises.
A fast relaxation toward monomorphism allows us to reduce the complexity of the
system of moments, using a slow-fast analysis. This reduction leads us to
complete, still in this regime, the analytical description of the bistable
asymmetrical equilibria numerically found in Ronce and Kirkpatrick 2001. More
globally, we provide explicit modelling hypotheses that allow for such local
adaptation patterns to occur.
