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When studying the dynamics of trait distribution of populations in aheterogeneous environment, classical models from quantitative genetics chooseto look at its system of moments, specifically the first two ones.Additionally, in order to close the resulting system of equations, they oftenassume that the trait distribution is Gaussian (see for instance Ronce andKirkpatrick 2001). The aim of this paper is to introduce a mathematicalframework 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 bythe infinitesimal model of segregation (Fisher 1919). We show that it allows usto derive a regime in which our model gives the same dynamics as when assuminga Gaussian trait distribution. To support that, we compare the stationaryproblems of the system of moments derived from our model with the one given inRonce and Kirkpatrick 2001 and show that they are equivalent under this regimeand do not need to be otherwise. Moreover, under this regime of equivalence, weshow that a separation bewteen ecological and evolutionary time scales arises.A fast relaxation toward monomorphism allows us to reduce the complexity of thesystem of moments, using a slow-fast analysis. This reduction leads us tocomplete, still in this regime, the analytical description of the bistableasymmetrical equilibria numerically found in Ronce and Kirkpatrick 2001. Moreglobally, we provide explicit modelling hypotheses that allow for such localadaptation patterns to occur.

Evolutionary dynamics of complex traits in sexual populations in a strongly heterogeneous environment: how normal?