  We consider nonlinear mutation selection models, known as replicator-mutator
equations in evolutionary biology. They involve a nonlocal mutation kernel and
a confining fitness potential. We prove that the long time behaviour of the
Cauchy problem is determined by the principal eigenelement of the underlying
linear operator. The novelties compared to the literature on these models are
about the case of symmetric mutations: we propose a new milder sufficient
condition for the existence of a principal eigenfunction, and we provide what
is to our knowledge the first quantification of the spectral gap. We also
recover existing results in the non-symmetric case, through a new approach.
