  This article deals with the emergence of a specific mating preference pattern
called homogamy in a population. Individuals are characterized by their
genotype at two haploid loci, and the population dynamics is modelled by a
non-linear birth-and-death process. The first locus codes for a phenotype,
while the second locus codes for homogamy defined with respect to the first
locus: two individuals are more (resp. less) likely to reproduce with each
other if they carry the same (resp. a different) trait at the first locus.
Initial resident individuals do not feature homogamy, and we are interested in
the probability and time of invasion of a mutant presenting this characteristic
under a large population assumption. To this aim, we study the trajectory of
the birth-and-death process during three phases: growth of the mutant,
coexistence of the two types, and extinction of the resident. We couple the
birth-and-death process with simpler processes, like branching processes or
dynamical systems, and study the latter ones in order to control the trajectory
and duration of each phase.
