  We model and study the genetic evolution and conservation of a population of
diploid hermaphroditic organisms, evolving continuously in time and subject to
resource competition. In the absence of mutations, the population follows a
3-type nonlinear birth-and-death process, in which birth rates are designed to
integrate Mendelian reproduction. We are interested in the long term genetic
behaviour of the population (adaptive dynamics), and in particular we compute
the fixation probability of a slightly non-neutral allele in the absence of
mutations, which involves finding the unique sub-polynomial solution of a
nonlinear 3-dimensional recurrence relationship. This equation is simplified to
a 1-order relationship which is proved to admit exactly one bounded solution.
Adding rare mutations and rescaling time, we study the successive mutation
fixations in the population, which are given by the jumps of a limiting Markov
process on the genotypes space. At this time scale, we prove that the fixation
rate of deleterious mutations increases with the number of already fixed
mutations, which creates a vicious circle called the extinction vortex.
