  For a continuous state branching process with two types of individuals which
are subject to selection and density dependent competition, we characterize the
joint evolution of population size, type configurations and genealogies as the
unique strong solution of a system of SDE's. Our construction is achieved in
the lookdown framework and provides a synthesis as well as a generalization of
cases considered separately in two seminal papers by Donnelly and Kurtz (1999),
namely fluctuating population sizes under neutrality, and selection with
constant population size. As a conceptual core in our approach we introduce the
selective lookdown space which is obtained from its neutral counterpart through
a state-dependent thinning of ``potential'' selection/competition events whose
rates interact with the evolution of the type densities. The updates of the
genealogical distance matrix at the ``active'' selection/competition events are
obtained through an appropriate sampling from the selective lookdown space. The
solution of the above mentioned system of SDE's is then mapped into the joint
evolution of population size and symmetrized type configurations and
genealogies, i.e. marked distance matrix distributions. By means of Kurtz'
Markov mapping theorem, we characterize the latter process as the unique
solution of a martingale problem. For the sake of transparency we restrict the
main part of our presentation to a prototypical example with two types, which
contains the essential features. In the final section we outline an extension
to processes with multiple types including mutation.

Probability