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

Evolving genealogies for branching populations under selection and competition