  We consider a population growth model given by a two-type continuous-state
branching process with immigration and competition, introduced by Ma. We study
the relative frequency of one of the types in the population when the total
mass is forced to be constant at a dense set of times. The resulting process is
described as the solution to an SDE, which we call the culled frequency
process, generalizing the $\Lambda$-asymmetric frequency process introduced by
Caballero et al. We obtain conditions for the culled frequency process to have
a moment dual and show that it is given by a branching-coalescing
continuous-time Markov chain that describes the genealogy of the two-type CBI
with competition. Finally, we obtain a large population limit of the culled
frequency process, resulting in a deterministic ordinary differential equation
(ODE). Two particular cases of the limiting ODE are studied to determine if
general two-type branching mechanisms and general Malthusians can lead to the
coexistence of the two types in the population.
