For organism A, its change in concentration is dA/dt = µa A - D A which says that it reproduces at a rate proportional to its concentration and washes out at a rate determined by the dilution rate, D.
The analogous equation for B is dB/dt = µb B - D B .
If one concentration, say that of A, reaches steady state, then dA/dt becomes zero and µa must equal D. This means that dB/dt cannot also equal zero except in the very rare case when they have the same µ. When the assumption of perfect mixing holds, the organism that grows most rapidly must win out while the other must wash out. Eventually only one can be present. You can experiment with the dilution rate and the growth rate coefficients with the following Java applet. These organisms compete for sugar. Organism A starts out alone in pure culture, and Organism B is introduced a short time later.