Back in the 70s, Star Trek was fallow while Who was active with one the the most popular versions of the Doctor. ( also the longest running).
In the 90s Star Trek was at its height when Doctor Who laid fallow. TNG started in 1987 and the original run of Who ended in 1989. Trek had the film series and four TV shows active in Doctor Who's downtime.
Trek dies in 2005 and Who comes back in 2005.
Now both franchises are active and popular. But as history shows... THERE CAN BE ONLY ONE!!!!!!