I suspect that the best approach is simply to not let them die in the first place. That is, put them on fresh medium every 3-4 weeks, so they happily keep dividing.
If you really need much higher concentrations for some reason, you may be able to gently spin down the cells, and then put all of them on fresh medium. Presumably, the reason why they stop growing eventually is because they are producing some byproduct that is limiting their own growth. That is, they are literally poisoning themselves in some way.
I've been contemplating doing some co-culture experiments by ordering a bunch of marine bacteria and growing them together with the dinoflagellates. Roseobacteria may be good candidates, because they tend to be highly abundant in the types of coastal waters where the dinos live, plus some Roseobacteria are known symbionts with other types of algae.
Patrik