Nope, no study, and no convergence. There are things you can tweak and control, but success depends more on how you advertise the forum, the forum topic, and how you moderate it than the number of users. A good small forum can live forever generating lots of great quality content with the right advertising, topic, and moderation. Likewise, a huge forum can thrash around and die even though it has hundreds of active participants due to bad advertising, poor topic, and/or bad moderation.
And the three are interrelated. If you have a great topic, but advertise it poorly or have terrible moderation, you’ll have lots of brief visits, and few people that will stick around.
My limited experience suggests that the best answer to your question is that if you have 15 people that visit the forum and generate content daily, and 100 people that lurk and contribute occasionally, then as long as the other aspects of the forum are good, then it will not die. It may not grow, either, but so much depends on factors other than participation and engagement.
Fewer than 15 daily content generators, and you’re likely to stagnate. I don’t know the anthropological reason, but my WAG is that those 15 people won’t be only friends, but friends of friends, which induces an amount of diversity that the topic at hand will be discussed from more than one perspective, which will incite cyclical response. Good moderation is needed not just to tamp flames down, but to encourage a warm fire that won’t get out of hand. You want minor disagreements. Otherwise it becomes “Opinion!” “yep” “uh huh” “True, dat!” etc. Of course you can have a unified forum as long as you also have one minor troll, but usually that requires more than 15 active users.
If you want hard data, I’d look toward the vast number of books that exist on building community.