There are two parts to this.
Competitions and contests that offer prizes can stimulate activity. But they're encumbered with major problems.
First, people don't interact with each other. They interact with you. This doesn't help the community. It simply provides short blips of activity (which usually mask bigger problems). It's not a tactic that has a long-term impact upon the community. Fun? Yes. Useful? No.
Second, there is no incentive to share it with anyone else. If I invite you to participate in a contest, it decreases the odds of me winning. In fact, it's in my interest to dissuade as many people as possible from participating. This leads to the third problem.
Third, the more people that might participate, the less the odds that I will win. The bigger the community, the less people are likely to participate.
Fourth, most people won't win. That means most people will be disappointed. There is only so long people will continue to lose every week/month (national lottery aside).
Fifth, prizes get boring. If people only interact for a prize, you will soon need bigger and better prizes it sustain their interest. This isn't sustainable.
It's perfectly fine to have a rare contest with a good prize, but it's one of the least effective tools in the community building toolbox.
If you are going down this path, I suggest not having a prize. Keep people in the mindset that they're participating for fun…or better, to increase their own reputation and social status within the community.