They will only expand into areas that have resources they need, and are willing to weight options. If you drain them of resources then they cannot gain resources.
Another point, if you have fleets ready to defend space, they will not simply ignore them. I imagine this will only happen if you are at war with them as there is no law preventing people from mining out or claiming space that other ships happen to be in.
We can calculate the resources they would be acquiring, we can set the rate at which they mine, and we can set the rate at which they expand. I don't think we could have an active dynamic universe without some of the work being done behind the scenes.