I don't know about any prophecies about overpopulation, but there are some about famine and huge numbers of people dying from war, disease, and starvation. Which isn't particularly impressive considering that such things had happened periodically before the bible was written and afterwards.
Based on my analysis of people in general, I think that whether or not it collapses or levels off depends on how much longer it can continue and whether or not the long overdue pandemic which kills off a portion of the population kills enough to make a difference without causing complete anarchy (we are about 70 years overdue for a plague which kills at least 1% of the afflicted countries' populations, last one I can think of was the Spanish Flu around World War 1). If we can go the next century without serious global famines which wipe out millions of people in all countries, I imagine it will level off. If within a couple decades we see that kind of die off, there is going to be a serious collapse. If it happens between 50-100 years from now, I think there will be serious famines, but the overall decline will be slow rather than a sudden collapse, which might give an opportunity to put measures in place to prevent the issue in the future.
However, chances are the energy crisis is going to hit faster than famine will.