Fertility rates are already plummeting across the planet, and most developed countries already need a steady influx of migrants just to keep their populations steady. The issue is that a baby born today will live for 70-ish years, so there's a large "hump" of overpopulation to deal with before world population growth goes into reverse around 2100.
War or famine would likely speed up reaching that point, not slow it down. Nevertheless, grim Malthusian predictions have been made since the 1700s, and have yet to come to pass.
Developed countries consume the most resources per capita, so their populations could do with a drop - there is no 'need' to keep them steady, but just the opposite.
