Population growth can be exponential or logistic, depends on the population's reproductive strategies and environment. Any ecology textbook covers this in much more detail than I will here.
Some species are what ecologists call r type, they grow exponentially, run out of resources, and crash to very small numbers. Most of these actually usually exist at low numbers until conditions are really good, then do that huge growth until condition go back to usual, and crash. Locusts are an obvious example. Dry weather, few eggs laid and few hatch, low populations, two years in a row of rains, lots of eggs laid, hatch, grow to adulthood, lay more eggs that hatch, and voila - locust swarms.
Other species are K type. Their populations are sensitive to density, so when they are at low density they grow quickly (look a lot like r type species). But when they get more crowded, they increase mortality or decrease births or both, so that by the time they reach the carrying capacity of their environment their births=deaths and the population is stable. Of course in reality the numbers go up and down, because the carrying capacity is not really a constant.
How does this apply to people? As far as we can tell, early humans were k type and at low numbers. Then agriculture was developed and increased the carrying capacity and populations grew, but leveled out again at a much higher number. So we do seem to act like a K type species. However, agriculture changed our reproductive behaviour (more babies, closer together). The industrial revolution (and I am lumping the green revolution in here) caused our carrying capacity to increase again, and our numbers went up again. Presumably we will level off again. The question is at what number? Or this time will we be an r type globally and overshoot and crash? It has happened locally many times - any time a group of people are pushing their environment (and for people the reasons can be economic, not just physical environment) they crash - think potato famine in Ireland, black death in Europe.
Ecologists tend not to talk too much about this, except a bit among themselves, because it is so depressing and most people tell us we are worrying too much. But if you look carefully, we tend to have few children.
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