The essential 'gist' of this --- is what we call the exponential growth model --- Nt+1 = λNt. As population numbers increase, the estimate of the intrinsic rate of natural increase; over a short period of time, tends to decrease. This means that there is a general relationship between the intrinsic rate of increase and population density. This is a cyclic phenomena, however, and the relationship between population growth is based on a cyclical model.
This is based on the concept of biological ecology population; when a particular species thrives and consumes its resources, and is devoid of natural predation, it will then experience an absolute lack of resources --- population will then drop. Eventually, with the taxing pressure on resources being abated, said resource will thrive yet again. Thus enabling the species to regrow numbers.
How this applies to human population? It does, quite frankly. If one studies the period of European Industrial Revolution during the 19th century, the American industrial revolution in mid 19th century, and the Japanese Industrial Revolution in the mid 19th century --- one thing that was apparent was the rise in population growth rates in said societies. As a society matures and transforms from production (manufacturing) oriented to say high service oriented, experiences high HDI praxis , then, it is only an eventuality that growth rate will whine down as there is a positive correlation with population growth rate and high quality of societal living.
Do note that high population growth rate is always a universally seen in war torn regions or in regions that are rebuilding from war --- population growth, specifically high breeding rates , is essential in repairing the populatory demographic.Please refer to image #1.
Image #1: Notice high birth rates in extremely poor , maldeveloped, war-torn regions; ergo, Africa, Central Asia, Middle East, Southeast Asia, and Latin America.
Regards,