One population growth model (called the logistic law of population growth) was developed by a Dutch mathematician-biologist named Verhulst in the mid 1800s. It recognizes that for small populations that populations will grow exponentially. However, when a population becomes large enough, competition for resources will begin to limit growth. This applies to deer, rats, bacteria, and humans. The model is:
dp/dt=ap-bp^2
where p is the population, dp/dt is the rate of population growth, and a and b are called vital coefficients.
Ecologists have estimated, for humans, that the value of a is 0.029. Population stats can be used to solve for b, which is about 2.941E-12. It can be shown that the population growth will reach zero (dp/dt=0) when the population reaches 9.86 billion, which is in good agreement with this article.
If this formula were valid, population densities like India and China wouldn't exist. Think how far the rest of the world has to go to reach their density. If the formula doesn't work in every setting, it's worthless. It's a theory that would make any mother proud, just so long as her son came up with it. But it doesn't stand up when exposed to reality.