Part of the problem with the simple growth equation (2) is that the rate of growth is constant and doesn't know anything about the population size. In a real population, you might expect that as population increases towards some mythical carrying capacity the growth rate will slow down as the death rate begins to match the birth rate. Whether this actually happens is unclear but if it did, one simple way to model this would be to modify the growth rate to look something like
where is the rate we would expect for small populations and K is the carrying capacity. Question: what happens when N=K?
If we do that, then our new model is only a bit more complicated and looks like
Note now that the growth rate depends both on the population and the square of the population. This is actually a non-linear problem and is much more difficult to solve analytically. Nevertheless, with the tricks we developed above, it is no harder to understand than the linear exponential growth problem. With that charge...do the following problems.