When does the exponential growth of a population stop?

When the area available reaches the limit given by its carrying capacity for that population.

Every habitat is an equilibrium among species. These can all share the same area as long as they have a constant behaviour as dictated by that equilibrium. Such unaltered behaviour however is impossible in nature where all living beings evolve and mutate, where conditions are affected by external factors such as whether, climate, volcanic eruptions and ebbing sea levels.

Hence as any one species evolves, and modifies the general living conditions to its advantage at the expense of those of all the others, so do the specific carrying capacities of all the others. Their growth suddenly is no longer exponential, it may go down to linear, decrease and eventually disappear.