Earth's population isn't growing exponentially or even linearly, though.
We know that the population can't grow exponentially forever, but other than a recent dip in fertility rates, we have seen a fairly steady exponential growth.
Who says so? Why would an observation that may (or may not) apply to bacteria or even to wolves apply to human beings?
Because it appears we're also DNA replicators.
But, let me play devil's advocate on your behalf, because I like your question. Let's assume that humans do realize their population problems and act to level growth, or reduce it. There's a strong argument that they would also act to reduce their resources, as well. Fewer people means less food necessary to feed the population; So why grow it?
We do have at least one significant difference from other species in our intelligence/self-awareness, but I challenge you to find any significantly large (compared to world population) group that acts even mostly out of rationality.
If a population stops growing, then clearly it's because it has run out of some resource, particularly if you go so far as to include things such as 'time' as a resource.
It also points out that it is the scarcest resource that limits growth. Increasing food will not allow the population grow if there isn't enough water, sun, etc.
My point was mostly to point out that we may have some other scarcity that isn't food which is limiting the growth rate in industrialized countries, while increasing the food in starving countries will just cause the population to grow until it hits the limit of some other resource.
I suspect that it isn't any sort of time scarcity that is limiting current human growth, but more likely the impact of contraceptives messing with our normal biological mechanisms.
Population growth world-wide is 1.14% per year.
This is due to the fact that if current trends continue, it will have taken 12.3 years for the world population to go up by a billion (however it should be noted, earlier estimates -1990 IIRC- suggested it should have hit 10 billion ~2012), which is a mild increase over the last billion of 12.25 years. I believe the next billions are expected to take 13, 15 and then IIRC 18 years.
There's many relevant factors effecting population growth. Many of the diseases that plagued our ancestors are almost completely cured in the west (I've never heard of someone dying from a broken arm, however 200 years ago it was a prominent risk).
We also have food preservation techniques that no one had before. Everywhere from penicillium being used in meats to prevent bacterial growth to refrigeration, vacuum sealing, freeze drying and even vacuum freezing (where the moisture is literally sucked out of food, meaning it can last unrefrigerated for up to 10 years). There's also irradiation techniques that, given proper storage, can enable food to last indefinitely.
There are also vast misconceptions about over population that plague these discussions. For example, China is less densely populated than the UK and Japan, yet has food struggles because they don't have the economy to import food on mass. Many African countries plague by starvation are desolate compared to most countries, yet neighbouring countries have populations almost one hundred times as populated (note that this even applies in the landlocked countries interior and isn't a comparison between a country with a large ocean border and one with no border).
This doesn't really make sense. Human population growth could potentially be modeled as a logistic curve, but I doubt that even really explains or will explain things adequately. More realistically, human population growth without new technology or social structures showing up is exponential, but those dramatically change its shape, to the point I'm not sure we can predict population more than 2-3 decades out.
Of course, this development is dependent upon the presence of natural resources which may interfere with the food supply (i.e. biodiesel or ethanol fuels).
From a peak growth rate of 2.2% on a base of 3.2 billion in 1962-1963, the world population growth rate declined to 1.2% on a base of 6.6 billion in 2007. That's about 79 million net new people in 2007.
Wikipedia elsewhere reports that 1989 was the year with the largest absolute increase in population, 87.8 million net new people. So shrughes' report of not-even-linear growth seems true, at least for the last 20 years.