Source? Numbers? I know that a few first-world countries like Japan have negative population growth, but is that the case everywhere? Are the countries that having negative population growth enough to offset those with positive?
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.