Yesterday, without fanfare, the United Nations announced a great triumph of human self-interest over the selfish gene. It published new, long-range population forecasts stretching out to 2150 which predict population growth will have drastically slowed down within a few decades. By the end of the next century it expects human numbers to have virtually stabilised at a little over 10 billion - less than double today's 5.9 billion.

What is going on? After three decades of global fretting about a population explosion, about a world with too many mouths to feed running short of basic resources, it looks as if the "demographic transition" is spreading rapidly around the globe.

Before this transition happens women have lots of babies but there is horrendous infant mortality and life expectancy is short. This is the wretched condition of most hunter-gatherer and primitive agricultural societies.

Then farming becomes more productive, cities, industry and commerce grow and basic sanitation and healthcare improve. Infant mortality falls and life expectancy lengthens. Fertility, however, remains very high with mothers having five or more children. So population soars, doubling in 25 years or less. That is just what has happened over the past quarter century in dozens of developing countries, so it is hardly surprising that pundits and politicians panicked.

But the transition moves on. As people become better educated and more prosperous, as the servitude of women is weakened, as access to contraception spreads, fertility starts to decline rapidly. Women marry later, wait longer after marriage to have their first child and have fewer over their lifetime. Once the fertility rate drops to about 2.1 children per woman and stays there population growth will end. Any lower and human numbers will begin to fall.

Amazingly, encouragingly, this demographic transition now seems to be well under way across most of the world. In developed countries such as Britain, it occupied about 150 years and ended roughly 70 years ago. In many developing countries, the transition seems to be taking place within 100 years and nearing completion already. In China and Thailand, for example, total fertility rates are already under 2.1.

It is hard to believe that people living in such different cultures, at different times, with different standards of living, should all conform to the same pattern. But, broadly speaking, they are (although in some countries governments have played a leading and sometimes oppressive role in managing the transition). The exceptions seem to prove the rule. It is only in the poorest nations and those where women are least emancipated that population growth is still frighteningly, dangerously rapid.

The demographic transition is among the most significant developments in human history and prehistory, every bit as important as writing and agriculture. Rapid, sustained population growth is the result of humanity separating itself from the rest of life through culture and invention, enabling itself to expropriate natural resources to a far greater extent than any other species.

But it cannot go on for ever. There comes a time when population growth must be slowed and then stopped.

When? The absolute limit on numbers imposed by living on a finite earth, with restricted fresh water, solar energy, and land surface, is unknown. With advances in technology the world could maybe support five or even 10 times today's population. Perhaps we will come to regard the most important constraints as aesthetic and psychological ones. We may want a planet with adequate space between people and cities, with remnants of countryside and wilderness.

Whatever the limits are, they exist - and if we are to avoid crashing into them catastrophically, at full tilt, we will have to get off the track of rapid population growth sooner or later. As it happens, we are doing it now.

The demographic transition can also be seen as one of humanity's greatest breakouts from the prison of biology, a victory of self interest over selfish genes. Population is stabilising not because we are looking out for posterity, nor for society as a whole, and certainly not for our genes, but because we are looking out for ourselves.

The world over people, rich and poor, are trying to juggle different aspects of their lives - work, family, friendships, pleasure - in the pursuit of happiness. For most of us, it now seems that having more than two or three children (indeed, for some of us, having any at all) is to risk letting those juggling balls crash to the floor.

We have a completely different mindset to our ancestors of just half a dozen generations ago. We no longer see children as bringing security in our old age. We know we can choose their number using fairly reliable and cheap contraception. Why incur the expense of a big family? And why, when the time and even the love we can put into parenting is finite, risk spreading those precious commodities too thinly?

Obvious, maybe, but all of this is all quite out of kilter with Darwinian evolution, even in its most sophisticated, "selfish gene" version. Over the past 30 years this theory has made great strides in explaining human societies and behaviour. By concentrating on natural selection at the level of the gene, not the individual, and by taking humanity's starting point as a group-living ape, it has offered evolutionary explanations for some human attributes which would seem to contradict crude Darwinism Why, if there is a struggle for existence in which only the fittest survive, do we have altruism and guilt? What is consciousness for and how did it come about?

Selfish gene theory has begun to explain such things but it cannot explain why, throughout our adult life, we no longer try to raise as many children and grandchildren as possible. Our new-found reproductive restraint is the fruit of reason and self-interest.

Of course the population story isn't over; the future will hold surprises. Demographers were startled at the low level fertility rates dropped to in Europe after the First World War in response to the uncertainty caused by the great depression. Then there came another surprise; fertility made a rapid recovery during the even more uncertain times of the Second World War and this continued through the post-war baby boom.

Today, in Italy, Russia and a dozen more former Communist countries, women are having so few children that numbers are falling. But that will probably change, sooner or later. At the other extreme some poor, densely populated nations with high growth rates such as Pakistan and Egypt seem headed for disaster. It is hard to see how they can supply twice as many people with the basics of food and fresh water.

World wide, however, the number of people being added to global population is falling with each passing year. The UN's Population Division believes that the highest global growth rates are behind us and we will never see their like again. This is our gift to the future.