The Population Division of the United Nations marked 31 October 2011 as the “Day of Seven Billion”. While that was a publicity gambit, world population is now above this figure and climbing. The global ecosystem is seriously stressed, and climate change is greatly aggravated by the expanding population. Accurate estimates of growth are essential for assessing our future well-being. This week, *That’s Maths* in *The Irish Times* ( TM034 ) is about population growth over this century.

**Past Growth**

Before the Black Death in the fourteenth century, there were about 450 million people. After the Plague this figure had fallen to 370 million. Since then there has been continuous growth. In percentage terms, the growth rate peaked at 2.2% per annum in 1963. It has now declined to about 1.1%. Population growth varies widely from place to place but in most countries there has been a demographic transition to smaller families in recent decades.

The UN estimated a growth rate of 75 million per annum in 2000, and the CIA’s World Factbook gave a comparable figure. Global life expectancy is about 67 years and growing. Death rates change dramatically due to disease, war and natural disasters. Better sanitation and medical advances reduce infant mortality and more efficient agricultural practice reduces the incidence of famine, so life expectancy is on the increase.

**Models**

Many mathematical models show an exponential growth: this implies a fixed time for the population to double. In 1798 Thomas Malthus predicted that the exponentially growing population would exhaust the world food supply within fifty years. But, although scary, this may not be pessimistic enough: during the second millennium, each doubling of population took half as long as the previous one. This pattern corresponds to “hyperbolic growth”, with the population becoming unbounded within a finite period, much more worrying than exponential growth.

At the more optimistic end of the spectrum we have “logistical growth”, where there are constraints on the maximum population. An early model for this was proposed by the Belgian demographer Pierre François Verhulst, who refined the model of Malthus, introducing a number called the carrying capacity. This is the maximum population that the environment can sustain. The resulting growth is initially exponential, but flattens out as it approaches the carrying capacity. Alas, there is little agreement about the Earth’s carrying capacity.

The various modes of growth are discussed on this blog.

**Prospects**

What lies ahead: a balanced plateau or a Malthusian catastrophe with mass starvation? Current projections all indicate an increasing population, but forecasts vary widely since they depend sensitively on the underlying statistical assumptions. There is considerable uncertainty so, in addition to actual numbers, probabilistic predictions provide a measure of confidence in the forecasts. The best current mathematical models give expected values for 2100 ranging from 9 to 13 billion, with the most probable value being 11 billion.