Population growth

How does a population grow over time? It depends on how much food and space is available.

Exponential growth happens when resources are unlimited. Each individual can reproduce to its full potential, and the population grows faster and faster. The rate of change is:

dN/dt = (b − d) × N = rN

where b is the birth rate, d the death rate (per capita), and r the intrinsic rate of natural increase: a measure of how fast a species can grow. Different species have very different values of r (the book gives figures for the Norway rat, the flour beetle and the human population of India). The integrated form is Nₜ = N₀ e^(rt).

Plotted on a graph, exponential growth gives a J-shaped curve. The old story of the chessboard shows how fast it goes: a king offered one grain of wheat on the first square, two on the second, four on the third, doubling each time. By the 64th square, the grains would be more than all the wheat in the world.

Logistic growth. In nature, resources are always limited, so exponential growth cannot last. Every habitat can support only a certain maximum number of a species: its carrying capacity, K. A population in a limited habitat grows like this:

• a lag phase at first, as the few individuals settle in;
• then acceleration, growing fast;
• then deceleration, as resources become scarce;
• finally an asymptote: the population levels off at K.

This gives an S-shaped (sigmoid) curve, described by the Verhulst-Pearl logistic equation:

dN/dt = rN × (K − N)/K

When N is small, (K − N)/K is close to 1, and growth is nearly exponential; as N approaches K, it falls towards zero. Because resources are always limited in the real world, the logistic model is considered more realistic.