Exponential growth

Unimpeded growth needs food and space. When a habitat's resources are unlimited, a species can realise its full innate potential to multiply, as Darwin noted, and the population grows exponentially (geometrically).

With per capita birth rate b and per capita death rate d, the change in N per unit time is dN/dt = (b − d) × N.

Writing r = b − d gives dN/dt = rN. Here r is the intrinsic rate of natural increase, a key measure for judging how any biotic or abiotic factor affects growth.

Some r values: Norway rat 0.015, flour beetle 0.12, and the human population of India in 1981, 0.0205.

Plotting N against time for this model gives a J-shaped curve.

In integral form, Nt = N0 e^(rt), where Nt is the density after time t, N0 the density at time zero, and e = 2.71828, the base of natural logarithms.

Under unlimited resources, even a slow breeder can reach huge numbers quickly. Darwin showed this for the elephant.

The chessboard story makes the point: one grain on square 1, doubling on each of the 64 squares. By the halfway mark the king saw that all the wheat of his kingdom would not be enough.

In the same way, one Paramecium dividing by binary fission once a day would reach an astonishing number in 64 days, if food and space never ran out.