Darwin realized that the great population pressures could cause extensive competition for scarce resources. A simple mathematical expression defines this type of population growth. The formula basically says that the rate of increase in population is directly proportional to the number of individuals at any given time. The constant of proportionality is the growth rate, r. If we plot the relationship between N and t using this formula we get the exponential or J-shape curve shown here.
Exponential growth is growth without limit.
Good examples of prolonged exponential growth are rare in the natural world, but can be observed experimentally in laboratory conditions of unlimited resources. The paucity of natural examples is due to the fact that there are population controls due to limits of resources such as food, light, shelter, water, etc., as well as to factors such as predation, climatic change, and disease. One example of experimentally verified exponential growth can be seen in a study made of the Goshawk population in Great Britain following reintroduction in 1959 from falconry escapes and releases. The graph shows the number of displaying pairs reported each year. The population follows a J-shaped curve fairly well, with a value of r of 0.37 growth rate per individual per year.