Population growth models

Exponential with no limits, logistic with them.

The idea

When resources are unlimited a population grows exponentially: the per-capita growth rate r (births minus deaths per individual per unit time) stays constant, N(t) = N₀eʳᵗ, and the doubling time is ln 2 / r ≈ 0.693 / r, giving a J-shaped curve. Real populations run into limits, so the logistic model multiplies rN by (1 − N/K), where K is the carrying capacity the environment can support; growth is fastest at N = K/2, slows as N approaches K, and the curve becomes S-shaped. Density-dependent factors — food, disease, predation, competition — are what set K, while density-independent factors such as drought or frost strike regardless of how crowded the population is. Species built for rapid growth (r-selected, like insects) and species that hold steady near K (K-selected, like elephants) ride these curves in different ways.

Work through an example

Fifty rabbits grow at r = 0.1 per month. Doubling time is 0.693 / 0.1 ≈ 6.9 months, and after 20 months N = 50 × e² ≈ 50 × 7.39 ≈ 369 rabbits. Now give the field a carrying capacity K = 1,000: at N = 50, growth is 0.1 × 50 × (1 − 0.05) ≈ 4.75 rabbits per month; at N = 500 it peaks at 0.1 × 500 × 0.5 = 25 per month; at N = 900 it has fallen to 0.1 × 900 × 0.1 = 9 per month. The rule of 70 gives the same doubling estimate quickly: 70 / (10% per month) = 7 months.

What to watch for

A population does not glide up to K and sit there. Real populations overshoot, crash, and oscillate, and K itself shifts with the seasons. The reindeer released on St. Matthew Island, Alaska, are the cautionary tale: 29 animals in 1944 became about 6,000 by 1963, then ate out the lichen and collapsed to 42 by 1966. Note too that the doubling time in exponential growth is constant — it depends on r, not on how big the population already is.