Key ideas
Populations grow in a few standard ways. Without limits growth is exponential, and with limited resources it levels off at the carrying capacity K. The equations tell how fast and how large.
- Per-capita growth rate r = birth rate − death rate (ignoring migration). Exponential growth follows dN ÷ dt = rN, with N = N₀eʳᵗ.
- The doubling time of an exponentially growing population is ln 2 ÷ r, or about 0.693 ÷ r.
- Logistic growth, dN ÷ dt = rN(1 − N ÷ K), slows as N approaches the carrying capacity K. It is fastest at N = K ÷ 2.
- The change in population size is births + immigration − deaths − emigration.
Key equations
N = N₀ e^(rt)dN ÷ dt = rNdN ÷ dt = rN(1 − N ÷ K)doubling time = ln 2 ÷ rr = births − deaths (per capita)
Worked example
A population of 100 grows at r = 0.05 per day. Its doubling time is 0.693 ÷ 0.05 = 13.9 days, and after 30 days N = 100 × e^(0.05 × 30) = 100 × e^1.5 = 448. With K = 1000 and r = 0.2, a population of 500 grows at 0.2 × 500 × (1 − 0.5) = 50 individuals per time unit, the maximum for this K.
Common mistakes
- Using exponential growth for a resource-limited population over long times.
- Forgetting that the logistic growth rate is zero at N = K, and highest at K ÷ 2.
- Mixing the units of r with the units of time, such as r per year with t in days.
Practice
Test yourself with practice questions on this topic: Populations and growth