Population regulation

J-shaped and S-shaped growth

Exponential (J-shaped) growth happens when resources are unlimited. Each individual reproduces at the full rate r, so: dN/dt = rN The curve rises slowly and then shoots up. It is seen when a species enters a new habitat with no competitors, like algae in a nutrient-rich pond. Such growth cannot last; it usually ends in a sudden crash (“boom and bust”).

Logistic (S-shaped or sigmoid) growth happens when resources are limited: dN/dt = rN (K − N)/K The curve has a lag phase (slow start), an exponential (log) phase, and a stationary phase where N levels off at the carrying capacity K. The term (K − N)/K is the “environmental resistance”: when N is small it is close to 1; when N reaches K it becomes 0 and growth stops.

Worked example: r = 0.2 per year, N = 200, K = 1,000. dN/dt = 0.2 × 200 × (1,000 − 200)/1,000 = 0.2 × 200 × 0.8 = 32 individuals per year. Growth is fastest when N = K/2 (here 500): 0.2 × 500 × 0.5 = 50 per year.

Factors that regulate population size

• Density-dependent factors act more strongly as density rises: competition for food and space, predation, parasitism, and infectious disease.
• Density-independent factors act regardless of density: floods, droughts, frost, fires, storms.

Predator–prey cycles (e.g. snowshoe hare and lynx in Canada) show regulation by predation.

Key points

• J-curve: dN/dt = rN; unlimited resources; ends in crash.
• S-curve: dN/dt = rN(K − N)/K; levels off at K.
• Fastest logistic growth at N = K/2.
• Density-dependent: competition, predation, disease. Density-independent: weather, disasters.

Exam tips

Substitute numbers carefully in the logistic equation. Questions also ask which curve shows carrying capacity (S) or which factor is density-independent (weather events).

Common mistakes

• Forgetting to divide (K − N) by K.
• Calling disease density-independent; it spreads faster in dense populations.
• Thinking J-shaped growth can continue for ever.