Density-dependent means the per capita rate shifts with crowding. That feedback is what holds a population near K.
A crowded population pushes back on itself. As density rises, each individual gets less, so the birth rate per individual falls and the death rate climbs until growth stalls at carrying capacity.
Key terms
- Carrying capacity (K) — the most individuals a habitat can support. The resource that runs short first sets it.
- Logistic growth — growth that slows as N nears K. The curve is S-shaped: slow, then steep, then flat at K.
- Density-dependent factor — a limit that hits harder per individual as crowding rises: competition, predation, disease, buildup of wastes.
- Density-independent factor — a limit that kills the same fraction no matter how crowded: a flood, a freeze, a fire.
- Intraspecific competition — members of one species competing for the same limited food, water or space. It's the classic density-dependent brake.
How it works
- At low density, resources are plentiful, and the population grows almost exponentially.
- As N rises, individuals compete for the same limited resources. Each one gets less food, water or space.
- Less per individual → slower growth, fewer offspring, more deaths. The per capita birth rate falls and the per capita death rate rises.
- Your formula sheet writes this as dN/dt = r_max N (K − N)/K. As N approaches K, the (K − N)/K term shrinks toward zero.
- The number added per year peaks at N = K/2, where there are many breeders and still room to grow. At K, births equal deaths.
- Above K, deaths exceed births and N falls back. This negative feedback holds populations near K. If the response lags, numbers overshoot and crash, cycling around K.
Why it matters
The exam tests this with experiments: add animals, cut the food, crowd a tank. Your job is to find which per capita rate responded, and how strongly. Compare percent changes, since births and survival come in different units. It's negative feedback, like a thermostat, at the scale of a population.
Exam traps
| Trap | Actually |
|---|
| Stocking extra animals into a habitat at K raises the population for good. | At K, extra animals mean more competition. Per capita survival drops, and numbers settle back near K. |
| Halve a population's main food and it dies out. | K drops, and numbers settle at a new, lower level. As numbers fall, each survivor gets enough again. |
| The population that reached the larger size must have the higher K. | Not if it's still climbing. A curve that hasn't bent toward a plateau hasn't revealed its K yet. |
| A flood that kills many voles is density-dependent, since it cut the population. | Ask whether the per capita death rate changes with crowding. A flood drowns about the same fraction at any density. |
Worked example
Sheep on an island have r_max = 0.4 per year and a carrying capacity of 800. Use the logistic equation to find how many sheep the population adds per year at N = 200, N = 400 and N = 700. At which size is it growing fastest?
- Start from the formula sheet: dN/dt = r_max N (K − N)/K.
- N = 200: 0.4 × 200 × (600/800) = 0.4 × 200 × 0.75 = 60 sheep per year.
- N = 400: 0.4 × 400 × (400/800) = 80 sheep per year. N = 700: 0.4 × 700 × (100/800) = 35 sheep per year.
- Divide each by N to get the per capita rate: 0.30, then 0.20, then 0.05. It falls steadily as crowding rises.
60, 80 and 35 sheep per year. Growth is fastest at N = 400, half of K.
The (K − N)/K term is the share of K still unused. It shrinks as the island fills, dragging the per capita rate down with it. The near miss is N = 700, because it has the most breeders; that's exponential thinking. Near K, each sheep gets less food, so births fall and deaths rise.