Short on time? This page is the recap. For explanations, figures and worked examples, use the full study guide.
Recap
- Density = individuals per unit area or volume. Resources set how dense a population can become; density sets how much of each resource each individual gets.
- Carrying capacity (K) is the sustainable population size supported by the ecosystem’s total available resources. It is set by the limiting resource and changes when the environment changes.
- As density rises, each individual gets less food, water, light or space, so per capita births fall and per capita deaths rise.
- Density-dependent factors: competition, predation, disease and parasites, waste build-up, territorial stress. They can hold N near K.
- Density-independent factors: frost, storms, floods, drought, fire. They remove about the same fraction at any density and cannot hold N steady.
- With these limits, growth usually becomes logistic: S-shaped, levelling off at K.
Key relationships
| Equation | Meaning |
|---|---|
| dN/dt = rₘₐₓN(K − N)/K | logistic growth; (K − N)/K = unused share of K |
| per capita rate = rₘₐₓ(K − N)/K | falls steadily as N rises |
| maximum dN/dt = rₘₐₓK/4 | at N = K/2 |
| Population size | (K − N)/K | Growth |
|---|---|---|
| N much less than K | close to 1 | nearly exponential |
| N = K/2 | 0.5 | fastest |
| N = K | 0 | zero (births balance deaths) |
| N > K | negative | population falls |
Calculation tips
- Mean = sum ÷ number of values. Percentage change = (new − original) ÷ original × 100.
- Ratio: divide one value by the other and write it as “x : 1”.
- In the logistic equation, work out (K − N)/K first, then multiply by rₘₐₓN.
Assumptions behind the model
- K and rₘₐₓ are constant; no immigration or emigration.
- Births and deaths respond instantly to density (real populations often overshoot or oscillate).
Mistakes to avoid
- Saying growth is fastest near K (it is fastest at K/2).
- Classifying a factor by the number killed instead of the fraction.
- Dividing by N instead of K in the logistic term.
- Saying births and deaths stop at K.
- Treating K as fixed for a species.
Quick self-check
- rₘₐₓ = 0.5 per month, K = 400, N = 100. dN/dt? (0.5 × 100 × 300 ÷ 400 = 37.5 per month)
- At which N is growth fastest when K = 400? (200)
- A frost kills 20% of a population at every density. Which type of factor? (Density-independent.)
Next: practice questions.