AP® Environmental Science review sheet from Aim for Five (aimforfive.com/enviro/units/3/3-5)
Unit 3 · Topic 3.5
3.5 Population Growth and Resource Availability
With plenty of resources, a population grows exponentially in a J-shaped curve. As resources become limited, growth slows and levels off near carrying capacity, forming an S-shaped logistic curve.
Key terms
- exponential growth
- logistic growth
- J-curve
- S-curve
- resource availability
Exponential growth (J-curve)
When resources are unlimited, a population grows by a constant percentage each time period. Because the base keeps getting bigger, the number added each period keeps getting bigger too. That's exponential growth. Plotted over time, it makes a J-shaped curve: flat at first, then rising more and more steeply.
Bacteria are a simple example. If they double every 20 minutes, one cell becomes 2, then 4, 8, 16 and so on. The equation used in ecology is dN/dt = rN, where N is population size and r is the growth rate per individual. You won't need to solve it, but it shows why growth speeds up as N gets bigger.
Exponential growth happens when a species enters a new area with abundant resources and few predators, like an invasive species, or after a disturbance clears space.
Exponential growth can look slow at first. A pond weed that doubles its coverage every day can go from covering just 1% of a pond to covering all of it in about a week, because seven doublings multiply its area by 2⁷ = 128. That's why invasive species and algal blooms can seem to appear suddenly.
Logistic growth (S-curve)
No population can grow exponentially forever. As the population grows, food, water and space become limited. This pressure against growth is called environmental resistance. Birth rates fall and death rates rise, so growth slows.
Logistic growth starts out looking exponential, then slows and levels off at carrying capacity (K), making an S-shaped curve. The population grows fastest in the middle of the S, at about half of K. Near K, the growth rate approaches zero because births about equal deaths.
Comparing the two
| Feature | Exponential (J) | Logistic (S) |
|---|---|---|
| Resources | Unlimited | Limited |
| Shape | J | S |
| Growth rate over time | Number added keeps increasing | Speeds up, then slows to zero |
| Levels off at K? | No | Yes |
| Example | Bacteria in fresh broth; invasive species just after arrival | Seals on an island; yeast in a closed flask |
Resource availability sets the limits
Whether a population grows exponentially or levels off depends on resource availability. When a resource becomes scarce, individuals compete for it, and the population responds with lower reproduction, more deaths or moving away. These effects grow stronger as the population gets denser.
Some species follow a boom-and-bust pattern instead of a smooth S. Their population grows exponentially, overshoots K and crashes. Algae in a lake fed by fertilizer runoff and insects with a short breeding season often show this pattern.
In logistic growth, carrying capacity acts as a ceiling set by the scarcest resource. If that resource increases, for example when a lake receives more nutrients or a grassland has a wet year, K rises and the population can grow again until it reaches the new limit.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Exponential doubling
A culture starts with 100 bacteria that double every 20 minutes, with unlimited resources. How many bacteria are there after 2 hours?
Show the solutionHide the solution
- Step 1: Find the number of doublings: 2 hours = 120 minutes. 120 ÷ 20 = 6 doublings.
- Step 2: Each doubling multiplies by 2, so 6 doublings multiply by 2⁶ = 64.
- Step 3: 100 × 64 = 6,400.
Answer: 6,400 bacteria.
- Example 2
When is growth fastest?
A fish population in a pond follows logistic growth with a carrying capacity of 2,000. At about what population size is it adding the most fish per year? What happens to growth as the population approaches 2,000?
Show the solutionHide the solution
- Step 1: In logistic growth, the growth rate is fastest in the middle of the S-curve, at about half of K.
- Step 2: Half of 2,000 is 1,000.
- Step 3: As the population approaches 2,000, resources become limiting, births and deaths become nearly equal, and growth slows toward zero.
Answer: About 1,000 fish; growth slows to nearly zero as the population approaches 2,000.
Common mistakes
- Thinking exponential growth means adding the same number each year. It means growing by the same percentage, so the number added keeps rising.
- Saying logistic growth is fastest near K. It's fastest at about half of K and slowest near K.
- Labeling any rising curve exponential. Check whether it bends and levels off; if so, it's logistic.
On the exam
- Questions often describe a graph and ask you to identify J or S growth and explain the shape using resource availability.
- Doubling calculations are common. Find the number of doublings first, then multiply by 2 that many times.
Connected topics
Videos
Check yourself
4 questions on 3.5 Population Growth and Resource Availability. Pick an answer to see if you got it, and why.
Yeast cells are grown in a flask of sugar solution. Their population rises slowly at first, then very quickly, and then levels off at about 800 cells per microliter. Which statement best explains the leveling off?
A population of 1,000 bacteria has unlimited resources and doubles every 30 minutes. How many bacteria will there be after 3 hours?
A population of deer follows a logistic growth curve with a carrying capacity of 1,200. At about what population size is the herd adding the most new deer per year?
Which pattern of population size over time is most typical of an r-selected species, such as a small insect?
0 of 4 answered