AP® Biology review sheet from Aim for Five (aimforfive.com/bio/units/8/8-3)
Unit 8 · Topic 8.3
8.3 Population Ecology
A population is all the individuals of one species living and interacting in an area. Its growth depends on birth rate, death rate and current size, and with unlimited resources it grows exponentially (dN/dt = rₘₐₓN), adding more individuals the bigger it gets.
Key terms
- population
- birth rate
- death rate
- exponential growth
- per capita growth rate
What a population is
A population is a group of individuals of the same species living in the same area and interacting with each other and their environment, competing for food, mating and sharing space. Many of a species' adaptations help it obtain and use energy and matter where it lives, and those adaptations shape how fast a population can grow.
Births minus deaths
The simplest growth model is dN/dt = B − D. Here N is population size, dN/dt is the change in population size over a period of time, B is the number of births in that time and D is the number of deaths. (Immigration and emigration are ignored.) If births exceed deaths, the population grows; if deaths exceed births, it shrinks.
Dividing by N gives the per capita growth rate, r: the change per individual. For example, a population of 1,000 deer with 150 births and 50 deaths in a year grows by 100 deer, so r = 100 ÷ 1,000 = 0.1 per year. Per capita rates let you compare populations of very different sizes.
Exponential growth
When resources are unlimited and nothing holds the population back, each individual reproduces at its maximum rate, rₘₐₓ (the maximum per capita growth rate). The population then grows exponentially: dN/dt = rₘₐₓN. Both equations are on the AP formula sheet.
The key feature is that rₘₐₓ stays the same, but the number of individuals added per unit time grows as N grows, because there are more individuals reproducing. Plotted with time on the x-axis and population size on the y-axis, exponential growth makes a J-shaped curve: slow at first, then steeper and steeper.
Exponential growth happens in real life only for limited times: bacteria in fresh culture medium, a species introduced to a new habitat with no predators, or a population recovering after a disaster. Eventually resources run short and growth slows (8.4).
| Term | Meaning | Units example |
|---|---|---|
| N | Population size | individuals |
| B, D | Births, deaths in a time period | individuals per year |
| dN/dt | Change in population size per time | individuals per year |
| rₘₐₓ | Maximum per capita growth rate | per year |
Graphing population data
Population data usually go on a line graph with time on the x-axis and population size on the y-axis, with labels and units. Because exponential growth gets so steep, scientists often plot it on a logarithmic y-axis, where exponential growth appears as a straight line. If you have repeated samples, show the mean at each time with error bars.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Using dN/dt = B − D
A rabbit population of 400 has 120 births and 40 deaths in one year. Calculate the change in population size and the per capita growth rate.
Show the solutionHide the solution
- Step 1: dN/dt = B − D = 120 − 40 = 80 rabbits per year.
- Step 2: Per capita growth rate r = (dN/dt) ÷ N = 80 ÷ 400 = 0.2 per year.
- Step 3: So each rabbit adds, on average, 0.2 rabbits to the population per year.
Answer: The population grows by 80 rabbits per year; r = 0.2 per year.
- Example 2
Exponential growth: the rate isn't constant (trap)
A bacterial population grows exponentially with rₘₐₓ = 0.1 per hour. Calculate dN/dt when N = 500 and when N = 5,000. A student says the growth rate stayed the same; explain what's right and wrong about that.
Show the solutionHide the solution
- Step 1: At N = 500: dN/dt = rₘₐₓN = 0.1 × 500 = 50 cells per hour.
- Step 2: At N = 5,000: dN/dt = 0.1 × 5,000 = 500 cells per hour.
- Step 3: The per capita rate, rₘₐₓ = 0.1, did stay the same.
- Step 4: But the population growth rate, dN/dt, increased tenfold because there are ten times as many cells dividing.
Answer: dN/dt = 50 cells/hour at N = 500 and 500 cells/hour at N = 5,000. The per capita rate is constant, but the overall growth rate rises with population size.
Common mistakes
- Confusing rₘₐₓ (growth per individual) with dN/dt (growth of the whole population). In exponential growth, rₘₐₓ is constant while dN/dt increases.
- Treating B and D as rates per individual in dN/dt = B − D. In this equation they're the numbers of births and deaths in the time period.
- Describing exponential growth as a straight line on a normal graph. It's J-shaped; it's only straight on a log scale.
On the exam
- Expect to calculate dN/dt or r from data and to graph population data correctly (labeled axes, units, appropriate scale).
- If asked when exponential growth occurs, mention unlimited resources and no limiting factors, and give a realistic example.
Connected topics
Videos
Check yourself
4 questions on 8.3 Population Ecology. Pick an answer to see if you got it, and why.
A population of 500 rabbits has a maximum per capita growth rate (rₘₐₓ) of 0.2 per month and unlimited resources. Using dN/dt = rₘₐₓN, how fast is the population growing right now?
Under which conditions is a population most likely to grow exponentially?
A population of 1,500 lizards has a birth rate of 0.20 per individual per year and a death rate of 0.08 per individual per year. Using dN/dt = B − D, how fast is the population growing?
A culture starts with 1,000 bacteria that have unlimited food and space. The population doubles every 20 minutes. How many bacteria will there be after 2 hours?
0 of 4 answered