Population Ecology — Free AP Biology Lesson

The idea, then worked examples solved step by step, then practice questions whose numbers change every time you try them.

The idea

What a population is

A population is a group of individuals of the same species living in the same place and interacting with one another and with their environment. They compete for food, find mates, and are eaten by the same predators. Ecologists ask one question of a population more than any other: why does its size, written N, go up, go down or stay the same?

Births, deaths and size

Population growth dynamics depend on birth rate, death rate and population size. The birth rate (B) is the number of organisms born during an interval of time, and the death rate (D) is the number that die during that interval. The change in the population is what is added minus what is lost:

\begin{aligned} \frac{dN}{dt} &= B - D \end{aligned}

Here $dN$ is the change in population size and $dt$ is the change in time. Think of a bathtub: births are the tap, deaths are the drain, and N is the water level. If the tap runs faster than the drain, N rises; if the drain wins, N falls; if they match, N holds steady. Notice that immigration and emigration are left out of this simple model, so it describes a population that gains and loses individuals only by birth and death.

If $120$ individuals are born in a year and $85$ die, the population changes by $120$ - $85$ = $35$ individuals that year. If deaths were greater than births, the result would be negative and the population would shrink.

Warning.

A common slip: a population with a high birth rate is not automatically growing. What matters is the birth rate compared with the death rate. A population with many births and even more deaths is declining.

Per capita rates and rmax

The number of births depends on how many parents there are. So ecologists often use a per capita rate, a rate for each individual. If each individual has a per capita birth rate b and a per capita death rate d, then B = bN and D = dN, and the change in the population is (b - d)N. The difference b - d is written r, the growth rate for each individual. A positive r means the population is increasing, a negative r means it is decreasing, and r = 0 means its size is not changing.

Different species have different potentials. The maximum per capita growth rate of a species is its biotic potential, written rmax. A bacterium can reproduce far more rapidly than a human, so a bacterial species has a much higher rmax. This is one of many adaptations related to obtaining and using energy and matter in a particular environment: a species that turns nutrients into new individuals quickly can add many individuals in a short time.

Exponential growth: reproduction without constraints

Suppose there is nothing to hold a population back: space, food and other resources are plentiful. Then reproduction goes on without constraints, and the population grows exponentially. The relevant equation is

\begin{aligned} \frac{dN}{dt} &= r_{max}\,N \end{aligned}

where N is the population size and rmax is the maximum per capita growth rate. The growth rate depends on N itself. The more individuals there are, the more offspring they produce, and those offspring soon produce offspring of their own. The classic case is bacteria in a flask with unlimited nutrients that divide once an hour.

Starting with 1000 bacteria, each hour every cell divides, so the population doubles each hour. The sizes at hours 0 to 5: $1000$ , $2000$ , $4000$ , $8000$ , $16000$ , $32000$ . The number added each hour is $1000$ , then $2000$ , then $4000$ , and so on. The important idea is that the number of organisms added in each generation keeps increasing. The population is not rising by a fixed amount each hour; it is rising by a fixed fraction of a bigger and bigger population.

The picture plots population size N against time for rmax = 0.5 per hour, starting from 100. The curve is J-shaped: it starts out gently, then climbs steeper and steeper. Steepness on this graph is the growth rate, and the curve gets steeper because the rate grows with N.

Species differ in rmax, and a bigger rmax makes a steeper J. Compare the same starting population with a smaller rmax.

Both populations start at 100 and run for 10 hours, but read the vertical scale: it is different in the two pictures. The population with rmax = 0.25 per hour ends far below the one with rmax = 0.5 per hour. A small difference in rmax produces a large difference in size after a long time, because the effect is compounded each hour.

At the start, the same population of 100 adds $25$ individuals per hour when rmax = 0.25 and $50$ individuals per hour when rmax = 0.5. The growth rate is dN/dt = rmax N, so with the same N, the larger rmax gives the larger rate.

What exponential growth does and does not tell you

Exponential growth is possible only when resources are not limiting. A population with its resources spread over few individuals can grow this way for a while. A population cannot do so forever, because the example above would reach many billions of bacteria in a day. The way real resources set a limit, and the leveling off at a carrying capacity, belong to the next topic. For this topic, remember the starting point: with no constraints on reproduction, the result is exponential growth.

Warning.

Three traps. First, exponential growth is not the same as "fast" growth: it means the rate of increase grows with N, which gives the J-shaped curve. Second, steady growth by the same number each time (100, 200, 300, 400) is not exponential; exponential growth adds more each time (1000, 2000, 4000, 8000). Third, the equation dN/dt = rmax N describes unconstrained reproduction, so it should not be used to predict the size of a population that is short of resources.

Adapted from Biology by OpenStax (Rice University), licensed under CC BY 4.0. Access for free at https://openstax.org/details/books/biology

Worked examples

Each example is the kind of question the AP exam asks on this topic, worked step by step.

Example.

1. Births and deaths. In one year a population of field mice has 340 births and 410 deaths. What is the change in population size for the year, and is the population growing or shrinking?

The change in a population is the births minus the deaths, $\frac{dN}{dt} = B - D$ .
Put in the numbers: $340$ - $410$ = $-70$ .
The result is negative, so more individuals were lost than were added.

Result.

The population changed by $-70$ mice, so it is shrinking.

Example.

2. Exponential growth in a flask. A flask has 500 bacteria and unlimited nutrients. Each hour every cell divides once, so the population doubles each hour. How many bacteria are there after 4 hours, and how many are added in the fourth hour?

Unlimited nutrients mean reproduction without constraints, so growth is exponential.
Each hour the population doubles. The sizes at hours 0 to 4: $500$ , $1000$ , $2000$ , $4000$ , $8000$ .
The number added in the fourth hour is $8000$ - $4000$ = $4000$ .
Check: the number added in the first hour was $500$ , so the number added each hour keeps increasing.

Result.

After 4 hours there are $8000$ bacteria, and $4000$ were added in the fourth hour.

Example.

3. The growth rate for a given size. A population of 800 insects with unlimited food has rmax = 0.25 per week. What is dN/dt now, and what is dN/dt when the population has grown to 1600?

With no constraints, $\frac{dN}{dt} = r_{max}N$ .
At N = $800$ : $0.25$ × $800$ = $200$ insects per week.
At N = $1600$ : $0.25$ × $1600$ = $400$ insects per week.
The population doubled, and so did the growth rate: the rate depends on N.

Result.

The growth rate is $200$ per week now and $400$ per week at 1600. The population grows faster the larger it is.

Example.

4. Reading the J-shaped curve. The graph shows a population with rmax = 0.4 per day, starting at 50. Describe the shape of the curve and find the population at day 5.

The curve is J-shaped: it rises slowly at first and then more and more steeply. That is the shape of exponential growth, where reproduction goes on without constraints.
The curve is steeper later because the growth rate grows with N: at the start dN/dt = $0.4$ × $50$ = $20$ per day, but at day $5$ (N about $369$ ) it is about $148$ per day.
Population size at day $5$ : $N_0 e^{r t}$ gives about $369$ .

Result.

The curve is J-shaped and steepens with time; at day 5 the population is about $369$ .

Example.

5. Compare two species. Under ideal conditions a bacterium has rmax = 0.8 per hour. Another species in the same flask conditions has rmax = 0.2 per hour. Both start at 100. Which has the larger growth rate at the start, and by how much?

Use $\frac{dN}{dt} = r_{max}N$ for each species, since both have unlimited resources.
First species: $0.8$ × $100$ = $80$ per hour. Second species: $0.2$ × $100$ = $20$ per hour.
The difference is $80$ - $20$ = $60$ per hour.
With the same N, the species with the larger rmax has the larger growth rate, and the gap widens as the populations grow.

Result.

The species with rmax = 0.8 adds $80$ individuals per hour at the start, $60$ more than the other.

Practice

Answer AP-style questions on population ecology. Each answer is checked, with the full solution, and every try brings new numbers. Practice needs a free account: it keeps your progress.