1Populations

A population is defined by three things: one species, one area, one time. The trout in a single lake form a population; the trout and the perch in that lake do not, because they are two species. The members of a population interact in complex ways: they compete for the same resources, they find mates among each other, they may cooperate or fight, and they share predators and diseases.

Many adaptations are about getting and using energy and matter in a particular environment. A desert plant's deep roots reach scarce water; a seal's layer of blubber stores energy and holds heat in cold seas. How well a population's members get energy decides how many offspring they can produce, which is why energy (8.2) and population growth are taught together.

The size of a population, written $N$, is the number of individuals in it. Population density is the number per unit area or volume, for example 40 snails per square metre.

2Births, deaths and the rate of change

The formula sheet gives

$$\frac{dN}{dt} = B - D$$

where $\frac{dN}{dt}$ is the rate of change of population size (the change in $N$, $dN$, over a change in time, $dt$), $B$ is the birth rate and $D$ is the death rate. On the sheet, and in most AP questions, $B$ and $D$ are the numbers of births and deaths in the population per unit time.

Worked example (invented data). A pond holds 400 water fleas. In one day, 120 are born and 40 die.

$$\frac{dN}{dt} = 120 - 40 = 80 \text{ water fleas per day}$$

The population is growing, by 80 individuals a day. If deaths had been 150, $\frac{dN}{dt}$ would be $-30$ per day: a negative rate means the population is shrinking.

You will also meet the per capita rate, the change per individual. Divide by $N$:

$$r = \frac{B - D}{N} = \frac{80}{400} = 0.2 \text{ per day}$$

so each water flea adds, on average, 0.2 of a new individual per day. Per capita rates let you compare populations of different sizes: a city growing by 5,000 people a year and a village growing by 50 may have the same per capita rate.

These equations treat the population as closed, with no individuals moving in or out. If the question gives immigration or emigration, add the arrivals and subtract the departures as well.

3Exponential growth

When a population has plenty of food, space and water and no predators or disease holding it back, each individual reproduces at the highest rate its biology allows. That rate is the maximum per capita growth rate, $r_{max}$. The formula sheet gives

$$\frac{dN}{dt} = r_{max}N$$

Because $r_{max}$ is constant and $N$ keeps rising, $\frac{dN}{dt}$ keeps rising too. A large population adds more individuals per day than a small one, purely because there are more individuals reproducing. This is exponential growth.

Worked example. The water fleas in the pond above grow exponentially with $r_{max} = 0.2$ per day.

• When $N = 400$: $\frac{dN}{dt} = 0.2 \times 400 = 80$ per day.
• When $N = 2{,}000$: $\frac{dN}{dt} = 0.2 \times 2{,}000 = 400$ per day.

The per capita rate is the same at both sizes; the number added per day is five times greater because the population is five times larger.

Worked example: a doubling population. An invented population of bacteria doubles every generation, starting from 100 cells.

The per capita increase is the same every generation (each cell becomes two), but the increase itself doubles each time. Figure 1 plots the table.

Exponential growth is seen when constraints are absent, and that is usually brief: bacteria transferred into fresh nutrient broth, a species reaching an island with no competitors, a population recovering after a fire has cleared space. It cannot last, because resources run out, which is where 8.4 begins.

Exponential growth means a constant per capita rate, so the number added per unit time rises as the population grows.

4Graphing population data

Skill 4.A, constructing a graph, is the skill the CED attaches to this topic. A population counted over time is a line graph (or a scatter plot with a trend line):

1. Type: time is continuous, so plot points joined by lines (or a smooth trend line), not bars.
2. Axes: time, the independent variable, on the x-axis; population size on the y-axis. Label both with units ("Time (days)", "Number of water fleas").
3. Scale: even steps that use most of the grid, starting at zero unless there is a clear reason not to.
4. Plotting: every point accurate; error bars if means of repeated counts are given.
5. Trend line: draw one only when the question asks or when the trend is clear.

A log scale for exponential data. If the y-axis uses a logarithmic scale (1, 10, 100, 1,000, each the same distance apart), exponential growth plots as a straight line. That is a quick test: data that fall on a straight line on a log scale are growing at a constant per capita rate; data that bend over are slowing down.