Population Ecology: Study Guide (Biology 8.3)

What a population is

A population is all the individuals of one species living in the same area at the same time. The members of a population are not isolated. They interact with each other: they compete for food and space, find mates, and may cooperate or spread disease. They also interact with their environment: they take in energy and matter, and they are affected by temperature, water and other species.

Three features help you tell a population apart from other groups:

• One species. All the fish in a lake form part of a community. All the perch in that lake form a population.
• One place. Members must be close enough to interact and, usually, to breed with each other.
• One time. A population is counted at a particular moment. Its size changes over time.

We write population size as N. Ecologists study how N changes and why. That is population ecology.

What changes population size

A population grows when individuals are added faster than they are removed. In the simplest model, the population is closed: no individuals move in or out. Then only two processes matter:

• births (B), the number of new individuals added per unit time;
• deaths (D), the number of individuals lost per unit time.

The rate of change of population size is:

dN/dt = B − D

where dN is the change in population size and dt is the change in time. The units are individuals per unit time, for example beetles per month.

• If B > D, dN/dt is positive and the population grows.
• If B = D, dN/dt is zero and the population size stays the same.
• If B < D, dN/dt is negative and the population shrinks.

Per capita rates: comparing populations of different sizes

A large population has more births than a small one simply because it has more parents. To compare populations fairly, divide by N to get per capita (“per head”) rates:

• per capita birth rate, b = B ÷ N
• per capita death rate, d = D ÷ N
• per capita growth rate, r = b − d = (B − D) ÷ N

So dN/dt = rN. This form shows the key idea of the topic: growth rate depends on birth rate, death rate and population size together. Two populations with the same r grow at different speeds if their sizes differ.

Energy, matter and births and deaths

Every birth costs energy and matter. A parent must gather enough food to stay alive, grow, and then build eggs, seeds or young. Every death that is avoided also depends on energy and matter: an animal that cannot find enough food starves or becomes easier prey. This is why many adaptations are tied to obtaining and using energy and matter, and why they show up in population numbers.

• Feeding adaptations (beak shape, root depth, hunting behaviour) set how much energy an individual can gather. More energy available for reproduction usually means more offspring, raising B.
• Energy budgets. Energy used for keeping warm, moving or repairing tissue is not available for reproduction. An endotherm spends much of its energy keeping a constant body temperature, so it usually needs more food than an ectotherm of similar mass.
• Timing. Many animals breed when food is most plentiful, so young are raised when energy is easy to find. This raises survival and lowers D.
• Saving energy in hard times. Dormancy, hibernation or dropping leaves reduce energy use when food or water is scarce, which lowers deaths.

So a change in the environment that changes food supply changes B and D, and through them dN/dt.

Exponential growth

Imagine a population with no constraints on reproduction: plenty of food, space and water, no build-up of waste and few predators or diseases. Each individual then reproduces at its highest possible rate. The per capita growth rate takes its largest value, called rₘₐₓ (the maximum per capita growth rate), and stays constant. The growth equation becomes:

dN/dt = rₘₐₓN

This is exponential growth. Read the equation carefully:

• rₘₐₓ is constant, so dN/dt is proportional to N. Double the population and you double the number added per unit time.
• As N grows, dN/dt grows too, so the curve of N against time gets steeper and steeper. Its shape is often called a J-shaped curve.
• Because the per capita rate is constant, the population takes the same time to double, whatever its size. A culture that doubles every 2 hours goes 1 → 2 → 4 → 8 → 16 units in 8 hours.

Exponential growth happens when resources are, for a while, far more than the population needs. Examples include bacteria or yeast added to fresh nutrient medium, a species reaching a new habitat with no competitors, or a population recovering after a disaster. It cannot continue forever: resources run short. What happens then is the subject of the next topic.

Representing population data on graphs

Population data are usually counts at different times, so the natural choice is a line graph (or a scatter plot of points joined by a line or trend line):

1. Type of graph. Time is continuous, so plot N against time as points joined by lines. Do not use a bar chart for a time series.
2. Axes and units. Time on the x-axis (the independent variable), population size or density on the y-axis, each with a label and a unit, for example “Cell density / 10⁴ cells mL⁻¹”. Add a legend if there is more than one data set.
3. Scaling. Choose even steps that use most of the grid. For exponential data, a log scale on the y-axis (1, 10, 100, …) gives equal spacing to equal ratios, so steady doubling plots as a straight line.
4. Plotting. Plot means. If you have repeated counts, add error bars (for example, showing the range, or a confidence interval as in later topics).
5. Trend line. Draw a smooth curve or straight line through the general trend, not “dot to dot” through every wobble, when the pattern is clear.

Worked example 1: births, deaths and growth in a beetle population

Question. A population of 256 leaf beetles colonises a new glasshouse crop. In the first month there are 96 births and 32 deaths. No beetles enter or leave.

(a) Calculate dN/dt for the first month. (b) Calculate the per capita birth, death and growth rates. (c) Assuming the per capita growth rate stays the same, project the population month by month for 4 months, and comment on the number added each month.

(a) dN/dt = B − D = 96 − 32 = 64 beetles per month.

(b)

1. b = B ÷ N = 96 ÷ 256 = 0.375 per month (each beetle produces, on average, 0.375 offspring per month).
2. d = D ÷ N = 32 ÷ 256 = 0.125 per month.
3. r = b − d = 0.375 − 0.125 = 0.25 per month. Check: 64 ÷ 256 = 0.25. ✓

(c) Use dN/dt = rN for each month, then add the increase to N.

Month	N at start	Added this month (0.25 × N)	N at end
1	256	64	320
2	320	80	400
3	400	100	500
4	500	125	625

Interpretation. The per capita rate never changes, yet the number added each month rises, from 64 to 125, nearly double (125 ÷ 64 ≈ 1.95). The increase grows because N grows. This is the signature of exponential growth.

A note on the model. dN/dt describes the rate at one instant. Stepping one month at a time treats each month’s growth as one jump, so it is an estimate. With smaller steps, the new beetles would start breeding sooner and the total after 4 months would be larger (about 696 if growth were truly continuous). For this course, a step-by-step projection is enough; just say it is an estimate.

Worked example 2: graphing and testing exponential growth

Question. A student adds yeast to fresh sugar solution in three flasks and counts the cells every 2 hours. Cell density (×10⁴ cells mL⁻¹):

Time / h	0	2	4	6	8	10
Flask 1	0.9	1.9	3.7	7.6	15.0	30.4
Flask 2	1.0	2.1	4.0	8.3	15.9	32.1
Flask 3	1.1	2.3	4.3	8.7	16.8	34.4

(a) Calculate the mean at each time and describe how to graph the data. (b) Use the means to test whether growth is exponential. (c) Estimate dN/dt and the per capita growth rate between 8 h and 10 h.

(a) Means. Add the three flasks and divide by 3. For example, at 10 h: (30.4 + 32.1 + 34.4) ÷ 3 = 96.9 ÷ 3 = 32.3. The means are 1.0, 2.1, 4.0, 8.2, 15.9 and 32.3.

Graph. Line graph of mean cell density against time; time on the x-axis in hours; cell density on the y-axis with its unit; error bars showing the range of the three flasks. On a linear axis (Figure 2A) the first four points are squashed near zero. A log axis (Figure 2B) spreads them out.

(b) Test. For exponential growth, the population should be multiplied by the same factor in each equal time interval.

Interval / h	0–2	2–4	4–6	6–8	8–10
Ratio of means	2.1	1.9	2.05	1.94	2.03

Each ratio is close to 2, so the culture doubles about every 2 hours. Over 10 h the density rises 32.3-fold, about five doublings (2⁵ = 32). The points lie close to a straight line on the log graph. The data support exponential growth over this period.

(c) Rates.

1. dN/dt ≈ change in N ÷ change in time = (32.3 − 15.9) ÷ 2 = 16.4 ÷ 2 = 8.2 × 10⁴ cells mL⁻¹ h⁻¹.
2. Mean density over the interval = (15.9 + 32.3) ÷ 2 = 24.1 × 10⁴ cells mL⁻¹.
3. Per capita growth rate ≈ 8.2 ÷ 24.1 = 0.34 per hour.

Interpretation. Between 0 and 2 h, dN/dt was only 0.55 × 10⁴ cells mL⁻¹ h⁻¹, but the per capita rate was similar (0.55 ÷ 1.55 ≈ 0.36 per hour). The absolute rate rose about 15-fold while the per capita rate stayed roughly constant: exactly what dN/dt = rₘₐₓN predicts. The sugar was plentiful, so rₘₐₓ ≈ 0.34–0.36 per hour for these conditions. With only three flasks and 10 hours of data, this is support, not proof; a longer run would show growth slowing as sugar runs out.

Common misconceptions

• “Exponential growth just means fast growth.” It means growth in proportion to N, with a constant per capita rate. A slow-breeding species can grow exponentially; it just has a smaller rₘₐₓ.
• “In exponential growth the growth rate is constant.” The per capita rate is constant. The number added per unit time (dN/dt) keeps rising.
• “B is the birth rate per individual.” In dN/dt = B − D, B and D are totals for the whole population per unit time. Divide by N to get per capita rates.
• “A population with more births per year must be growing faster.” Compare per capita rates: a large population can have more births but a lower growth rate per individual.
• “Deaths stop a population growing.” The population still grows as long as B > D. It only shrinks when deaths outnumber births.
• “All the animals in a forest are a population.” Several species together form a community. A population is one species.
• “A straight line on a log graph means linear growth.” On a log scale, a straight line means steady multiplication, which is exponential growth.
• “Populations can grow exponentially forever.” Exponential growth needs unlimited resources. Sooner or later, food, space or other limits slow it down.

Where this leads

This topic builds on how energy moves through ecosystems (Energy Flow Through Ecosystems). Next, you will see what happens when resources run short and the population’s own density slows its growth: Effect of Density on Populations. Test yourself with the practice questions, then use the revision notes and the checklist.