Population math
Population questions are the only place in Biology 30 where you calculate, and the equations are printed on your data pages. What is not printed is which one to use, what the answer means, and the order of operations for Hardy–Weinberg — which is what this page is for.
- 1. The symbols and equations
- 2. Change, rate and per capita rate
- 3. The two growth curves
- 4. Hardy–Weinberg, step by step
- 5. Traps
The symbols and equations
The idea: these are all on the data pages. Your job is choosing among them and getting the units right.
| Quantity | Equation | Units |
|---|---|---|
| Population density | Dp = N/A, or N/V for water or air | individuals per m², km² or m³ |
| Change in population | ∆N = (births + immigration) − (deaths + emigration) | individuals |
| Growth rate | gr = ∆N/∆t | individuals per unit time |
| Per capita growth rate | cgr = ∆N/N | none — it is per individual |
| Hardy–Weinberg | p + q = 1 and p² + 2pq + q² = 1 | frequencies between 0 and 1 |
Growth rate answers "how many more per year?" Per capita growth rate answers "how many more per year, per individual already there?" — which is why it has no units and why it is the fair way to compare a herd of 50 with a herd of 5 000.
Change, rate and per capita rate
The idea: find ∆N first, then divide by time for a rate, or by N for a per capita rate.
Worked: a deer population
A population of 2 000 deer has 400 births and 180 deaths in a year, with 60 deer immigrating and 120 emigrating.
- ∆N = (400 + 60) − (180 + 120) = 460 − 300 = 160 deer.
- gr = ∆N/∆t = 160 ÷ 1 year = 160 deer per year.
- cgr = ∆N/N = 160 ÷ 2 000 = 0.08.
On a numerical-response question, 0.08 is recorded with the leading zero. As a percentage, the population grew 8% that year.
Worked: density
1 800 beetles occupy 12 m² of forest floor: Dp = 1 800 ÷ 12 = 150 beetles/m². Choose area for a surface, volume for water or air, and state the units — markers look for them.
The two growth curves
The idea: unlimited resources give a J; limited resources give an S that flattens at carrying capacity.
| Term | Means |
|---|---|
| Biotic potential | The maximum rate a population could reproduce under ideal conditions |
| Environmental resistance | Everything that holds it below that: food, space, predators, disease |
| Carrying capacity (K) | The population an environment can sustain over time; the population fluctuates around it |
| Density-dependent factor | Hits harder when crowded: competition, disease, predation, parasitism |
| Density-independent factor | Hits the same proportion at any density: drought, fire, a hard frost |
r-selected species have many offspring with little care and boom and crash — insects, weeds. K-selected species have few offspring with heavy investment and sit near carrying capacity — whales, humans, oak trees.
Hardy–Weinberg, step by step
The idea: p is the frequency of the dominant allele and q the recessive one. The only phenotype that names a genotype outright is the recessive one, so every question starts there.
| Term | Is the frequency of |
|---|---|
| p | the dominant allele |
| q | the recessive allele |
| p² | homozygous dominant individuals |
| 2pq | heterozygous individuals — the carriers |
| q² | homozygous recessive individuals — the ones you can see |
Worked: 16% of a population shows the recessive phenotype
- Start at q²: the recessive phenotype is the only genotype you can read directly, so q² = 0.16.
- Take the square root: q = √0.16 = 0.4.
- Use p + q = 1: p = 1 − 0.4 = 0.6.
- Carriers: 2pq = 2(0.6)(0.4) = 0.48, so 48% are heterozygous.
- Check: p² + 2pq + q² = 0.36 + 0.48 + 0.16 = 1.00 ✓
Notice that carriers outnumber affected individuals more than three to one — which is why a recessive allele does not disappear from a population.
The equation only holds when the population is large, mating is random, and there is no mutation, no migration and no natural selection. Real populations break at least one of those, so the real use of Hardy–Weinberg is as a baseline: if the observed frequencies drift away from the predicted ones, something is acting on the population.
Traps
The idea: the biology here is easier than the bookkeeping. Nearly all lost marks are one of these.
- Mistaking q² for q. The percentage showing the trait is q², and q needs the square root.
- Using percentages in the equation. Convert to decimals: 16% is 0.16, not 16.
- Carriers are 2pq, not pq. There are two ways to be heterozygous.
- Mixing up gr and cgr. Divide by time for a rate, by population for a per capita rate.
- Dropping units, or reporting a density without saying per what.
- Forgetting emigration in ∆N — immigration is in, emigration is out, and questions include both on purpose.