AP Biology Unit 8: Ecology — Worked Examples

Maximum Population Growth Rate and Carrying Capacity

Hard

A population of deer in an enclosed nature reserve follows logistic growth. The carrying capacity (K) of the reserve is 1000 deer and the maximum per capita growth rate ( $r_{max}$ ) is 0.5 per year. At what population size will the population be adding the most individuals per year?

250 deer, because growth rate is highest at K/4
500 deer, because the population growth rate $\\frac{dN}{dt}$ is maximized at $N = K/2$ ✓ Correct
1000 deer, because the population is largest at carrying capacity
100 deer, because the per capita growth rate is highest when the population is small

Solution

The logistic growth equation is $\\frac{dN}{dt} = r_{max}N\\left(\\frac{K-N}{K}\\right)$ . This is a quadratic function of N that is maximized at $N = K/2$ . At $N = 500$ : $\\frac{dN}{dt} = 0.5 \\times 500 \\times \\frac{500}{1000} = 125$ deer/year. Choice A (250) incorrectly uses K/4. Choice C (1000) confuses maximum population size with maximum growth rate — at K, growth rate is zero because (K-N)/K = 0. Choice D (100) confuses per capita growth rate (highest when N is small) with population growth rate (the total number added per year).

Trophic Efficiency and Ecological Pyramids

Hard

In a grassland ecosystem, the net primary productivity of grasses is 10,000 kcal/m^2/year. Assuming 10% ecological efficiency at each trophic transfer, a researcher wants to know how much energy is available to tertiary consumers (e.g., hawks). The food chain is: Grasses -> Grasshoppers -> Frogs -> Hawks. How much energy is available to hawks, and what happens to the other 99.9% of the original energy?

10 kcal/m^2/year; the rest is lost as heat from cellular respiration at each trophic level ✓ Correct
100 kcal/m^2/year; the rest is stored in biomass at lower trophic levels
1,000 kcal/m^2/year; energy is conserved so only 90% is lost total
10 kcal/m^2/year; the rest is recycled back to producers through decomposition

Solution

With 10% efficiency at each transfer: Producers 10,000 -> Primary consumers 1,000 -> Secondary consumers 100 -> Tertiary consumers 10 kcal/m^2/year. Only $10,000 \\times 0.1^3 = 10$ kcal reaches hawks. The lost energy at each level is primarily converted to heat through cellular respiration (metabolic processes). This is why energy does not cycle in ecosystems — it flows unidirectionally and is lost as heat at each level. Choice B correctly calculates 10% but then says 100 — incorrect multiplication and wrong fate (biomass storage, not heat). Choice C miscalculates by only applying one 10% loss instead of three. Choice D gets the energy amount right but incorrectly claims energy is recycled — nutrients cycle, but energy flows one way and exits as heat.