Put a few bacteria in a fresh petri dish and their numbers explode — doubling, doubling again, a J-shaped rocket to the sky. But the dish is finite. Food runs out, waste piles up, and the curve bends over into a lazy S, leveling off at the most the dish can support. Almost every population in nature tells some version of this story: unchecked growth when resources are plentiful, then the brakes of a finite world. This lesson gives you the two curves that describe it, the vocabulary for the brakes (limiting factors and carrying capacity), and the two life strategies — live fast or play the long game — that species use to cope. It's also a math lesson: growth rate and doubling time are guaranteed FRQ calculations.
When resources are unlimited, a population grows exponentially — a constant percentage increase per time step produces a J-shaped curve that gets steeper and steeper.
Growth rate (r) = (births − deaths) / N (per individual, per time), and population change is ΔN = r × N.
Exponential growth appears in newly colonized habitats, after a disturbance, or in species with few natural checks (invasive species, bacteria, human population for the last few centuries).
[DIAGRAM: J-shaped exponential curve — population (y) vs. time (x) rising with ever-increasing steepness, no upper limit. Label "unlimited resources."]
Resources are never truly unlimited. As a population grows, limiting factors slow it until it levels off at the carrying capacity (K) — the maximum population size the environment can sustainably support. This produces an S-shaped (sigmoidal) curve.
[DIAGRAM: S-shaped logistic curve — slow start, steep middle (near-exponential), then leveling at a horizontal dashed line labeled "carrying capacity (K)." Note growth rate is fastest at the inflection point (~K/2) and approaches zero at K.]
The logistic model: ΔN = r × N × [(K − N)/K]. The bracket term shrinks as N approaches K, slowing growth to zero at K.
Limiting factors cap populations. They come in two types: - Density-dependent — their effect intensifies as population density rises: competition for food/space, disease, predation, waste accumulation. - Density-independent — affect the population regardless of density: weather, natural disasters, temperature, fire.
If a population grows past K (often due to a reproductive lag before limiting factors kick in), it overshoots carrying capacity. Depleted resources then cause a dieback (crash) back toward or below K. Repeated overshoot/dieback appears as oscillations around K. Severe resource destruction can even lower K itself (environmental degradation).
Species evolve toward one of two ends of a strategy spectrum:
| Trait | r-selected | K-selected |
|---|---|---|
| Reproduction | Many offspring, early, once or few times | Few offspring, later, repeatedly |
| Parental care | Little/none | High |
| Body size / lifespan | Small / short | Large / long |
| Population near | Fluctuates, often below K (J-curve booms) | Stable, near K |
| Examples | Insects, weeds, bacteria, rodents, dandelions | Elephants, whales, humans, oak trees |
| Survivorship | Type III (Lesson 8) | Type I |
r-strategists thrive in unstable/disturbed environments and as pioneer/invasive species (link to succession, Lesson 5). K-strategists dominate stable environments and are more vulnerable to overharvesting and extinction because they reproduce slowly.
Two guaranteed calculations:
r (%) = [(births + immigration) − (deaths + emigration)] / N × 100. A simpler exam version: growth rate (%) = (CBR − CDR)/10 where CBR/CDR are crude birth/death rates per 1,000. (Detailed in Lesson 9.)doubling time (years) = 70 / (percent growth rate). This is the single most-tested APES formula.Recognizing J vs. S curves, explaining carrying capacity and limiting factors, and computing growth rate and doubling time cover a big slice of Unit 3 — the largest of the early units. The r/K framework ties populations back to succession and forward to conservation and human demographics (Lesson 9).
A population of rabbits introduced to an island grows slowly, then rapidly, then levels off at about 5,000. Which growth model, and what is 5,000?
Solution: The leveling S-shape is logistic growth; 5,000 is the carrying capacity (K).
Interpretation: Leveling off = logistic; the plateau value = K.
A deer population of 2,000 has 300 births and 100 deaths in a year (ignore migration). Find the growth rate (%) and the number added.
Strategy: r = (births − deaths)/N; ΔN = births − deaths.
Solution:
r = (300 − 100)/2,000 = 200/2,000 = 0.10 = 10%
ΔN = 300 − 100 = 200 deer added
Answer: 10% growth; 200 deer added.
Interpretation: Growth rate is the per-capita net change; ΔN is the raw number.
A population grows at 2% per year. How long until it doubles? What about at 5%?
Strategy: doubling time = 70 / percent growth rate.
Solution:
At 2%: 70 / 2 = 35 years
At 5%: 70 / 5 = 14 years
Answer: 35 years (2%); 14 years (5%).
Interpretation: Small changes in percent rate produce big changes in doubling time — the power of exponential growth.
A reindeer herd introduced to an island soars to 6,000, far above the island's ~2,000 carrying capacity, then crashes to 500. Explain using overshoot, limiting factors, and density-dependence.
Solution: The herd overshot K because reproduction outpaced the lag in resource depletion. Once overpopulated, density-dependent limiting factors — food shortage from overgrazing, starvation, disease — caused a dieback (crash). Overgrazing degraded the vegetation, likely lowering K itself, so the population crashed below the original carrying capacity.
Interpretation: Overshoot → resource depletion → dieback, and severe damage can lower K.
70 / percent rate, using the rate as a whole number (2, not 0.02). 70/2 = 35, not 70/0.02.70/3.5 = 20 years.(500−250)/5,000 = 250/5,000 = 0.05 = 5%.(B) Density-independent = same effect regardless of density.
Growth cannot continue indefinitely because resources are finite. Two limiting factors: food/nutrient depletion and accumulation of toxic waste (also space, oxygen). Once these limiting factors take hold, growth slows and levels off at carrying capacity, converting the J-shaped exponential curve into an S-shaped logistic curve.
(a) 70/2 = 35 years. (b) After one doubling: 40 × 2 = 80 million. (c) First-year addition ≈ 40,000,000 × 0.02 = 800,000 people.
FRQ rubric (10 pts):
- (a) 1 pt setup (2,240 − 2,000)/2,000; 1 pt = 0.12 = 12%. (2)
- (b) 1 pt setup 70/12; 1 pt ≈ 5.8 years. (2)
- (c) 1 pt exponential (J-shaped); 1 pt because resources are abundant and there are no predators/limiting factors yet. (2)
- (d) 1 pt names density-dependent factor (food competition, disease, space); 1 pt explains it intensifies as density rises, slowing growth toward K. (2)
- (e) 1 pt names a valid action (introduce controlled predator/native competitor, targeted removal/fishing, biocontrol, prevent further introductions); 1 pt justification tied to reducing the invasive population or its impact. (2)
An invasive fish is introduced to a lake with no natural predators. Its population grows from 2,000 to 2,240 in one year.
(a) Calculate the population growth rate as a percentage. Show your work. (2 pts) (b) Using the rule of 70, calculate the doubling time. Show your work. (2 pts) (c) Identify the type of growth curve expected in the short term and explain why. (2 pts) (d) Explain what will eventually limit this population, naming one density-dependent factor. (2 pts) (e) Propose one management action to control the invasive fish and justify it. (2 pts)
MC:
1. (B) J-shape = exponential growth.
2. (B) K = max sustainable population.
3. (A) 70/3.5 = 20 years.
4. (C) Competition for food is density-dependent. (A)/(B)/(D) are density-independent.
5. (B) Many offspring, little care = r-selected.
6. (B) Fastest near K/2 (the steep middle). Zero at K.
7. (B) (500−250)/5,000 = 250/5,000 = 0.05 = 5%.
8. (B) Exceeding K and depleting resources → crash.
9. (C) Elephant — few offspring, long life, high care.
10. (B) Density-independent = same effect regardless of density.
Growth cannot continue indefinitely because resources are finite. Two limiting factors: food/nutrient depletion and accumulation of toxic waste (also space, oxygen). Once these limiting factors take hold, growth slows and levels off at carrying capacity, converting the J-shaped exponential curve into an S-shaped logistic curve.
(a) 70/2 = 35 years. (b) After one doubling: 40 × 2 = 80 million. (c) First-year addition ≈ 40,000,000 × 0.02 = 800,000 people.
FRQ rubric (10 pts):
- (a) 1 pt setup (2,240 − 2,000)/2,000; 1 pt = 0.12 = 12%. (2)
- (b) 1 pt setup 70/12; 1 pt ≈ 5.8 years. (2)
- (c) 1 pt exponential (J-shaped); 1 pt because resources are abundant and there are no predators/limiting factors yet. (2)
- (d) 1 pt names density-dependent factor (food competition, disease, space); 1 pt explains it intensifies as density rises, slowing growth toward K. (2)
- (e) 1 pt names a valid action (introduce controlled predator/native competitor, targeted removal/fishing, biocontrol, prevent further introductions); 1 pt justification tied to reducing the invasive population or its impact. (2)
⭐ Exam strategy: Memorize doubling time = 70 / (% growth rate) cold — it appears nearly every year. Always use the rate as a whole-number percent (70/2 = 35), and always write your units. And remember: J-curve = exponential, S-curve = logistic, plateau = K.
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