What are Population Growth Models?
Population growth models describe how populations change over time. Two main models explain real-world growth: exponential growth (unlimited resources → J-shaped curve) and logistic growth (limited resources → S-shaped curve with carrying capacity). Understanding these models is critical for conservation, fishery management, and predicting ecosystem collapse.
Exponential growth (Nt = N₀e^rt) assumes unlimited resources, leading to ever-faster population increase and a J-shaped curve. Logistic growth (dN/dt = rN[K−N]/K) includes carrying capacity K, slowing growth as population approaches resource limits, forming an S-shaped curve.
Step-by-step worked examples
A bacterial colony starts with 1,000 cells and doubles every hour under unlimited food. What is the population after 5 hours? Predict the model.
This is exponential growth: Nt = N₀ × 2^t where t = hours. N₀ = 1,000 N₅ = 1,000 × 2^5 = 1,000 × 32 = 32,000 cells after 5 hours. Pattern: 1h = 2,000; 2h = 4,000; 3h = 8,000; 4h = 16,000; 5h = 32,000 → J-shaped curve (doubles forever, unrealistic for long term). If food becomes limited after 10 hours, population slows and follows logistic growth instead.
A lake's fish population grows logistically with r = 0.5/year and carrying capacity K = 5,000 fish. Starting from 100 fish, find population at year 2.
Logistic formula: Nt = K / (1 + [K−N₀]/N₀ × e^(−rt)) N₀ = 100, K = 5,000, r = 0.5, t = 2 Nt = 5,000 / (1 + [5,000−100]/100 × e^(−0.5×2)) Nt = 5,000 / (1 + 49 × e^(−1)) Nt = 5,000 / (1 + 49 × 0.368) Nt = 5,000 / (1 + 18.03) Nt = 5,000 / 19.03 ≈ 262 fish at year 2. Population is growing but slowing as it approaches K = 5,000 (S-curve).
A deer population in a forest follows logistic growth with K = 2,000. If poachers remove 400 deer/year, will the population be stable at 2,000?
At K = 2,000, dN/dt = 0 (stable, no growth). But if 400 deer are harvested annually, the population cannot stay at 2,000. New equilibrium: dN/dt = rN(K−N)/K − 400 = 0 If r = 0.3, then 0.3 × N × (2,000−N)/2,000 = 400 Solving: 0.3N(2,000−N) = 800,000 → 600N − 0.3N² = 800,000 → N ≈ 1,500 to 1,600 fish. Population crashes below K and stabilizes lower unless harvesting reduces.
Flashcards
Quick quiz
Q1.A population starts at 100 and triples every year. After 3 years, how many?
Q2.What shape does exponential growth form?
Q3.In logistic growth, what limits population size?
Q4.A population is at carrying capacity K = 1,000. What is dN/dt (rate of change)?
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Common mistakes
Exponential growth is always harmful and must be stopped. — Correct: Exponential growth is natural early in a population's life. It becomes unsustainable only without resource limits.
Carrying capacity is a fixed number that never changes. — Correct: Carrying capacity changes with environmental conditions: drought lowers K, abundant rain raises K.
When population reaches K, it stays at K forever. — Correct: Population fluctuates around K due to environmental variability, predation, and disease — rarely stays exactly at K.
All real populations follow logistic growth. — Correct: Some populations (bacteria in lab, invasive species early on) follow exponential growth for years before resource limits kick in.
FAQ
What is the difference between exponential and logistic growth?
Exponential: unlimited resources, continuous doubling, J-curve, unsustainable. Logistic: limited resources, slows near carrying capacity, S-curve, sustainable.
Why do real populations follow logistic growth more than exponential?
Real environments have finite resources (food, water, space). As population grows, competition and resource scarcity slow growth.
Can a population exceed carrying capacity?
Yes, briefly — if food suddenly becomes abundant or predators disappear. But the population crashes back as resources deplete.
How does harvesting affect logistic population growth?
Sustainable harvest: remove ~50% of growth rate per year (below dN/dt at half K). Unsustainable: remove more → population crashes.