Population Growth Models: Exponential and Logistic Guide

A population growth model is a mathematical rule for how the number of individuals in a population changes over time. The two classic models are exponential growth, where numbers multiply at a constant rate, and logistic growth, where growth slows as the population approaches the environment's limit.

These models matter because the same mathematics describes bacteria in a dish, fish in a lake, and human cities. Wildlife managers, doctors and city planners all use them to predict whether a population will explode, stabilize or collapse.

Key Concepts

• Population size N(t) — number of individuals at time t.
• Growth rate r — the per-person birth rate minus death rate, per unit time (units: 1/time, e.g. per year).
• Carrying capacity K — the maximum population the environment can sustain long-term.
• Exponential model — unlimited resources; growth is proportional to current size.
• Logistic model — growth slows as N approaches K; the standard model for real constrained populations.

Formulas

• Exponential: dN/dt = rN, solution N(t) = N₀e^(rt) — constant per-capita growth. Applies only while resources are effectively unlimited; predicts infinite growth, which is its famous limitation.
• Logistic: dN/dt = rN(1 − N/K), solution N(t) = K / (1 + Ae^(−rt)) where A = (K − N₀)/N₀ — growth rate shrinks linearly as N nears K, producing the S-shaped curve.
• Doubling time (exponential): t_double = ln(2)/r ≈ 0.693/r — time for the population to double. Valid only in the exponential regime.
• Equilibria of logistic model — N = 0 (unstable: a few individuals grow away from it) and N = K (stable: populations near K settle onto it).

Worked Example

Bacteria in a dish: N₀ = 1,000 cells, growth rate r = 0.5 per hour. Compare exponential and logistic (K = 1,000,000) predictions at t = 6 hours.

1. Step 1: Exponential: N(6) = 1,000 × e^(0.5×6) = 1,000 × e³ ≈ 1,000 × 20.086 = 20,086 cells.
2. Step 2: Logistic: A = (1,000,000 − 1,000)/1,000 = 999. N(6) = 1,000,000 / (1 + 999 × e^(−0.5×6)) = 1,000,000 / (1 + 999 × 0.0498) = 1,000,000 / 50.74 ≈ 19,708 cells.
3. Step 3: Compare — at 6 hours both agree (population far below K). At t = 30 hours, exponential predicts 1,000 × e^15 ≈ 3.3 billion (absurd), while logistic saturates near 1,000,000.
4. Step 4: Lesson — exponential works for early growth; logistic is needed once crowding matters.

Common Misconceptions

• "Exponential growth goes on forever." Nothing does. Every real population hits limits — food, space, disease — which is exactly why the logistic model exists.
• "K is a hard ceiling the population never crosses." Populations can overshoot K temporarily (e.g. a bumper breeding season), then crash back — the logistic equation describes the pull toward K, not an unbreakable wall.
• "r is the birth rate." r is births minus deaths per individual. High births with high deaths can still give r near zero.

Real-World Applications

• Fisheries management — logistic models set sustainable catch quotas so fish stocks stay near productive levels.
• Epidemiology — early outbreak case counts follow exponential growth; r is estimated from the doubling time.
• Conservation — logistic fits guide how many animals a reserve can support before reintroduction harms the habitat.

Exponential growth captures the "multiply freely" phase; logistic growth adds the reality of limits through the carrying capacity K. Together they are the starting vocabulary of population biology. Whenever you see an S-shaped curve in nature — from yeast to cities — the logistic equation is probably behind it.