Logistic Growth and the Limits of Exponentials

You’ll be able to

Describe the shape of a logistic curve and identify its carrying capacity
Explain why exponential models fail for bounded populations
Locate the point of fastest growth on a logistic curve and interpret it

Nothing grows exponentially for long

Exponential growth assumes the growth rate is proportional to the current amount, with nothing pushing back. Real systems always push back: food runs out, space fills, susceptible hosts get infected, the market saturates. A logistic model keeps exponential behavior while the population is small and bends it over as the population approaches a ceiling — the carrying capacity L.

Logistic model

P(t) = L / (1 + a·e^(−kt)), with carrying capacity L

As t → ∞ the term a·e^(−kt) → 0 and P → L. As t → −∞ that term dominates and P → 0. So the curve runs from 0 up to L.

The S-shape, read as concavity

A logistic curve is increasing throughout, but its concavity changes exactly once. Early on, growth accelerates — each interval adds more than the last, and the curve is concave up, nearly indistinguishable from an exponential. Later, growth decelerates as the ceiling binds, and the curve is concave down, flattening toward L. The switch is the point of inflection, and it is where the population is growing fastest.

Worked example

A fish population is modeled by P(t) = 2400/(1 + 11·e^(−0.35t)), t in years. Find P(0), the carrying capacity, and the population after 10 years.

1.P(0) = 2400/(1 + 11·e⁰) = 2400/12 = 200.
2.Carrying capacity: as t → ∞, e^(−0.35t) → 0, so P → 2400/1 = 2400 fish.
3.At t = 10: e^(−3.5) ≈ 0.030197, so 11(0.030197) ≈ 0.33217.
4.P(10) = 2400/1.33217 ≈ 1801.6.

Answer: The population starts at 200, has a carrying capacity of 2,400 fish, and reaches about 1,802 after 10 years — roughly 75% of capacity. Note that the parameter 11 is exactly (L − P₀)/P₀ = (2400 − 200)/200, which is how a is determined from an initial value.

Fastest growth at half capacity

The logistic curve's point of inflection sits at P = L/2, exactly half the carrying capacity. That is the moment of maximum growth rate. Before it, growth is still accelerating; after it, growth slows steadily toward zero. The interpretation is intuitive: growth requires both plenty of individuals reproducing and plenty of room left, and the product of those two is largest halfway up.

Inflection point of a logistic curve

occurs at P = L/2, at time t = ln(a)/k

Setting L/(1 + a·e^(−kt)) = L/2 requires a·e^(−kt) = 1, giving e^(kt) = a and t = ln(a)/k.

Checkpoint

For P(t) = 2400/(1 + 11·e^(−0.35t)), when is the population growing fastest?

Watch out

Early logistic data is nearly indistinguishable from exponential data — which is why extrapolating early growth is so dangerous. Fitting an exponential to the first few points of an epidemic or a product launch will overpredict badly once the curve turns.

Choosing between exponential and logistic

Ask whether a ceiling exists and whether the data has reached enough of it to be visible. Unconstrained early growth — a new infection in a large population, a bacterial culture in fresh medium — is well served by an exponential. Anything with a known bound — a fixed habitat, a finite market, a total number of susceptible people — needs a logistic, and the data will show it by bending over. A residual plot from an exponential fit gives the tell: a run of overpredictions at the high end.

Checkpoint

An exponential model is fit to the first eight weeks of an outbreak and predicts more cases than actually occur in weeks 9 through 12. What is the most likely explanation?

Answer the 2 checkpoints as you read.