Logistic Population Growth

From exponential to logistic

Start with Malthus. A population of N individuals reproduces at per-capita rate r, so the total rate of new births minus deaths is rN. That gives:

dN/dt = rN

Solve and you get exponential growth — N(t) = N₀ e^rt. It works for the first few generations of bacteria in a flask, the first weeks of an algal bloom, the first decade of an introduced species. Then it always fails. Resources run out, predators numerically respond, disease spreads.

Verhulst's correction is to multiply rN by a brake term that closes as N grows:

dN/dt = rN · (1 − N/K)

When N ≪ K the brake is near 1 and growth is essentially exponential. When N = K the brake is exactly 0 and growth halts. When N > K the brake goes negative and the population shrinks back. K is therefore a stable equilibrium and zero is an unstable one.

The closed-form solution

The logistic ODE separates and integrates. Define the dimensionless variable u = K/N − 1; then du/dt = −ru and u(t) = u₀ e^−rt. Back-substituting gives:

N(t) = K / (1 + ((K − N₀)/N₀) · e^−rt)

This is the S-curve. At t = 0, N = N₀. As t → ∞, the exponential decays to zero and N → K. The midpoint t* where N = K/2 is t* = (1/r) ln((K − N₀)/N₀) — the inflection of the curve.

Worked example

Suppose a herd of 20 mountain goats is introduced to a 50 km² ridge. From related populations r ≈ 0.2 per year, K ≈ 250 goats. Plug into the closed form:

• t = 0: N = 20.
• t = 5 years: 20 → 50 (ramp phase).
• t = 10 years: ≈ 109 — past inflection at K/2 = 125 not yet reached.
• t = 15 years: ≈ 180.
• t = 25 years: ≈ 235 — within 6 percent of K.
• t = 50 years: ≈ 250 — indistinguishable from K.

The maximum yearly increment occurs around year 12–13 when the herd hits 125 — the inflection. After that, the brake kicks in and yearly gains shrink. Real goat populations would scatter around this curve year-to-year because of weather, but the average trajectory matches the logistic well over decade timescales.

Maximum sustainable yield

Fisheries and wildlife management use the logistic to compute the largest harvest a population can absorb indefinitely. If you remove H individuals per year, the modified equation is:

dN/dt = rN(1 − N/K) − H

The population has zero net growth when harvest equals natural increase: H = rN(1 − N/K). The right side is a parabola in N, peaking at N = K/2 with maximum value H_MSY = rK/4.

That is the textbook MSY result. Set the harvest at rK/4, hold the population near K/2, and you get the largest indefinite catch. In practice, environmental noise, age structure, and economic pressure conspire to push fisheries past N = K/2 and the population spirals toward zero. The collapse of the Atlantic cod fishery off Newfoundland in 1992 is the canonical cautionary tale.

Logistic vs alternatives

	Exponential	Logistic	Allee-effect
Per-capita growth at low N	r (constant)	r (constant)	Negative below threshold A
Per-capita growth at high N	r (constant)	Approaches 0	Approaches 0
Number of equilibria	1 (unstable at 0)	2 (0 unstable, K stable)	3 (0 stable, A unstable, K stable)
Curve shape	J-curve	S-curve	S-curve with extinction trap
Recovers from low density?	Yes	Yes	Only above A
Predicts overshoot?	No (no ceiling)	No (smooth approach to K)	No
Empirical fit	Bacteria, invasions, first weeks	Yeast, Paramecium, mid-term	Cooperative breeders, schooling fish

Time-lagged versions of the logistic — dN/dt = rN(1 − N(t − τ)/K) — generate cycles or chaos for large rτ and capture lemming irruptions, larch budmoth outbreaks, and other oscillatory populations the smooth logistic cannot.

Empirical fits

• Yeast in glucose flask (Pearl, 1925). Total cell counts trace a near-perfect sigmoid; r and K both extracted within 2 percent of physiological predictions.
• Paramecium aurelia (Gause, 1934). The textbook fit. Carrying capacity scales linearly with bacteria food supply.
• Antarctic blue whale (post-whaling recovery). Annual census from 1965 to 2025 traces a logistic with r ≈ 0.07/yr, K ≈ 17,000 — though uncertainty in current K is large.
• Galápagos tortoise on Pinzón. Population reintroduction from 100 in 1965 to 1500 by 2020 lies near a logistic curve with K ≈ 2000, r ≈ 0.06/yr.
• Failed fits — vole cycles, snowshoe hare cycles, lynx cycles. Multi-year oscillations the constant-K logistic cannot reproduce. Time-lagged or predator-prey models needed.

Diagram sketch

• Panel A. N(t) on linear time axis: starts low, S-curves up, asymptotes at K. Mark the inflection at K/2 and the value t* = (1/r) ln((K − N₀)/N₀).
• Panel B. dN/dt vs N: parabola with zeros at 0 and K, peak at N = K/2 of height rK/4. The peak is the MSY point.
• Panel C. Phase portrait on the N axis: arrows pointing right (growth) for N < K, arrows pointing left for N > K, fixed points at 0 (unstable) and K (stable).

Pitfalls

• Constant K assumption. Real K shifts with weather and habitat dynamics. The logistic curve smoothly converges; real populations oscillate around a moving K.
• Ignores age structure. A juvenile-heavy population at N = K is poised for a baby boom; an old-adult-heavy population at N = K is poised to crash. Leslie matrix or McKendrick-von Foerster models needed when demographics matter.
• Assumes well-mixed population. Spatial heterogeneity and dispersal limitations break the global density signal. Patches at K coexist with empty patches that have not been colonized yet.
• No time lag in density dependence. Real populations respond to crowding with a delay — by the time competition lowers reproduction, the population may already be far above K. Time lags drive cycles and chaos.
• Continuous time approximation. Many populations breed in discrete pulses. The discrete-time analogue, the logistic map N_t+1 = rN_t(1 − N_t/K), produces chaos for r > 3 and is famously sensitive to initial conditions.
• r and K not independent in reality. Mathematical r and K are independent parameters, but biologically they often trade off — r-selected vs K-selected life histories.

Variants

• Allee effect. Adds (N/A − 1) factor producing extinction below threshold A. Critical for endangered species and some marine fish stocks.
• Theta-logistic. dN/dt = rN(1 − (N/K)^θ). Tunes the curvature near K independently of r and K. θ > 1 gives sharper braking; θ < 1 gentler.
• Stochastic logistic. Adds white-noise environmental forcing. Predicts variance proportional to K and quasi-stationary distributions for finite populations.
• Discrete logistic map. Cousin equation in discrete time. Generates the period-doubling route to chaos studied by Robert May (1976) — a classical example of complex behaviour from a simple recurrence.
• Logistic with harvest. dN/dt = rN(1 − N/K) − H. Foundation of fisheries MSY theory; H = rK/4 is the upper bound.