7.9 Logistic Models with Differential Equations

The logistic differential equation

The logistic model describes growth limited by resources: dP/dt = kP(1 − P/L), where L is the carrying capacity. When P is small, growth is nearly exponential; as P approaches L, growth slows toward zero. This models populations, spread of disease, and other bounded growth phenomena.

Key features of logistic growth

Growth rate dP/dt is maximized when P = L/2 (the inflection point of the S-shaped curve). As t → ∞, P approaches L regardless of starting value (as long as P > 0). If P > L, the population decreases toward L; if 0 < P < L, it increases toward L.

Solving the logistic equation

The general solution is P(t) = L / (1 + Ae^(−kLt)), where A depends on the initial condition. On the exam, you often analyze behavior (equilibrium solutions, max growth rate) using the differential equation directly rather than solving explicitly, since separation of variables here involves partial fractions.

Key terms

Logistic differential equation

dP/dt = kP(1 − P/L), where L is the carrying capacity and k is the growth constant.

Carrying capacity (L)

The maximum sustainable population value; P(t) approaches L as t → ∞.

Where is growth rate maximized?

At P = L/2, the inflection point of the logistic curve.

Equilibrium solutions of logistic equation

P = 0 and P = L (where dP/dt = 0).

General solution form

P(t) = L / (1 + Ae^(−kLt)), with A determined by the initial condition.

Try an AP-style question

For dP/dt = 0.2P(1 − P/500), what is the carrying capacity?

1. 0.2
2. 250
3. 100
4. 500