1.2 The Logistic Equation

Exponential growth cannot last. A population that doubles every hour eventually runs out of food, space, or patience, and a model that ignores this fact answers questions about the far future with nonsense. The classical fix is to make the per-capita growth rate fall as the population rises. With carrying capacity $8$ (in whatever units the habitat supports), the logistic equation is

$\frac{d P}{d t} = b P (1 - \frac{P}{8}) .$

Read it before solving it. When $P$ is small the factor $1 minus P / 8$ is close to $1$ and the equation is nearly $P^{'} = b P$ — exponential growth. When $P$ is near $8$ the factor is near $0$ and growth stalls. Between them, the growth rate $b P open parenthesis 1 minus P / 8 close parenthesis$ is largest at $P equals 4$ , halfway up. And there are two population levels where nothing changes at all: $P equals 0$ and $P equals 8$ , the equilibria, where the right-hand side vanishes and a population that starts there stays there forever.

The equation can be solved in closed form, and the solutions are

$P (t) = \frac{8}{1 + a e^{- b t}},$

where the constant $a$ encodes the starting value: $P (0) = 8 / (1 + a)$ , so large $a$ means a small starting population and $a$ near zero means starting almost at capacity.

A red S-shaped logistic curve rising from its starting value toward a dashed horizontal line at height 8, with a second dashed line along height 0. The slider a moves the starting population P(0) between about 0.4 and 7.3, and the slider b sharpens or relaxes how fast the curve climbs toward the dashed capacity line, which it approaches but never crosses. — Logistic solutions $P (t) = 8 / (1 + a e^{- b t})$ with carrying capacity $8$ (upper dashed line) and the extinction equilibrium $P equals 0$ (lower dashed line). The slider $a$ sets the starting population and $b$ sets the growth rate.

Explore

Observe. Set $a equals 7$ , $b equals 0.8$ , so $P open parenthesis 0 close parenthesis equals 1$ . The curve is S-shaped: slow, then fast, then slow. At roughly what height is the curve steepest? Compare with the halfway level $P equals 4$ .
Predict. If you increase $a$ to $20$ , the starting population drops to about $0.4$ . Will the curve still reach toward the dashed line at $8$ , or level off lower? Decide, then drag the slider.
Verify. With any settings, zoom your attention to the right edge of the window. Does the red curve ever touch or cross the dashed line at $8$ ? Explain from the equation why it cannot: what would $d P / d t$ be at the moment of touching?
Observe. Drag $a$ down to $0.1$ , so the population starts near $7.3$ , close to capacity. The S-shape disappears. Which part of the S did this starting value skip?
Predict, then verify. The equilibrium $P equals 0$ is also a solution. Judging from the curves with large $a$ , does a tiny population move toward $0$ or away from it? What does that say about the stability of the extinction equilibrium?

Connect

The two dashed lines behave differently, and the difference is the most important qualitative fact about the model. Solutions that start just above $0$ flee from it: $P equals 0$ is unstable, an equilibrium in the sense of a pencil balanced on its tip. Solutions from every positive starting value climb or descend toward $8$ and flatten against it: $P equals 8$ is stable, an equilibrium like a marble at the bottom of a bowl. You verified both facts with sliders, but they were readable from the equation alone — the sign of $b P open parenthesis 1 minus P / 8 close parenthesis$ is positive for $0 less than P less than 8$ and negative for $P greater than 8$ , so arrows on a vertical $P$ -axis point away from $0$ and toward $8$ . That sign analysis, not the closed formula, is what generalizes to equations nobody can solve.

An original work of XYZ Homework, built around interactive XYZ 3D figures, following the standard first-course sequence (first-order equations through systems and the phase plane). Distinct from the LibreTexts-sourced Differential Equations for Engineers (Lebl) edition, which has its own attribution. License: CC-BY-NC-SA-4.0.