Logistic equation

Example

Let $r > 0$ and $K > 0$ be constants. The logistic equation is $$P'(t) = r\, P(t) \left(1 - \frac{P(t)}{K}\right).$$ For an initial value $P(0) = P_0 > 0$, its solution is $$P(t) = \frac{K}{1 + \left(\frac{K - P_0}{P_0}\right) e^{-r t}}, \qquad t \ge 0.$$ With $r = K = 1$, the function $f(t) = 1 / \left(1 + e^{-t}\right)$ is the standard logistic function. It satisfies $f' = f(1 - f)$ and $f(0) = 1/2$.

Remarks

The constant functions $P = 0$ and $P = K$ are the equilibrium points. For each $P_0 > 0$, $P(t) \to K$ as $t \to \infty$. For small $P$, the equation is close to $P' = r P$, which is exponential growth. If $0 < P_0 < K/2$, the graph of $P$ is an S-shaped curve with its inflection point at the value $P = K/2$. The equation is separable, and it is a Bernoulli equation with exponent $2$.

Explanations

L1 human

The logistic equation is a model of a population with limited resources. When the population is small, it grows almost exponentially. When it is close to the capacity $K$, the growth slows down and stops. The constant $K$ is the carrying capacity: the largest population that the environment can support.