Logistic Models

Exponential growth can't last. A population that outgrows its food or space slows down, and its size levels off at the most the environment can support. The logistic differential equation models that kind of limited growth.

The right side is $0$ when $P = 0$ and when $P = L$, so those are the equilibrium solutions. For $0 < P < L$ both factors are positive and $P$ increases. For $P > L$ the factor $L - P$ is negative and $P$ decreases. The right side and its derivative with respect to $P$ are continuous, so no solution curve crosses $P = L$. A solution's slope approaches $0$ only as $P$ approaches $0$ or $L$. So every solution with $P(0) > 0$ approaches $L$:

As a function of $P$, the rate $kP(L - P)$ is a downward-opening parabola with zeros at $0$ and $L$. Its largest value is at the vertex, $P = \frac{L}{2}$. So a population that starts between $0$ and $\frac{L}{2}$ grows fastest when it reaches half the carrying capacity. One that starts between $\frac{L}{2}$ and $L$ grows fastest at the start.

Differentiating with the chain rule gives the concavity:

For $0 < P < \frac{L}{2}$ this is positive, for $\frac{L}{2} < P < L$ it's negative, and for $P > L$ it's positive again. A solution that starts between $0$ and $\frac{L}{2}$ is concave up until it reaches $\frac{L}{2}$ and concave down after, so its graph has an inflection point where $P = \frac{L}{2}$. A solution that starts between $\frac{L}{2}$ and $L$ is concave down throughout. One that starts above $L$ decreases and is concave up.

A logistic model is often written as $\frac{dP}{dt} = rP\left(1 - \frac{P}{L}\right)$. Here $r = kL$, and dividing by $P$ shows what it means. The relative growth rate $\frac{dP/dt}{P} = r\left(1 - \frac{P}{L}\right)$ is close to $r$ when $P$ is small, and it falls to $0$ as $P$ approaches $L$.