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  "body": "\n## Example\n\nLet $r \u003e 0$ and $K \u003e 0$ be constants. The logistic equation is\n$$P'(t) = r\\, P(t) \\left(1 - \\frac{P(t)}{K}\\right).$$\nFor an initial value $P(0) = P_0 \u003e 0$, its solution is\n$$P(t) = \\frac{K}{1 + \\left(\\frac{K - P_0}{P_0}\\right) e^{-r t}}, \\qquad t \\ge 0.$$\nWith $r = K = 1$, the function $f(t) = 1 / \\left(1 + e^{-t}\\right)$ is the standard logistic function. It satisfies $f' = f(1 - f)$ and $f(0) = 1/2$.\n\n## Remarks\n\nThe constant functions $P = 0$ and $P = K$ are the [[equilibrium points|0N8NGKDZDZ]]. For each $P_0 \u003e 0$, $P(t) \\to K$ as $t \\to \\infty$.\nFor small $P$, the equation is close to $P' = r P$, which is [[exponential growth|00SVP6W3CB]].\nIf $0 \u003c P_0 \u003c K/2$, the graph of $P$ is an S-shaped curve with its inflection point at the value $P = K/2$.\nThe equation is separable, and it is a [[Bernoulli equation|0T92VGGVWP]] with exponent $2$.\n\n## Explanations\n\n### L1 human\n\nThe logistic equation is a model of a population with limited resources.\nWhen the population is small, it grows almost exponentially. When it is close to the capacity $K$, the growth slows down and stops.\nThe constant $K$ is the carrying capacity: the largest population that the environment can support.\n",
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