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  "body": "\n## Definition\n\nLet $D \u003e 0$ and $r \u003e 0$ be constants.\n*Fisher's equation* is the [[reaction–diffusion equation|TAAHWE917M]] for a function $u(x, t)$ of $x \\in \\R$ and $t \u003e 0$:\n$$u_t = D\\, u_{xx} + r\\, u (1 - u).$$\nHere $u_t$ and $u_{xx}$ are [[partial derivatives|09EGC6F8PT]].\nThe same name is used for $u_t = D \\Delta u + r u(1 - u)$ with $x \\in \\R^n$.\n\n## Remarks\n\nThe constants $u \\equiv 0$ and $u \\equiv 1$ are solutions: both sides vanish.\nThe state $0$ is unstable and the state $1$ is stable for the reaction $u' = r u(1 - u)$.\nA [[travelling wave|M37M411KP7]] is a solution $u(x, t) = U(x - ct)$ with $U(-\\infty) = 1$ and $U(+\\infty) = 0$.\nSuch a wave with $0 \\le U \\le 1$ exists if and only if $c \\ge 2\\sqrt{rD}$.\nA non-example is the heat equation $u_t = D u_{xx}$: it has no reaction term, so it is not Fisher's equation for any $r \u003e 0$.\nThe rescaling $x \\mapsto x\\sqrt{r/D}$, $t \\mapsto r t$ reduces the equation to $u_t = u_{xx} + u(1 - u)$.\n",
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