{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $m \\ge 2$, let $D$ be a diagonal $m \\times m$ matrix with positive diagonal entries, and let $f \\colon \\R^m \\to \\R^m$ be smooth.\nConsider the [[reaction–diffusion system|TAAHWE917M]] $\\partial_t u = D \\Delta u + f(u)$ for $u \\colon \\R^n \\times [0, \\infty) \\to \\R^m$, where $\\Delta$ is the [[Laplacian|0E0B6RA8K0]].\nLet $u^* \\in \\R^m$ satisfy $f(u^*) = 0$ and let $J$ be the [[Jacobian matrix|047NDTEPCV]] of $f$ at $u^*$.\nThe uniform state $u^*$ is *Turing unstable*, or has a *diffusion-driven instability*, if:\n\n1. every [[eigenvalue|0BQAA2BK3Q]] of $J$ has negative real part, so $u^*$ is stable without diffusion;\n2. there is $k \u003e 0$ such that $J - k^2 D$ has an eigenvalue with positive real part.\n\nA *Turing pattern* is a stationary, spatially non-constant solution near $u^*$ that bifurcates from a Turing unstable uniform state.\nIt is typically periodic, with a wave number $k$ as in clause 2.\n\n## Remarks\n\nThe perturbation $u = u^* + e^{\\lambda t} e^{i q \\cdot x} w$ with a wave vector $q \\in \\R^n$, $\\abs{q} = k$, solves the linearised system exactly when $\\lambda$ is an eigenvalue of $J - k^2 D$ with eigenvector $w$, which explains clause 2.\nExample: $m = 2$, $J = \\begin{pmatrix} 1 \u0026 -1 \\\\ 3 \u0026 -2 \\end{pmatrix}$, $D = \\operatorname{diag}(1, 10)$.\nThen $\\tr J = -1 \u003c 0$ and $\\det J = 1 \u003e 0$, so clause 1 holds.\nFor $k^2 = 0.3$, $\\det(J - k^2 D) = (0.7)(-5) + 3 = -0.5 \u003c 0$, so $J - k^2 D$ has a positive real eigenvalue and clause 2 holds.\nNon-example: if $D$ is a multiple $d I$ of the identity, the eigenvalues of $J - k^2 D$ are those of $J$ shifted by $-d k^2$, so clause 2 fails whenever clause 1 holds.\nFor $m = 2$, a necessary condition is that the two diffusion coefficients differ.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "RM94KBV8P7",
    "kind": "definition",
    "msc": [
      "35B36",
      "35K57",
      "92C15"
    ],
    "name": "Turing pattern",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Turing_pattern"
      }
    ],
    "tag": "KSA55H8Q8N",
    "uses": [
      "TAAHWE917M",
      "047NDTEPCV",
      "0BQAA2BK3Q",
      "0E0B6RA8K0"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/KSA55H8Q8N/",
  "links": [
    {
      "field": "defines",
      "target": "RM94KBV8P7"
    },
    {
      "field": "uses",
      "target": "TAAHWE917M"
    },
    {
      "field": "uses",
      "target": "047NDTEPCV"
    },
    {
      "field": "uses",
      "target": "0BQAA2BK3Q"
    },
    {
      "field": "uses",
      "target": "0E0B6RA8K0"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/KSA55H8Q8N.md",
  "msc": [
    "35B36",
    "35K57",
    "92C15"
  ],
  "name": "Turing pattern",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/KSA/KSA55H8Q8N.md",
  "status": null,
  "tag": "KSA55H8Q8N",
  "uri": "m2://KSA55H8Q8N",
  "url": "https://tamnd.github.io/math2/wiki/turing-pattern-definition/",
  "writing": {
    "tag": "KSA55H8Q8N",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}