31:Te70,{ // Immigration I(S) = m * (1 - S/P)^α // Extinction E(S) = e * (S/P)^β const immigration = (S, m, P, alpha) => m * Math.pow(1 - S / P, alpha); const extinction = (S, e, P, beta) => e * Math.pow(S / P, beta); /* Scenarios (Near vs Far; Small vs Large) */ const scenarios = [ { Isolation: "Near", Size: "Small", m: I0 * 1.0, e: E0 * 1.0 }, { Isolation: "Far", Size: "Small", m: I0 * farFactor, e: E0 * 1.0 }, { Isolation: "Near", Size: "Large", m: I0 * 1.0, e: E0 * largeFactor }, { Isolation: "Far", Size: "Large", m: I0 * farFactor, e: E0 * largeFactor } ]; /* Numerical equilibrium S*: Solve f(s) = m*(1 - s)^α - e*s^β = 0 for s in [0,1], then S* = P*s. We use a robust bisection with a fallback mini-grid if endpoints don't bracket. */ function equilibriumSstar(P, m, e, alpha, beta) { const f = (s) => m * Math.pow(1 - s, alpha) - e * Math.pow(s, beta); let a = 0, b = 1, fa = f(a), fb = f(b); // If not bracketed, pick the closest point from a small grid as a proxy if (fa * fb > 0) { let bestS = 0, bestVal = Math.abs(fa); for (let i = 1; i <= 400; i++) { const s = i / 400; const val = Math.abs(f(s)); if (val < bestVal) { bestVal = val; bestS = s; } } const Sstar = P * bestS; const Tstar = immigration(Sstar, m, P, alpha); // equals extinction at eq. return { Sstar, Tstar }; } // Bisection for (let i = 0; i < 80; i++) { const c = 0.5 * (a + b); const fc = f(c); if (Math.abs(fc) < 1e-9) { const Sstar = P * c; const Tstar = immigration(Sstar, m, P, alpha); return { Sstar, Tstar }; } if (fa * fc < 0) { b = c; fb = fc; } else { a = c; fa = fc; } } const s = 0.5 * (a + b); const Sstar = P * s; const Tstar = immigration(Sstar, m, P, alpha); return { Sstar, Tstar }; } /* Lines & equilibria */ const S_values = d3.range(0, P + 1, Math.max(1, Math.floor(P / 200))); const lineData = scenarios.flatMap(({ Isolation, Size, m, e }) => S_values.flatMap((S) => [ { Isolation, Size, kind: "Immigration", S, rate: immigration(S, m, P, alpha) }, { Isolation, Size, kind: "Extinction", S, rate: extinction(S, e, P, beta) } ]) ); const equilibria = scenarios.map(({ Isolation, Size, m, e }) => { const { Sstar, Tstar } = equilibriumSstar(P, m, e, alpha, beta); return { Isolation, Size, Sstar, Tstar, label: `S*=${Sstar.toFixed(1)}, T*=${Tstar.toFixed(2)}` }; }); /* ============ Plot ============ */ return Plot.plot({ width: 950, height: 600, marginLeft: 60, marginBottom: 50, grid: true, facet: { data: lineData, x: "Isolation", y: "Size", marginX: 80, marginY: 60 }, x: { label: "Number of species on island (S)", domain: [0, P] }, y: { label: "Rate (per unit time)" }, color: { legend: true, domain: ["Immigration", "Extinction"] }, marks: [ Plot.lineY(lineData, { x: "S", y: "rate", stroke: "kind", strokeWidth: 2 }), Plot.ruleX(equilibria, { x: "Sstar", stroke: "#888", strokeDasharray: "4,3" }), Plot.dot(equilibria, { x: "Sstar", y: "Tstar", r: 4, fill: "black" }), Plot.text( equilibria.map((d) => ({ ...d, y: d.Tstar })), { x: "Sstar", y: "y", text: "label", dy: -8, textAnchor: "middle" } ) ] }); }32:[] 2f:{"title":"Species-Area relationships","cells":[{"id":0,"value":"# Species-Area relationships","pinned":false,"hidden":false,"mode":"md"},{"id":37,"value":"In general there are more species present on larger islands (or area) than smaller ones - however, this relationship is not linear. The increase in species richness (S) is larger on a smaller island compared to a larger island. \n\nS = cA^z\n\nWe can interpret the key parameters as follows:\n\n- c (intercept / scaling): Sets the baseline richness. If your area unit is km², then c is the expected richness at 1 km². Larger c shifts the line up.\n\n- z (scaling exponent): Controls how fast richness increases with area. On the log–log plot, the line’s slope is z.\n\n- Small z → richness increases slowly with area\n\n- Large z → richness increases quickly with area\n\nTypical empirical z values often fall somewhere around 0.1–0.4 for contiguous habitats; island systems can be higher. Tweak the parameters below to see how they impact the graph.","pinned":false,"hidden":false,"mode":"md"},{"id":3,"value":"viewof c = Inputs.range([0.01, 100], {\n label: \"c (richness at A=1)\",\n step: 0.01,\n value: 10\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":8,"value":"viewof z = Inputs.range([0.001, 1], {\n label: \"z (scaling exponent)\",\n step: 0.01,\n value: 0.25\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":26,"value":"Plot.plot({\n width: 800,\n height: 420,\n marginLeft: 60,\n marginBottom: 56,\n grid: true,\n x: { label: \"Area A (e.g., km²)\" },\n y: { label: \"Species richness S\" },\n marks: [\n Plot.line(series, { x: \"A\", y: \"S\", strokeWidth: 2 }),\n // a label with the current equation\n Plot.text(\n [\n {\n x: series[series.length - 1].A,\n y: series[series.length - 1].S,\n label: `S = ${c.toFixed(2)}·A^${z.toFixed(2)}`\n }\n ],\n { x: \"x\", y: \"y\", text: \"label\", dx: -8, dy: -8, textAnchor: \"end\" }\n )\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":52,"value":"## Species-Isolation Relationships","pinned":false,"hidden":false,"mode":"md"},{"id":54,"value":"Similarly to species-area relationships, we can look at how isolation from a source impacts species richness, S(d), on an island. The shape of the species-isolation curve should be a function of dispersal for the potential source colonist pool.\n\nS(d) = S₀e^−kd\n\nParameter explanations:\n\n- S₀: maximum potential richness at the source (distance = 0).\n\n- k: isolation decay rate.\n\n- Small k → richness declines slowly with distance (species disperse well).\n\n- Large k → richness declines steeply with distance (dispersal limitation strong).\n\n- d: isolation distance (to mainland, or between patches).","pinned":false,"hidden":false,"mode":"md"},{"id":58,"value":"viewof S0 = Inputs.range([1, 200], {\n label: \"S₀ (richness at d = 0)\",\n step: 1,\n value: 100\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":56,"value":"viewof k = Inputs.range([0.001, 0.5], {\n label: \"k (decay rate with distance)\",\n step: 0.001,\n value: 0.05\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":80,"value":"Plot.plot({\n width: 800,\n height: 420,\n marginLeft: 60,\n marginBottom: 56,\n grid: true,\n x: { label: \"Distance d (km)\" },\n y: { label: \"Species richness S\" },\n marks: [\n Plot.line(sirSeries, { x: \"d\", y: \"S\", strokeWidth: 2 }),\n // show the equation as a label\n Plot.text(\n [\n {\n x: dmax,\n y: sirSeries[sirSeries.length - 1].S,\n label: `S = ${S0}·e^(-${k.toFixed(3)}·d)`\n }\n ],\n { x: \"x\", y: \"y\", text: \"label\", dx: -8, dy: -8, textAnchor: \"end\" }\n )\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":96,"value":"## The full MacArthur/Wilson model","pinned":false,"hidden":false,"mode":"md"},{"id":142,"value":"Immigration starts high as there is room for many colonists (see the blue line always starts top left). Extinction (yellow line) always starts low as there are no species, and gradually increases as the island is colonised. There is an equilibrium where these two lines cross and the species turnover is the common rate at that point (black dot).\n\n### Island size\n\nSmaller islands have lower equilibrium diversity and higher equilibrium turnover, along with higher extinction rates (steeper yellow lines).\n\n### Island Isolation\n\nNear islands have more species, with higher equilibrium rates of turnover. Immigration rates are higher and steeper for a near island (blue lines) and lower/shallower for a distant island.","pinned":false,"hidden":false,"mode":"md"},{"id":97,"value":"// Mainland species pool (max possible richness)\nviewof P = Inputs.range([20, 1000], {label: \"Mainland pool P\", value: 200, step: 10})","pinned":false,"hidden":false,"mode":"ojs"},{"id":101,"value":"// Base max immigration (at S=0) before isolation multiplier\nviewof I0 = Inputs.range([0.2, 5], {\n label: \"Base immigration I₀\",\n value: 1.0,\n step: 0.1\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":104,"value":"// Base max extinction (at S=P) before size multiplier\nviewof E0 = Inputs.range([0.2, 5], {\n label: \"Base extinction E₀\",\n value: 1.0,\n step: 0.1\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":107,"value":"// Isolation multipliers: Near = 1.0 (reference), Far < 1 (reduces immigration)\nviewof farFactor = Inputs.range([0.2, 1], {\n label: \"Far isolation multiplier (on I₀)\",\n value: 0.3,\n step: 0.05\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":109,"value":"// Size multipliers: Small = 1.0 (reference), Large < 1 (reduces extinction)\nviewof largeFactor = Inputs.range([0.2, 1], {\n label: \"Large island multiplier (on E₀)\",\n value: 0.3,\n step: 0.05\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":134,"value":"Plot.plot({\n width: 950,\n height: 600,\n marginLeft: 60,\n marginBottom: 50,\n grid: true,\n facet: {\n data: lineData,\n x: \"Isolation\",\n y: \"Size\",\n marginX: 80,\n marginY: 60\n },\n x: {\n label: \"Number of species on island (S)\",\n domain: [0, P]\n },\n y: {\n label: \"Rate (per unit time)\"\n },\n color: {\n legend: true,\n domain: [\"Immigration\", \"Extinction\"]\n },\n marks: [\n // Immigration & Extinction curves\n Plot.lineY(lineData, { x: \"S\", y: \"rate\", stroke: \"kind\", strokeWidth: 2 }),\n // Vertical line at S*\n Plot.ruleX(equilibria, {\n x: \"Sstar\",\n stroke: \"#888\",\n strokeDasharray: \"4,3\"\n }),\n // Equilibrium point\n Plot.dot(equilibria, { x: \"Sstar\", y: \"Tstar\", r: 4, fill: \"black\" }),\n // Label at the equilibrium\n Plot.text(\n equilibria.map((d) => ({ ...d, y: d.Tstar })),\n { x: \"Sstar\", y: \"y\", text: \"label\", dy: -8, textAnchor: \"middle\" }\n )\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":152,"value":"## A more complex model with curves","pinned":false,"hidden":false,"mode":"md"},{"id":161,"value":"Interpreted exactly as above, this is the plot most often seen in textbooks etc.","pinned":false,"hidden":false,"mode":"md"},{"id":153,"value":"viewof alpha = Inputs.range([0.5, 3], {label: \"α (immigration curvature)\", value: 1.5, step: 0.1});","pinned":false,"hidden":false,"mode":"ojs"},{"id":157,"value":"viewof beta = Inputs.range([0.5, 3], {label: \"β (extinction curvature)\", value: 1.5, step: 0.1});","pinned":false,"hidden":false,"mode":"ojs"},{"id":158,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":113,"value":"### Full model data/functions","pinned":false,"hidden":false,"mode":"md"},{"id":114,"value":"immigration = (S, m, P) => m * (1 - S / P)","pinned":false,"hidden":false,"mode":"ojs"},{"id":119,"value":"extinction = (S, e, P) => e * (S / P)","pinned":false,"hidden":false,"mode":"ojs"},{"id":121,"value":"// Scenarios (Near vs Far; Small vs Large)\nscenarios = [\n { Isolation: \"Near\", Size: \"Small\", m: I0 * 1.0, e: E0 * 1.0 },\n { Isolation: \"Far\", Size: \"Small\", m: I0 * farFactor, e: E0 * 1.0 },\n { Isolation: \"Near\", Size: \"Large\", m: I0 * 1.0, e: E0 * largeFactor },\n { Isolation: \"Far\", Size: \"Large\", m: I0 * farFactor, e: E0 * largeFactor }\n]","pinned":false,"hidden":false,"mode":"ojs"},{"id":123,"value":"S_values = d3.range(0, P + 1, Math.max(1, Math.floor(P / 200)))","pinned":false,"hidden":false,"mode":"ojs"},{"id":125,"value":"// ~200 points per line\nlineData = scenarios.flatMap(({ Isolation, Size, m, e }) =>\n S_values.flatMap((S) => [\n { Isolation, Size, kind: \"Immigration\", S, rate: immigration(S, m, P) },\n { Isolation, Size, kind: \"Extinction\", S, rate: extinction(S, e, P) }\n ])\n)","pinned":false,"hidden":false,"mode":"ojs"},{"id":132,"value":"equilibria = scenarios.map(({ Isolation, Size, m, e }) => {\n const Sstar = (P * m) / (m + e);\n const Tstar = (m * e) / (m + e);\n return {\n Isolation,\n Size,\n Sstar,\n Tstar,\n label: `S*=${Sstar.toFixed(1)}, T*=${Tstar.toFixed(2)}`\n };\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":62,"value":"### Species-Isolation data/functions","pinned":false,"hidden":false,"mode":"md"},{"id":64,"value":"speciesIsolation = (d, S0, k) => S0 * Math.exp(-k * d);","pinned":false,"hidden":false,"mode":"ojs"},{"id":68,"value":"dmax = 500","pinned":false,"hidden":false,"mode":"ojs"},{"id":70,"value":"dRange = d3.range(0, dmax + 1, 1);","pinned":false,"hidden":false,"mode":"ojs"},{"id":72,"value":"sirSeries = dRange.map(d => ({ d, S: speciesIsolation(d, S0, k) }));","pinned":false,"hidden":false,"mode":"ojs"},{"id":44,"value":"### Species-Area data/functions","pinned":false,"hidden":false,"mode":"md"},{"id":10,"value":"speciesArea = (A, c, z) => c * Math.pow(A, z)","pinned":false,"hidden":false,"mode":"ojs"},{"id":14,"value":"minExp = 0","pinned":false,"hidden":false,"mode":"ojs"},{"id":12,"value":"maxExp = 6;","pinned":false,"hidden":false,"mode":"ojs"},{"id":16,"value":"stepExp = 0.01 // smoothness of the line","pinned":false,"hidden":false,"mode":"ojs"},{"id":20,"value":"areaRange = d3.range(minExp, maxExp + 1e-9, stepExp).map((e) => 10 ** e)","pinned":false,"hidden":false,"mode":"ojs"},{"id":22,"value":"series = areaRange.map((A) => ({ A, S: speciesArea(A, c, z) }))","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W32","stdlib":"1","titling":"auto"}