A flexible model for thermal performance curves

FlexTPC is a mathematical model for thermal performance curves that aims to be both flexible and interpretable. We show an interactive visualization of flexTPC and its parameters below:

viewof T_min = Inputs.range(
  [-5, 20], 
  {value: 10, step: 0.2, label: "Tmin:"}
)
viewof T_max = Inputs.range(
  [30, 50], 
  {value: 35, step: 0.2, label: "Tmax:"}
)

viewof r_max = Inputs.range(
  [0, 1.2], 
  {value: 1, step: 0.1, label: "rmax:"}
)

viewof alpha = Inputs.range(
  [0, 1], 
  {value: 0.8, step: 0.02, label: "alpha:"}
)

viewof beta = Inputs.range(
  [0, 1], 
  {value: 0.3, step: 0.02, label: "beta:"}
)

function incrementalArray(start, end, step) {
    var arr = [];
    // convert count to an integer to avoid rounding errors
    var count = +((end - start) / step).toFixed();
    for (var j = 0; j <= count; j++) {
        var i = start + j * step;
        arr.push(i);
    }
    return arr;
}

function flexTPC(x, T_min, T_max, r_max, alpha, beta) {
  if (x < T_min) {
  return 0.0
  }
  if (x > T_max) {
  return 0.0
  }
  return r_max * Math.exp((alpha * (1.0 - alpha) / beta**2) * (alpha *    Math.log( (x - T_min) / alpha)  + 
      (1.0 - alpha) * Math.log( (T_max - x) / (1.0 - alpha)) -
      Math.log(T_max - T_min)))
}

xvals = incrementalArray(-5, 50, 0.1)
yvals = xvals.map((x) => flexTPC(x, T_min, T_max, r_max, alpha, beta))
zip = (a, b) => a.map((k, i) => [k, b[i]]);
data = zip(xvals, yvals)

Plot.plot({
  width: 600,
  height: 400,
  y: { domain: [-0.01, 1.25] },
  marks: [
    Plot.line(
      data,
      {
        strokeWidth: 3,
        stroke: "steelblue",
        fontSize: 14
      }
    ),
    Plot.ruleX([0]),
    Plot.ruleY([0]),
    Plot.axisX({label: "Temperature [°C]", fontSize: 14, marginBottom: 40}),
    Plot.axisY({label: "trait performance", fontSize: 14, x: 0 })
  ]
})

FlexTPC is parametrized in terms of biologically interpretable quantities:

• \(T_{\min}\) is the minimum temperature,
• \(T_{\max}\) the maximum temperature,
• \(r_{\max}\) the maximum trait value/performance of the TPC,
• \(\alpha \in (0,1)\) determines where the optimal temperature \(T_{\mathrm{opt}}\) is relative to \(T_{\min}\) and \(T_{\max}\) through the equation \[ T_{\mathrm{opt}} = \alpha T_{\max} + (1 - \alpha) T_{\min} \] (where, for example, \(\alpha = 1/2\) corresponds to a symmetric TPC where \(T_{\mathrm{opt}} = (T_{\min} + T_{\max}) / 2\)), the limit \(\alpha \rightarrow 0\) corresponds to \(T_{\mathrm{opt}} \rightarrow T_{\min}\), and the limit \(\alpha \rightarrow 1\) corresponds to \(T_{\mathrm{opt}} \rightarrow T_{\max}\), and
• \(\beta > 0\) determines the breadth of the TPC near its peak.

Code examples

We also provide example code that shows how to fit flexTPC with various methodologies.

Reference

If you use flexTPC, please cite

Cruz-Loya M, Mordecai EA, Savage VM. 2025. A flexible model for thermal performance curves. Ecology (in press) (biorXiv preprint) (data and code)