The Bĕlehrádek/Ratkowski, also known as the square root model, was originally developed to describe and quantify the temperature dependence of organisms’ growth rate, in terms of a simple power-law expression that contains a threshold growth temperature. The model’s original version implied that the rate rises monotonically with temperature. Therefore it has been modified to account for peak growth at the organism’s optimal temperature, by adding a falling multiplying factor. The modified model has been extended to post-peak growth temperatures, and all the way to where growth turns into thermal inactivation. In principle, the model can also be extended to low temperatures where growth not only ceases, as the original model implies, but turns into mortality, albeit at a much slower pace. The performance of the original and extended model’s versions can be visualized with three freely downloadable interactive Wolfram Demonstrations available on the Internet. The three versions can be combined into a single temperature-dependence model that covers the entire temperature range from lethal cold to death by heat. This combined model can be incorporated into a general dynamic (non-isothermal) growth/mortality model, in the form of a differential rate equation, which can describe oscillations between growth, no growth, and mortality modes during a population’s thermal history. All the Bĕlehrádek/Ratkowski model’s variants can be replaced by a single continuous algebraic expression that eliminates the need to have “If statements” in their equations. The continuous version too can be incorporated into a dynamic rate model to describe growth, no growth, and mortality, and transitions between them.

Bĕlehrádek/ Ratkowski，也称为平方根模型，最初是根据包含阈值生长温度的简单幂律表达式来描述和量化生物生长速率的温度依赖性的。该模型的原始版本暗示速率随温度单调上升。因此，已对其进行了修改，以通过添加下降的乘数来解决生物体最佳温度下的峰值增长。修改后的模型已扩展到峰值后的生长温度，一直到生长变成热失活的地方。原则上，该模型还可以扩展到低温，在这种情况下，增长速度不仅慢了，正如原始模型所暗示的那样停止了，而且变成了死亡率。原始模型和扩展模型版本的性能可以通过Internet上的三个可免费下载的交互式Wolfram演示进行可视化。这三个版本可以组合成一个单一的温度依赖性模型，该模型涵盖从致死致冷到因热致死的整个温度范围。该组合模型可以以差分速率方程的形式并入一般的动态（非等温）增长/死亡率模型，该模型可以描述人口热史期间增长，无增长和死亡率模式之间的振荡。所有Bĕlehrádek/ Ratkowski模型的变体都可以由单个连续的代数表达式代替，从而消除了方程式中包含“ If语句”的需要。