Expanded Discrete and Continuous Bĕlehrádek/Ratkowski Models of Microbial Growth Rates Under Oscillating and Extreme Storage Temperatures

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.