Mathematical elucidation of the Kaufmann effect based on the thermodynamic SSI model

Abstract

The development rate of insects at hourly fluctuating temperatures is not infrequently different from that at constant temperatures even when the averages of these temperatures are the same. This temperature-dependent development phenomenon has been known as the Kaufmann effect. However, its theoretical analysis has not yet been successfully carried out owing to the insufficiency of mathematical insight especially into quantitative expressions. In insect development, the interrelationships among the three environmental temperatures, namely, the constant and alternating temperatures controlled in the laboratory and the hourly fluctuating temperatures in the natural environment, have not been clarified. Here, we completely succeeded in analyzing this phenomenon and in elucidating the interrelationships by introducing the components of the nonlinear SSI development model, the second derivative, the cosine-wave model of hourly fluctuating temperatures and their variance, and Taylor series. As a result, it has been possible to predict the development rate at fluctuating temperatures in the natural environment using prospective daily maximum, minimum and average temperatures and the development rate at constant temperatures without conducting experiments at alternating temperatures.