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.

Appendix: derivation of the SSI model for fluctuating temperatures

In the text, the SSI model for fluctuating temperatures is derived employing that for constant temperatures. The model equation provides the expected value of a function, the variable of which is random.

Let f be a smooth function. The Taylor series of f(x) at point x ₀ is given by the form

$$ f(x)\,\; = \,f(x_{0} )\; + \;f^{\prime } (x_{0} )(x - x_{0} )\; + \;\frac{{\,f^{\prime \prime } (x_{0} )\;}}{2!}(x - x_{0} )^{2} + \;\frac{{\;f^{\prime \prime \prime } (x_{0} )\;}}{{3{\kern 1pt} \,!}}(x - x_{0} )^{3} + \; \cdots. $$

(24)

By ignoring the third and higher order terms, it follows that in the neighborhood of x ₀,

$$ f(x)\,\; \approx \,f(x_{0} )\; + \;f^{\prime } (x_{0} )(x - x_{0} )\; + \;\frac{{\,f^{\prime \prime } (x_{0} )\;}}{2!}(x - x_{0} )^{2} . $$

(25)

Let X be a random variable and let us denote E[X] with the expected value of X, as the variance V [X]. Using the above argument, in the neighborhood of E[X], f(X) can be approximately written by

$$ f(X)\,\; = \,f(E\,[X])\; + \;f^{\prime } (E\,[X])(X - E\,[X])\; + \;\frac{{\,f^{\prime \prime } (E\,[X])\;}}{2!}(X - E\,[X])^{2} . $$

(26)

Note that f (X) is a random variable. Taking the expected values of both sides of Eq. 26, we find

$$ \begin{gathered} E\,[f(X)]\,\; = \,\;E\,[f(E\,[X])]\; + \;E\,[f'(E\,[X])(X - E[X])]\; + \;E\,[\frac{{\,f^{\prime \prime } (E{\kern 1pt} [X])\;}}{2!}(X - E[X])^{2} ] \hfill \\ \quad \quad \quad \quad = f(E\,[X])\; + \;f^{\prime } (E\,[X])E\,[X - E\,[X]]\; + \;\frac{{\;f^{\prime \prime } (E\,[X]\;)}}{2!}E\,[(X - E\,[X])^{2} ] \hfill \\ \quad \quad \quad \quad = f(E\,[X]) + f^{\prime } (E\,[X])(E\,[X] - E\,[X]) + \frac{{f^{\prime \prime } (E\,[X])}}{2!}E\,[(X - E\,[X])^{2} ] \hfill \\ \quad \quad \quad \quad = f(E\,[X])\; + \;\frac{1}{2}f^{\prime \prime } (E\,[X])\,V\,[X]. \hfill \\ \end{gathered} $$

(27)

If V[X] is invariant for any X, E[f(X)] is expressed as a function of E[X] by Eq. 27.

Let us denote T _f as fluctuating temperatures; T _f is a random variable. Write T _Af = E[T _f]. In the SSI model for fluctuating temperatures, r _f(T _Af) is obtained by replacing f and X of Eq. (27) with r _c and T _f, respectively, that is,

$$ r_{\text{f}} (T_{\text{Af}} )\,\; = \,\;r_{\text{c}} (T_{\text{Af}} )\; + \;\frac{1}{2}\,r_{\text{c}}^{\prime \prime } (T_{\text{Af}} )\,\,V[T_{\text{f}} ]. $$

(28)

Jensen’s inequality

In order to explain the relationship between the graphs of the constant temperatures model r _c(T _c) and the fluctuating temperatures model r _f(T _Af), we introduce the following theorem known as “Jensen’s inequality.” For details, refer to Feller (1966).

Theorem

Let X be a random variable. If f is a convex function, then

$$ E\left[ {f\left( X \right)} \right] \, \ge f\left( {E\left[ X \right]} \right). $$

If f is a concave function, then

$$ E\left[ {f\left( X \right)} \right] \, \le f\left( {E\left[ X \right]} \right). $$

When the two graphs are plotted using the common horizontal axis T, “Jensen’s inequality” implies that r _f(T) ≥ r _c(T) for T to satisfy r ^″_c (T) > 0; r _f(T) ≤ r _c(T) for r ^″_c (T) < 0.