Organisms exhibiting phenological events in early spring often experience several cold days
during their development. In this case, Arrhenius' equation (Shi et al., 2017a, b,
and references therein) has been recommended to describe the effect of the absolute temperature
(\(T\) in Kelvin [K]) on the developmental rate (\(r\)):
$$r = \mathrm{exp}\left(B - \frac{E_{a}}{R\,T}\right),$$
where \(E_{a}\) represents the activation free energy (in kcal \(\cdot\) mol\({}^{-1}\));
\(R\) is the universal gas constant (= 1.987 cal \(\cdot\) mol\({}^{-1}\) \(\cdot\) K\({}^{-1}\));
\(B\) is a constant. To maintain consistency between the units used for \(E_{a}\) and \(R\), we need to
re-assign \(R\) to be 1.987\(\times {10}^{-3}\), making its unit 1.987\(\times {10}^{-3}\)
kcal \(\cdot\) mol\({}^{-1}\) \(\cdot\) K\({}^{-1}\) in the above formula.
\(\qquad\)In the accumulated developmental progress (ADP) method, when the annual accumulated developmental
progress (AADP) reaches 100%, the phenological event is predicted to occur for each year.
Let \(\mathrm{AADP}_{i}\) denote the AADP of the \(i\)th year, which equals
$$\mathrm{AADP}_{i} = \sum_{j=S}^{E_{i}}\sum_{w=1}^{24}\frac{r_{ijw}}{24},$$
where \(E_{i}\) represents the ending date (in day-of-year), i.e., the occurrence time of a pariticular
phenological event in the \(i\)th year. \(r_{ijw}\) is the developmental rate (per hour), which is
transferred to \(r_{ij}\) (per day) by dividing 24. If the temperature-dependent developmental rate follows
Arrhenius' equation, the AADP of the \(i\)th year is equal to
$$\mathrm{AADP}_{i} = \sum_{j=S}^{E_{i}}\sum_{w=1}^{24}\left\{\frac{1}{24}\mathrm{exp}\left(B - \frac{E_{a}}{R\,T_{ijw}}\right)\right\},$$
where \(T_{ijw}\) represents the estimated mean hourly temperature of the \(w\)th hour of the
\(j\)th day of the \(i\)th year (in K). This packages takes the method proposed by
Zohner et al. (2020) to estimate the mean hourly temperature (\(T_{w}\)) for each of 24 hours:
$$T_{w} = \frac{T_{\mathrm{max}} - T_{\mathrm{min}}}{2}\, \mathrm{sin}\left(\frac{w\pi}{12}-
\frac{\pi}{2}\right)+\frac{T_{\mathrm{max}} + T_{\mathrm{min}}}{2},$$
where \(w\) represents the \(w\)th hour of a day, and \(T_{\mathrm{min}}\) and \(T_{\mathrm{max}}\)
represent the minimum and maximum temperatures of the day, respectively.
\(\qquad\)In theory, \(\mathrm{AADP}_{i} = 100\%\),
i.e., the AADP values of different years are a constant 100%. However, in practice, there is
a certain deviation of \(\mathrm{AADP}_{i}\) from 100%. The following approach
is used to determine the predicted occurrence time.
When \(\sum_{j=S}^{F}\sum_{w=1}^{24}\left(r_{ijw}/24\right) = 100\%\) (where \(F \geq S\)), it follows that \(F\) is
the predicted occurrence time; when \(\sum_{j=S}^{F}\sum_{w=1}^{24}\left(r_{ijw}/24\right) < 100\%\) and
\(\sum_{j=S}^{F+1}\sum_{w=1}^{24}\left(r_{ijw}/24\right) > 100\%\), the trapezoid method (Ring and Harris, 1983)
is used to determine the predicted occurrence time.
\(\qquad\)The argument of expr can be any an arbitrary user-defined temperature-dependent
developmental rate function, e.g., a function named myfun,
but it needs to take the form of myfun <- function(P, x){...},
where P is the vector of the model parameter(s), and x is the vector of the
predictor variable, i.e., the temperature variable.