Kevin Gross, Anthony R. Ives, and Erik V. Nordheim. 2005. Estimating fluctuating vital rates from time-series data: a case study of aphid biocontrol. Ecology 86:740–752.

Appendix B. Calculation of growth rates. See Appendix H for references cited.

To calculate aphid and wasp development rates as functions of temperature, we first fit Logan et al.'s (1976) model for development at constant temperatures

(B.1)

where T is the temperature in degrees Centigrade. We chose the Logan model over others (Stinner et al. 1974; Lamb 1992; Sharpe and DeMichele 1977), as it fits Lamb's (1992) extensive data on pea aphid development at constant temperatures on broad beans, and it has a mechanistic, biochemical motivation. 

For aphid development, we used the data from Hutchison and Hogg (1984) (pea aphids reared on alfalfa) and the constant temperature data from Siddiqui et al. (1973) (pea aphids reared on peas). For mummy development (rate of mummy formation for parasitized aphids), data were combined from Thiboldeaux (1986) (Aphidius ervi on Acyrthosiphon pisum), Botto et al. (1998) (Aphidius ervi on Acyrthosiphon pisum on broad bean; development rates used are the inverse of the weighted average of the development times for the two populations), Hofsvang and Hagvar (1985) (Aphidius ervi on green peach aphid, Myzus persicae), Sigsgaard (2000) (Aphidius ervi on cereal aphids), and ChristiansenWeniger and Hardie (1997) (Aphidius ervi on Acyrthosiphon pisum on tick beans). For wasp development (time from mummification to wasp emergence), the same sources were used as for mummy development, except ChristiansenWeniger and Hardie (1997). For each data set, the parameters in the Logan curve were estimated with unweighted nonlinear least squares (Fig. B1, Table B1).  For mummy development, a T_M parameter was included for consistency with the other models, even though development rates apparently do not decrease in the range of the data.

For hyperparasitoid development (time from hyperparasitism to emergence of the hyper wasp), fig. 3 of Walker and Cameron (1981) shows that hyperparasitoid development is nearly linear over the range of observed temperatures. Further, these data are not at odds with S. Schooler's anecdotal note that Dendrocerus carpenteri takes 16 days to develop (the species studied by Walker and Cameron). Schooler also notes that Asaphes lucens takes 21 days to develop, however. We used the linear curve reported in fig. 3 of Walker and Cameron.

Siddiqui et al. (1973) and Hagstrum and Milliken (1991) provide clear evidence that developmental time in insects is different at fluctuating temperatures than at constant temperatures. To derive time series of developmental rates at fluctuating temperatures, we first generated daily temperature profiles by using daily high and low temperatures from Dane Co. Regional Airport assuming that temperatures fluctuated sinusoidally. We then numerically integrated r(T) to obtain average daily development rates.

Table B1: Parameter estimates (and standard errors) for Eq. B.1.

FIG. B1. Data and nonlinear regression fits of Eq. B.1 for development times of (A) adult aphids, (B) mummies, and (C) adult wasps. Plot symbols correspond to particular studies: panel (A): H: Hutchison and Hogg (1984), S: Siddiqui et al. (1973); panels (B) and (C): B: Botto et al. (1988), C: ChristiansenWeniger and Hardie (1997), H: Hofsvang and Hagvar (1975), S: Sigsgaard (2000), T: Thiboldeaux (1986).