Life table analysis
Based on the age-stage, two-sex life table procedure (Chi and Liu 1985; Chi 1988), all data collected from our study were analyzed using the computer program TWOSEX-MSChart (Chi 2022b). Because the insects were group-reared, the total number of individuals that survived to age x in each stage were recorded (Chang et al. 2016). The age-stage specific survival rate (sxj) was calculated as:
$${\text{s}}_{\text{xj}}\text{= }\frac{{\text{n}}_{\text{xj}}}{{\text{n}}_{\text{0}}}$$
where n0 is the number of eggs finally included (were calculated) in the life table trials and nxj is the number of individuals that survived to age x and stage j. Because we recorded the total number of eggs (Ex) produced by all living females (the fourth life stage) at age x, the female age-specific fecundity (fx4) was calculated as:
$${\text{f}}_{\text{x}\text{4}}\text{=}\frac{{\text{E}}_{\text{x}}}{{\text{n}}_{\text{x}\text{4}}}$$
The net reproductive rate (R0) is the total number of offspring that an individual can produce during its lifespan and was calculated as:
$${\text{R}}_{\text{0}}\text{=}\sum _{\text{x}\text{=0}}^{\text{∞}}\sum _{\text{j}\text{=1}}^{\text{k}}{\text{s}}_{\text{xj}}{\text{f}}_{\text{xj}}$$
where k is the number of life stages. The age-specific survival rate (lx) is the probability that a newly laid egg will survive to age x, which is obtained by pooling all surviving individuals of different stages, and was calculated as:
$${\text{l}}_{\text{x}}\text{=}\sum _{\text{j=}\text{1}}^{\text{k}}{\text{s}}_{\text{xj}}$$
The age-specific fecundity (mx), which is the mean fecundity of survival individuals at age x, was calculated as follows:
$${\text{m}}_{\text{x}}\text{=}\frac{\sum _{\text{j=}\text{1}}^{\text{k}}{\text{s}}_{\text{xj}}{\text{f}}_{\text{xj}}}{\sum _{\text{j=}\text{1}}^{\text{k}}{\text{s}}_{\text{xj}}}$$
The intrinsic rate of increase (r) was calculated by using the Euler-Lotka formula with the age indexed from 0 (Goodman 1982) as:
$$\sum _{\text{x}\text{=0}}^{\text{∞}}{\text{e}}^{\text{-}\text{r}\text{(}\text{x}\text{+1)}}{\text{l}}_{\text{x}}{\text{m}}_{\text{x}}\text{=1}$$
The finite rate of increase (λ) was calculated by the equation:
$$\text{λ=}{\text{e}}^{\text{r}}$$
The mean generation time (T) is the length of time that a population requires to increase to R0 fold of its size at the stable age-stage distribution. It was calculated as:
$$\text{T=}\frac{\text{ln}{\text{R}}_{\text{0}}}{\text{r}}$$
Because some individuals entered diapause at 20°C, in order to assess the contribution of non-diapausing individuals vs. diapausing individuals to the life table parameters, we derived a means of calculating the contribution similar to the method used by Özgökçe et al. (2018). The survival rate and fecundity of the total cohort was separated into non-diapausing cohort and diapausing cohort using the formula:
$${\text{l}}_{\text{x}}\text{ = }{\text{l}}_{\text{x,non-diapause }}\text{+ }{\text{l}}_{\text{x,diapause}}$$
$${\text{m}}_{\text{x}}\text{ = }{\text{m}}_{\text{x,non-diapause}}\text{+}{\text{ }\text{m}}_{\text{x,diapause}}$$
where lx,non−diapause and lx,diapause are the age-specific survival rates of non-diapausing and diapausing cohorts, respectively; and mx,non−diapause and mx,diapause are the age-specific fecundity of non-diapausing and diapausing cohorts. The equation of r can then be reformulated as:
$$\sum _{\text{x}\text{=0}}^{\text{∞}}{{\text{e}}^{\text{-r}\text{(}\text{x}\text{+1)}}\text{l}}_{\text{x}}{\text{m}}_{\text{x,total}}\text{ = }\sum _{\text{x}\text{=0}}^{\text{∞}}{\text{e}}^{\text{-}\text{r}\text{(}\text{x}\text{+1)}}\left({\text{l}}_{\text{x,non-diapause}}\text{+}{\text{ }\text{l}}_{\text{x,diapause}}\text{) (}{\text{m}}_{\text{x,non-diapause}}{\text{+}\text{ }\text{m}}_{\text{x,diapause}}\right)$$
Because \({\text{l}}_{\text{x,non-diapause}}{\text{m}}_{\text{x,diapause}}\text{ }\text{=}\text{ }\text{0}\) and \({\text{l}}_{\text{x,diapause}}{\text{m}}_{\text{x,non-diapause}}\text{ }\text{=}\text{ }\text{0}\), the equation can be simplified to:
\(\sum _{\text{x}\text{=0}}^{\text{∞}}{{\text{e}}^{\text{-}\text{r}\text{(}\text{x}\text{+1)}}\text{l}}_{\text{x}}{\text{m}}_{\text{x,total}}\text{ = }\sum _{\text{x}\text{=0}}^{\text{∞}}{\text{e}}^{\text{-}\text{r}\text{(}\text{x}\text{+1)}}\left({\text{l}}_{\text{x,non-diapause}}{\text{m}}_{\text{x,non-diapause}}\text{ }\text{+}\text{ }{\text{l}}_{\text{x,diapause}}{\text{m}}_{\text{x,diapause}}\right)\text{ }\text{=}\text{ }\text{ }\sum _{\text{x}\text{=0}}^{\text{∞}}{\text{e}}^{\text{-}\text{r}\text{(}\text{x}\text{+1)}}{\text{l}}_{\text{x,non-diapause}}{\text{m}}_{\text{x,non-diapause}}\text{ }\text{+}\text{ }\sum _{\text{x}\text{=0}}^{\text{∞}}{\text{e}}^{\text{-}\text{r}\text{(}\text{x}\text{+1)}}{\text{l}}_{\text{x,diapause}}{\text{m}}_{\text{x,diapause}}\) = 1
The term \(\sum _{\text{x}\text{=0}}^{\text{∞}}{\text{e}}^{\text{-}\text{r}\text{(}\text{x}\text{+1)}}{\text{l}}_{\text{x,non-diapause}}{\text{m}}_{\text{x,non-diapause}}\) is the proportion of the contribution to r made by non-diapausing individuals and \(\sum _{\text{x}\text{=0}}^{\text{∞}}{\text{e}}^{\text{-}\text{r}\text{(}\text{x}\text{+1)}}{\text{l}}_{\text{x,diapause}}{\text{m}}_{\text{x,diapause}}\) is the proportion of the contribution made by diapausing individuals. Similarly, the contribution of non-diapausing individuals and diapausing individuals to R0 can be calculated as:
$${\text{R}}_{\text{0}}\text{=}\sum _{\text{x}\text{=0}}^{\text{∞}}{\text{l}}_{\text{x}}{\text{m}}_{\text{x,total}\text{ }}\text{=}\text{ }\sum _{\text{x}\text{=0}}^{\text{∞}}\left({\text{l}}_{\text{x,non-diapause}}{\text{m}}_{\text{x,non-diapause}}\text{ }\text{+}{\text{ }\text{l}}_{\text{x,diapause}}{\text{m}}_{\text{x,diapause}}\right)\text{=}\text{ }\text{ }\sum _{\text{x}\text{=0}}^{\text{∞}}{\text{l}}_{\text{x,non-diapause}}{\text{m}}_{\text{x,non-diapause}} \text{+}\text{ }\sum _{\text{x}\text{=0}}^{\text{∞}}{\text{l}}_{\text{x,diapause}}{\text{m}}_{\text{x,diapause}} \text{=}\text{ }{\text{R}}_{\text{0,}\text{non-diapause}}\text{ }\text{+}\text{ }{\text{R}}_{\text{0,}\text{diapause}}$$
$$1 \text{=}\text{ }\sum _{\text{x}\text{=0}}^{\text{∞}}{\text{l}}_{\text{x}}{\text{m}}_{\text{x}}\text{/}{\text{R}}_{\text{0}}\text{ }\text{==}\text{ }\frac{{\text{R}}_{\text{0,}\text{non-dipause}}}{{\text{R}}_{\text{0}}}\text{ }\text{+}\text{ }\frac{{\text{R}}_{\text{0,}\text{diapause}}}{{\text{R}}_{\text{0}}}$$
The terms R0,non−diapause/R0 and R0,diapause/R0 are the proportions of the contribution to R0 made by non-diapausing individuals and diapausing individuals, respectively
To estimate the standard errors of the population parameters using the bootstrap technique, the longevity data for each individual and daily fecundity of each female adult are needed. Because we recorded the age at death of each individual during the experimental period, the longevity and sex of each individual could be calculated. Additionally, the information from the daily number of eggs laid by each female was used to assign the mean fecundity to each female. This procedure does not affect the lx and mx due to the relationship between R0 and F as proven by Chi (1988). Therefore, this practice will not affect the population parameters. Because of the variable longevity of females, this practice can still reveal the variability of fecundity found in female adults. The variances and standard errors of all parameters were estimated by using the bootstrap technique with 100,000 resampling (Efron and Tibshirani 1993; Huang et al. 2018). Differences among treatments were then compared using the paired bootstrap test (Wei et al. 2021).