Abstract:  【Aim】The bootstrap technique has been widely used to estimate the variances, standard errors, and confidence intervals of life table parameters, while the paired bootstrap test (PBT) has been used to compare the differences in life table parameters between treatments. In our research, we present life tables of Helicoverpa armigera reared on corn grains and tea leaves as an example to further explain the application of set theory, Cartesian products, and multinomial theorem in research on pest population biology. 【Methods】 Age-stage, two sex life table was used to analyze the population parameters of H. armigera reared on corn grains and tea leaves. A population statistical analysis was presented in an explicitly mathematical way using set theory, Cartesian products, and multinomial theorems in order to detect all possible bootstrap samples, to accurately compute the confidence intervals of the population parameters as well as the confidence intervals of the differences between treatments, and to compute the probability of fertile and sterile samples. 【Results】 The preadult survival rate, intrinsic rate of increase (r), finite rate of increase (λ), net reproductive rate (R₀), and mean generation time (T) of H. armigera reared on tea leaves were significantly lower than those of H. armigera reared on corn grains, indicating that tea leaf is not a suitable host plant for H. armigera. Because of the large number of sterile samples of the H. armigera population reared on tea leaves, the difference in the population parameters between the original cohort and estimated one was less than 5% when the bootstrap technique was applied and both fertile and sterile samples were accepted. If only fertile samples were accepted in the bootstrap sampling, the results were significantly different from the original values, and there would be obvious errors (>5%). The use of the Cartesian paired test (CPT) to compare differences in R₀ between H. armigera populations fed on the two kinds of foods allowed the calculation of precise confidence intervals for all possible differences in the bootstrap sampling results of the two treatments, whereas the use of the paired bootstrap test usually resulted in an overestimated or underestimated confidence intervals, especially when the number of bootstrap samples was small. The multinomial theorem could reveal the results of bootstrap sampling for both fertile and infertile and kept the bootstrap sampling records for further application in subsequent analysis. 【Conclusion】This study further clarified the mathematical foundation of two-sex life table theory, and also provided mathematical support for the application of life table techniques in entomological research.

Key words: Helicoverpa armigera, life table, set theory, Cartesian product, multinomial theorem, infertile bootstrap sample