Cucumber (*Cucumis sativa* L., cv. Negin; Cucurbitales: Cucurbitaceae) and broad bean (*Vicia faba* L.; Fabales: Fabaceae) seedlings were grown from seeds purchased from Caoxian County, Shandong, China. The seeds were sown in 30 × 25 cm diameter regular pots filled with (3:1) soil: manure. The seedlings were maintained under greenhouse conditions of 12–26°C, 45–55 % RH, and 16:8 h (Light: Dark) photoperiod, and subsequently used for rearing and conducting functional response assays. The plant materials used were obtained with prior permission, and the present study is in compliance with relevant guidelines and legislation.

For establishing *A. pisum* culture, the initial populations of aphid collected from unsprayed alfalfa fields were subsequently brought to the laboratory and reared on broad bean plants inside net cages (20 × 10 × 30 cm height). The stock culture of *H. axyridis* was developed from a pre-established laboratory colony, already available in the same laboratory. The predator was reared on *A. pisum* infested bean plants (7–8 leaves) inside net cages (60 × 42 × 30 cm height) for three consecutive generations at laboratory conditions of 24 ± 1°C, 65 ± 5 % RH and 16:8 h (Light: Dark) photoperiod. Bean plants were checked daily for predator eggs. Egg batches when found were carefully removed, placed on tissue paper in Petri dishes (9 cm), and transferred to a computer-operated growth chamber, maintained at settings of 25 ± 1°C, 65 ± 5 % RH and 16:8 h (Light: Dark) photoperiod. The post-emergence larvae were separated and reared in Petri dishes containing aphid as their diet, refreshed daily. The whole culture was maintained at the Department of Plant Protection, Huazhong Agricultural University, China.

The experimental arena consisted of clear Petri dishes (9 cm diameter), with a micromesh screen over the top for ventilation and bottom covered with clean cucumber leaf disk. The desiccation of cucumber leaf disc was prevented by adding 1% agar solution95. The assays were performed with *H. axyridis* larvae (i.e., 1st instar, 2nd instar, 3rd instar, 4th instar) and adults (male, female) at constant temperatures (i.e., 15, 20, 25, 30, 35°C). The homogeneity of predator age was maintained within each tested growth stage. The first instar larvae were separated one by one shortly after hatching to avoid sibling cannibalism. Hatchlings were reared in Petri dishes (9 cm diameter) until maturity on 4th instar nymphs (100–150 aphids/day). Female *H. axyridis* included mated individuals72. First instar larvae were starved for about 6h, whereas subsequent instars/stages were starved for 24 h to standardize hunger level, according to Islam, et al. 50. The moist cotton roll offered humidity to all predators during starvation. The use of 4th instar aphid was ensured throughout the experiments as a way to prevent predator preference switch according to prey size96. Using a fine camel hairbrush, the aphids at different densities (i.e., 2, 4, 8, 16, 32, 64, 128, and 160 aphids) were transferred in Petri dishes, allowed for 30 minutes to uniformly spread and settle over the substrate, and thereafter were transferred to a computerized growth chamber at various fixed temperatures (i.e., 15, 20, 25, 30, 35°C), 70 ± 5 % RH and 16:8h (Light: Dark) photoperiod. The whole experiment was replicated 10 times for each prey density, growth stage, and temperature. The numbers of aphid consumed were recorded every 24th h. Control replicates were kept free from *H. axyridis* to account for natural mortality and correct *A. pisum* consumption by the predator as a function of natural mortality. Predation mortality data were corrected for control mortality by applying Abbott's correction97.

The control mortality data were analyzed between temperatures, aphid densities, and their interaction, by using Univariate Analysis of Variance (ANOVA) in SPSS (version 21), fitting the above three variables as fixed factors against the dependent variable (i.e., host mortality). Significant (*P* < 0.05) effects were further compared by using Tukey’s Honestly Significant Difference (HSD) multiple comparisons Test. Prior to analysis, the mortality data were tested for normality and homogeneity of error variance (i.e., homoscedasticity) by using Shapiro-Wilk and Levene tests, and Y = √ x + 1 transformed to improve compliance with these assumptions. All means and standard errors in text and figures are calculated with untransformed data.

Aphid consumption by *H. axyridis* for temperature, growth stage, density, and their two-way and three-way interactions were analyzed by using Generalized Linear Models (GLM) in SPSS (version 21). Kolmogorov-Smirnov test confirmed non-normal distributions of data (*P* > 0.05), and due to over-dispersion, the data were fitted with negative binomial distribution and a log link function, and factors and interaction effects were analyzed by using the Wald Chi-Square test for a confidence level (CI) of 95%. If needed, the multiple follow up tests were run to analyze the temperature and growth stage effects, separately, at each aphid density, and the significance for each test was adjusted by following Bonferroni correction to avoid Type 1 error.

Analysis of the functional response was done in two different phases17: first phase involved the determination of type and estimation of the parameters of the functional response curve. It is compulsory to find the type of functional response for calculating the functional response parameters using a proper model. The type was determined by applying logistic regression of the proportion of prey eaten as a function of initial prey density offered. A polynomial logistic regression equation assuming a binomial distribution of data to define the type of functional response17 (Eq. 1) was fitted as under:

Where *N*a and *N*o indicate the number of prey consumed and the initial prey density offered, respectively, and is the proportion of prey consumed. The *P*o, *P*1, P2, and *P*3 are the regression parameters representing intercept or constant, linear, quadratic, and cubic coefficients, respectively. The coefficients were calculated by using the maximum likelihood method. The values of the linear and quadratic coefficients indicate the nature of functional response either it is Type II or Type III. When the value of a linear parameter is negative, the functional response is Type II, and if it is positive with a negative quadratic coefficient, then response is of Type III. The Type II response shows that the proportion of prey consumption decreases as the prey density increases, and a Type III response represents that the proportion of prey consumed increases until an inflection point and then decreases17. Once the functional response type was determined, the second phase started where functional response parameters were determined. For which, data were fitted to Rogers’ type II random predator equation, with the help of non-linear least square regression, and determined and analyzed the parameters of functional response. As the prey was not changed or replaced during the entire experiment, the random predator equation was determined to be more appropriate for such a dataset98. The attack rate (*a*) and handling time (*Th*) were calculated by using the random predator model as under: (Eq. 2).

Where, *a* is the attack rate, *T*h is the handling time, *T* is time available for predator during the experiment. Here, “*glm*” function was used to fit the logistic regression, and the parameters (attack rate *a* and handling time *T*h of functional response were estimated by using *FRAIR* (Functional Response Analysis in R, version 4.0.0)99 in the R statistical environment100. The maximum predation rate is the ratio between *T*/*Th*101 and estimates the maximum amount of prey that a predator can consume in a given time frame.