Parasitism experiments and analyses
To examine the parasitoid wasp’s host preference and the effect of the 1st degree and the 2nd degree HOIs on the beetle’s parasitism and sex ratio, we conducted a laboratory experiment in insect tents (60cm x 60cm x 60cm) with three treatments: 1) no ants (no HOIs but only the wasp and the beetle larvae), 2) ants (1st degree HOI), and 3) ants and phorids (1st and 2nd degree HOIs) (Fig. 1-B). We randomly assigned insect tents to each treatment in each trial, and the tents for each treatment were also shuffled in each trial. All beetle larvae used for these experiments were reared in the laboratory for at least two generations from freshly collected beetle adults. In each tent we placed a coffee branch with 4–6 leaves infested with approximately 100 scale insects inside a plastic container at the center of an insect tent. The set up for the three treatments of species combinations were as follows: 1) 4–5 third or fourth instar beetle larvae and a parasitoid wasp; 2) 4–5 third or fourth instar beetle larvae, a parasitoid wasp, and about 60–80 ant workers; 3) 4–5 third or fourth instar beetle larvae, a parasitoid wasp, about 60–80 ant workers and 3–4 phorid flies. To allow for acclimation, we introduced organisms into the tents in the following order: first, we introduced the coffee branch containing scales, immediately followed by the ants (in treatments 2 and 3). After the ants settled down and started tending the scale insects, we introduced the beetle larvae. Once the larvae began moving on the coffee leaves, we introduced the phorids (in treatment 3). When the three treatments were established, and the organisms exhibit normal behavior, we released one lab-reared female parasitoid wasp (H. shuvakhinae) in each tent (treatments 1, 2, and 3). We allowed the organisms to interact for 24 hours. After 24 hours, we collected all beetle larvae in each treatment and reared them with sufficient scale insects as food, until beetle adults emerged or parasitism symptoms appeared (parasitized larvae turned into hardened black mummies). The treatments of no HOI and 1st + 2nd degree HOI were repeated for 10 consecutive times, and the treatment of 1st degree HOI was repeated for 11 consecutive times, with new individuals of each organism. We recorded parasitism instances and beetle sexes upon emergence. To estimate the sex ratio without parasitoid influence, 78 randomly selected beetle individuals were reared on coffee leaves with scale insects without any interaction with other organisms.
To analyze the effect of the parasitoid, the ant and the phorid fly on the parasitism rate and the sex ratio of the beetle, we developed a nested model, starting from
$$logit\left(\widehat{P}\left(S\right)\right)=a+bA$$
where \(\widehat{P}\left(S\right)\) is the probability of an individual being parasitized, A is a binary variable, standing for the absence (0) and presence (1) of ants, a is the baseline probability of parasitism, and b is the magnitude of parasitism altered by ants in the logistic function. We further hypothesized that phorid attacks modify the strength of the interaction modification that ants exert upon the host-parasitoid interaction. Therefore,
where P is another binary variable, standing for the presence (1) and absence (0) of phorids. Substituting b, we obtain the following function,
$$logit\left(\widehat{P}\left(S\right)\right)=a+gA+hAP$$
where g represents the effect of ants on the parasitism rate of A. orbigera larvae, and h represents the effect of the fly’s facilitation, via interfering with the ant’s interference on the parasitism rate of A. orbigera larvae. We used binary responses (1: survival; 0: parasitized) of all available beetle individuals across the three treatments. We performed model selection based on the Akaike Information Criterion (AIC) and likelihood ratio tests. For the latter, we started model selection by fitting the full model and preceding each step by eliminating the term that had the least significance (the greatest p-value) on the explanation of the dependent variable. The analysis was performed with the application of the bbmle package in R. By doing this, we determined the maximum likelihood estimates of survival probability of the beetle, \(\widehat{P}\left(S\right)\), in the three treatments: (1) A = 0, AP = 0 (no TMII); (2) A = 1, AP = 0 (one TMII: ant interference) and (3) A = 1, AP = 1 (interacting TMIIs: phorid interference with ant interference), and errors associated with these estimates.
The same idea applies to the sex ratio of the beetle under the influence of various organisms. We developed the following equation,
$$logit\left(\widehat{P}\left(F\right|S)\right)= r+mA+nAP$$
where \(\widehat{P}\left(F\right|S)\) is the probability of a parasitism survivor being female. A and P are both binary variables. Respectively, they represent the ant and the phorid fly, and the numeric attributes, 0 and 1, denote their absence and presence. As before, model selection and parameter estimates were conducted with AIC. By doing this, we determined \(\widehat{P}\left(F\right|S)\), the estimate of being a female beetle given survival, for the three treatments: (1) A = 0, AP = 0 (no TMII); (2) A = 1, AP = 0 (one TMII: ant interference) and (3) A = 1, AP = 1 (interacting TMIIs: phorid interference with ant interference), and errors associated with these estimates. We employed the mle2 function in the bbmle package in R to estimate the female probability (1) in the absence of HOI (the beetle and the parasitoid alone), (2) in the presence of the 1st degree HOI (the beetle, the parasitoid and the ant), and (3) in the presence of the 1st and the 2nd degree HOIs (the beetle, the parasitoid, the ant and the phorid fly).
Probabilities of per capita female and per capita male survival from parasitism under the influence of ant and the phorid fly
To test whether the sex ratio of beetle survivors’ population is due to sex-differential survival probability, Bayes’ theorem was employed. Per capita female survival probability from parasitism in each treatment of the parasitism experiment was derived based on \(\widehat{P}\left(F\right)\), \(\widehat{P}\left(F|S\right),\)and \(\widehat{P}\left(S\right)\), and per capita male survival probability was derived based on \(\widehat{P}\left(M\right)\), \(\widehat{P}\left(M|S\right),\)and \(\widehat{P}\left(S\right)\). According to the Central Limit Theorem, the estimates of proportions, \(\widehat{P}\left(S|F\right)\) and \(\widehat{P}\left(S|M\right)\), are approximately normally distributed,
$$\widehat{P}\left(S|F\right)\tilde N\left(\widehat{P}\left(S|F\right), \sqrt{\frac{\widehat{P}\left(S\right|F)\times \left(1-\widehat{P}\left(S|F\right)\right)}{{n}^{*}}}\right)$$
$$\widehat{P}\left(S|M\right)\tilde N\left(\widehat{P}\left(S|M\right), \sqrt{\frac{\widehat{P}\left(S\right|M)\times \left(1-\widehat{P}\left(S|M\right)\right)}{{n}^{*}}}\right)$$
with means \(\widehat{P}\left(S|F\right)\) and \(\widehat{P}\left(S\right|M)\), and standard deviations \(\sqrt{\frac{\widehat{P}\left(S|F\right)\times (1-\widehat{P}\left(S|F\right))}{{n}^{*}}}\) and \(\sqrt{\frac{\widehat{P}\left(S|M\right)\times (1-\widehat{P}\left(S|M\right))}{{n}^{*}}}\), where \(\widehat{P}\left(S\right|F)\) and \(\widehat{P}\left(S\right|M)\), respectively, are the population proportions of females and males. Here we employ n*, the smallest sample size among those of the three variables in the Bayesian formulas for males and females. Since the three variables have different sample sizes, n* guarantees a conservative estimate of standard error, and thus confidence interval, of each derived probability.