## 4 Numerical Results for the Male Random Flight Model
### Night-by-Night Mating Probabilities  
For a more nuanced understanding of the mating process, we study the night-by-night mating probability for each of the four strategies shown in Figure 5. We use the same calling time sequences \(t_{\text{call}}(k)\) as in Section 4.3 but consider a wider range of values for the male characteristics \(R\) and \(D\). In particular, we evaluate the mating probability \(\mathcal{M}_{\ell}\) that a female mates on night \(\ell\) given that no male reached her on prior nights, defined as follows:  
\[\mathcal{M}_{1} =C_{1}, \tag{17}\] \[\mathcal{M}_{\ell} =C_{\ell}\prod_{k=1}^{\ell-1}(1-C_{k}),\quad\ell=2,\ldots 7. \tag{18}\]  
We also calculate the total mating probability \(\mathcal{M}\) over the one-week mating period given by  
\[\mathcal{M}=C_{1}+C_{2}(1-C_{1})+\cdots+C_{7}\prod_{k=1}^{6}(1-C_{k})=\sum_{j =1}^{7}C_{j}\prod_{k=1}^{j-1}(1-C_{k}). \tag{19}\]  
The results are shown in Figures 7-10. We find that all four calling strategies result in qualitatively similar trends for \(\mathcal{M}_{\ell}\). Specifically, the mating probability for the first night, \(\mathcal{M}_{1}\), is an increasing function of \(D\) and \(R\) and is largest for high-quality moths, as can be expected. However, the maximum probability on successive nights, \(\mathcal{M}_{\ell}\) with \(\ell>1\), declines in magnitude and shifts to lower diffusivity \(D\) and smaller relative radius \(R\). Compared to high-quality moths, those with medium \(\{R,D\}\) do not display a large mating likelihood on the first night; however, their mating likelihood remains relatively large on successive nights. This implies that, if only intermediate quality males are present, the female can increase her mating likelihood and partner with one of them if the call extends further than one night. The last panels of Figures 7-10 show the total mating probability, \(\mathcal{M}\), at the end of the one-week mating process, confirming that mating occurs with large probability for almost all moth qualities considered. This suggests that a female moth can employ a conservative strategy such as the _increase_ strategy to maximize the quality of a mate early in the calling process without risking not copulating at all after the one-week mating period.  
Figure 6: Effectiveness (mating probability/time spend calling) of the four female calling strategies. Each dot corresponds to a night. The _increase_ strategy is the most effective for small calling times, while all strategies have similar effectiveness for large calling times.  
Figure 8: Night-by-night and total mating probabilities under _decrease_ calling. Note the different likelihood scales for each panel. Similar to Figure 7, intermediate-quality moths (medium \(\{R,D\}\)) display relatively large mating likelihoods \(\mathcal{M}_{\ell}\) on the second and successive nights.  
Figure 7: Night-by-night and total mating probabilities for the _increase_ calling strategy. Note the different likelihood scales for each panel. On the first night, the mating likelihood \(\mathcal{M}_{1}\) is highest for high-quality moths (large \(\{R,D\}\)), and lower for intermediate and low-quality moths (medium and low \(\{R,D\}\)). The latter reaches the female with relatively large likelihoods on the second and successive nights, as can be seen for \(\mathcal{M}_{\ell}\) with \(\ell>1\). The last panel displays the total capture and mating likelihood at the end of the one-week mating period.