## 4 Numerical Results for the Male Random Flight Model
### Multiple Female Calling Periods  
So far, females have been modeled to call males via a single calling period of time \(t_{\text{sim}}\). In this section, we model calling over a typical one-week period where calling times can be adjusted nightly according to the chosen _constant_, _increase_, _decrease_, or _hat_ strategies described in Section 2.1 and Figure 1. The female still generates a stationary, circular plume that attracts male moths, and male quality will still be associated with large diffusivities \(D\) and large sensitivity radii \(r_{\text{mate}}\). For each of the \(k\) nights, with \(1\leq k\leq 7\), we denote the time spent calling by \(t_{\text{call}}(k)\), while the radius of the circular plume per call is fixed at \(r_{\text{mate}}\). Regardless of female strategy, the values of \(t_{\text{call}}(k)\) are chosen between a minimum of 0.5 and maximum of 8 h as reported in Section 2.1 from previous experimental observations, such that the total allotted total calling time \(T_{\text{call}}=\sum_{k=1}^{7}t_{\text{call}}(k)\) does not change among strategies. For \(1\leq k\leq 7\), we thus consider  
\[\text{Increase:}\quad t_{\text{call}}(k) =\{0.5000,\ 1.7500,\ 3.0000,\ 4.2500,\ 5.5000,\ 6.7500,\ 8.0000\},\] \[\text{Decrease:}\quad t_{\text{call}}(k) =\{8.0000,\ 6.7500,\ 5.5000,\ 4.2500,\ 3.0000,\ 1.7500,\ 0.5000\},\] \[\text{Constant:}\quad t_{\text{call}}(k) =4.2500,\] \[\text{Hat:}\quad t_{\text{call}}(k) =\{2.5870,\ 3.8804,\ 5.1739,\ 6.4674,\ 5.1739,\ 3.8804,\ 2.5870\}.\]  
We set \(r_{0}=1\) m and utilize two possible values for \(r_{\text{mate}}\): \(r_{\text{mate}}=3/4\,r_{0}\) for a higher quality male and \(r_{\text{mate}}=1/4\,r_{0}\) for a lower quality male. Finally, male diffusivities are set at \(D=10^{-3/2}\) or \(D=10^{-7/2}\) m\({}^{2}\)/s, resulting in four possibilities:  
\[\{R,D\}=\{0.75,10^{-1.5}\text{m}^{2}/\text{s}\},\{0.75,10^{-3.5}\text{m}^{2}/ \text{s}\},\{0.25,10^{-1.5}\text{m}^{2}/\text{s}\},\text{ or }\{0.25,10^{-3.5}\text{m}^{2}/\text{s}\}.\]  
These parameter values are chosen as typical values from Figure 3.  
To calculate the capture and mating probabilities \(C_{k}\) on each night, we evaluated \(C_{k}\equiv C(t_{\text{call}}(k))\) from Equation (5) with \(t=t_{\text{call}}(k)\) and the four \(\{R,D\}\) parameters specified above. We find that the capture and mating probability \(C_{k}\) for male moths with the highest diffusivities and the largest mating radius \(r_{\text{mate}}\) remains elevated for all \(k\) values and relatively constant across all strategies, as can be seen in Figure 5.  
Moths with lower diffusivities and where \(r_{\text{mate}}\) is smaller display capture and mating probabilities that vary with \(k\) and that follow the same trends as calling time strategy used, so that increasing calling time generally increases the mating probability from one night to the next. Intermediate capture and mating probabilities emerge for small diffusivities \(D\) but high \(r_{\text{mate}}\)  
Figure 4: Male arrival time distributions evaluated in \(\{R,D\}\) phase space, where \(R=r_{\text{mate}}/r_{0}\). **(A)** Mean first arrival times \(\mathbb{E}[t_{a}]\) calculated from Equation (13a) for \(N=1000\) moths. **(B)** Mean first arrival times \(\mathbb{E}[t_{a}]\) calculated from numerical simulations with \(N=1000\) arrivals and \(t_{\text{call}}=1800\) s (Section 4.1). **(C)** Mean arrival times \(\mathbb{E}[t]\) calculated from numerical simulations. Mean first arrival times \(\mathbb{E}[t_{a}]\) are the lowest for large diffusivities \(D\) and for \(R\to 1\), as can be expected. Note the nine orders of magnitude variation for \(\mathbb{E}[t_{a}]\) across the \(\{R,D\}\) ranges considered. The mean arrival time \(\mathbb{E}[t]\) is generally much larger than \(\mathbb{E}[t_{a}]\) and only spans three orders of magnitude in the same \(\{R,D\}\) space. and vice versa for large diffusivities \(D\) and low \(r_{\text{mate}}\), with the latter scenario yielding slightly more favorable outcomes.