## 4 Numerical Results for the Male Random Flight Model
### Multiple Female Calling Periods
Figure 5 suggests that the _increase_ or _hat_ strategies may be best for a female to successfully attract a high-quality male through the least calling effort while still finding a mate within the one-week mating period. At onset of the mating process, both the _increase_ or _hat_ strategies yield a large capture and mating probability for high-quality males, with high \(D\) and large \(r_{\text{mate}}\), and a relatively lower probability for low-quality males, with low \(D\) and small \(r_{\text{mate}}\). At this stage, the female employs short calling times \(t_{\text{call}}(k)\), ensuring that she is being selective while minimizing her calling effort. As the mating period continues and \(t_{\text{call}}(k)\) increases, the likelihood of attracting lower quality males increases, so that overall mating success with males of any quality increases. This observation is consistent with field observations where it is shown that female moths preferentially employ the _increase_ strategy to optimize their mating likelihood ([PERSON] et al. (2017)). We confirmed that the _increase_ and _hat_ strategies are the most effective for females. To do so, we defined the effectiveness of the strategy on the \(k\)th night, \(E_{k}\), as the ratio of the cumulative total mating probability and total time spent calling (Figure 6). Using the calling times \(t_{k}\) specified above for each strategy, we calculated the mating probability \(C_{k}=C(t_{k})\) in the \(\{R,D\}\) parameter plane as in Figure 3B. This defines the total mating probability \(M_{k}\), which is calculated as the sum of \(C_{k}\) over all \(R\) and \(D\) values. The effectiveness is thus calculated as the ratio of the cumulative total mating probability to the total time spent calling  
\[E_{k}=\frac{\sum_{i=1}^{k}M_{i}}{\sum_{i=1}^{k}t_{i}}. \tag{16}\]  
In Figure 6, we plot \((\sum_{i=1}^{k}t_{i},E_{k})\) for each night \(k=1,\ldots,7\) and for each strategy. The results show that the _increase_ strategy is the most effective, since a female employing this strategy has high total mating probability obtained with a relatively short calling time. On the other hand, the _decrease_ strategy is the least effective, since a female has a higher total mating probability than in the increase strategy at the cost of a long calling time. \[\text{Increase:}\quad t_{\text{call}}(k) =\{0.5000,\ 1.7500,\ 3.0000,\ 4.2500,\ 5.5000,\ 6.7500,\ 8.0000\},\]0 Effectiveness of each of the four female calling strategies described in Section 2.1 and \[\text{Increase:}\quad t_{\text{call}}(k) =\{0.5000,\ 1.7500,\ 3.0000,\ 4.2500,\ 5.5000,\ 6.7500,\ 8.0000\},\]1 Capture and mating probability \(C(t)\) for four representative moths with \(\{R,D\}=\{0.75,10^{-1.5}\) m\({}^{2}\)/s\(\}\) (blue curve), \(\{0.75,10^{-3.5}\) m\({}^{2}\)/s\(\}\) (red curve), \(\{0.25,10^{-1.5}\) m\({}^{2}\)/s\(\}\) (yellow curve), and \(\{0.25,10^{-3.5}\) m\({}^{2}\)/s\(\}\) (purple curve). Calling times \(t_{\text{call}}(k)\) for \(1\leq k\leq 7\) nights are specified in Section 4.3. Male moths of the highest quality display similar capture and mating probability across female strategies, with \(C(t)\simeq 1\) in all cases. For intermediate quality moths \(C(t)\) depends on the chosen female strategy.