## 4 Numerical Results for the Male Random Flight Model
### First Arrial Time  
We now evaluate the expected time for a male moth to first arrive at the plume. The mean first arrival time \(\mathbb{E}[t_{\mathrm{a}}]\) is analytically estimated in Equation (13a) and is a function of male diffusivity \(D\), plume radius \(r_{\mathrm{mate}}\), and \(N\). Numerically, it will also depend on the duration of the calling time, which poses an upper limit for the time available to males to reach the female. Here, we we set \(t_{\mathrm{call}}\) to be the total simulation time, \(t_{\mathrm{call}}=t_{\mathrm{sim}}\). We repeatedly simulate independent male moths performing random flights with diffusivity \(D\) and record the time \(t_{\mathrm{a}}\) for each to reach the female-generated plume at \(r_{\mathrm{mate}}\) starting from an initial radius \(r_{0}\) until the number of arrivals is \(N\) or the maximum number \(M\) of simulations is reached. For those males who are unable to reach the female within \(t_{\mathrm{call}}\), no entry is tallied and the attempt is discarded and not used when calculating the mean first arrival time. If \(R\simeq 1\) and \(t_{\mathrm{call}}\) is large enough, we expect the number of discarded events to be few, and a male reaching the female at times larger than \(t_{\mathrm{call}}\) is expected to be a rare event. However, if \(R\ll 1\) and \(t_{\mathrm{call}}\) is not large enough, many males will fail to reach the female within the allotted time. The mean first arrival time we evaluate here is thus conditioned on the male reaching the female within \(t_{\mathrm{call}}\).  
For most \(D\) and \(R\) parameter values, simulating \(N=1000\) arrivals is straightforward since the capture/mating probability \(C(t)\) is relatively large; however, for those parameter values with small \(C(t)\), many simulations are required. When \(t_{\mathrm{call}}=1800\) s, \(\min_{R,D}C(t)\approx 0.0175\). Thus, we choose the maximum number of simulations \(M=60000\) to obtain approximately \(MC(t)=1050>N=1000\) arrivals. From these \(N=1000\) arrival times, we then simply obtain the minimum and average values.  
Theoretical results are shown in Figure 4A, where we present a contour plot of the mean first mating time \(\mathbb{E}[t_{\mathrm{a}}]\) as a function of \(D\) and \(R\) based on Equation (13a). Simulation results for the mean first mating time conditioned on arrival prior to \(t_{\mathrm{call}}=t_{\mathrm{sim}}\) are presented in Figure 4B. There is good agreement between the two, indicating that moths reaching the female plume at times larger than \(t_{\mathrm{call}}\) are few and do not affect \(\mathbb{E}[t_{\mathrm{a}}]\). Ranges are \(\mathbb{E}[t_{\mathrm{a}}]\simeq 10^{-5}\) to \(10^{3}\), depending on parameter choices. In addition to the first arrival time, we also calculate the mean arrival time which is displayed in Figure 4C. As can be expected for all parameters, this quantity is larger than the expected mean first arrival time, with typical values ranging between \(\mathbb{E}[t]\approx 1\) to \(10^{3}\).  
Figure 4 reveals that the mean first arrival and mating time \(\mathbb{E}[t_{\mathrm{a}}]\) is lowest for moths with large diffusivity \(D\) and large values of \(R\simeq 1\), with mating radius \(r_{\mathrm{mate}}\simeq r_{0}\). Similar trends are observed for the mean arrival time \(\mathbb{E}[t]\). One key observation is that the difference between the mean first mating time and the average arrival time can be several orders of magnitude in \(\{R,D\}\) phase space.  
Figure 3: The probability for a male moth to reach a stationary, circular female-generated plume after \(t_{\mathrm{sim}}=100\) s in the Male Random Flight Model. (**A**) Capture and mating probability \(C(t)\) as a function of \(R=r_{\mathrm{mate}}/r_{0}\), calculated from Equation (5b) for male diffusivities \(D=10^{-1}\), \(10^{-2}\), \(10^{-3}\), and \(10^{-4}\). Numerical results obtained by simulating \(N=5000\) independent moths as described in Section 4.1 are overlaid. (**B**) Capture and mating probability \(C(t)\) plotted via contours in \(\{R,D\}\) space.