## 3 Male Random Flight Model
### First Arrival and Mating Time  
The capture/mating probability \(C(t)=1-P(t)\) that an individual male moth reaches a female by time \(t\) and the corresponding arrival time distribution \(S(t)\), evaluated via Equation (5a,b), are important overall metrics in determining the mating success of a single moth. However, competition arises when multiple males attempt to reach the same female and only the first one can mate. In this section, we consider \(N\) competing moths and adapt some known results on first arrival times ([PERSON] [68]) to give analytical estimates for the average time it takes the first male to reach the female. These results are valid in the \(N\gg 1\) limit.  
In accordance with our previous discussion, we identify the male's first arrival time to the female-generated plume as the first mating time. [PERSON] [68] analyzes the probability \(P_{\rm a}(t_{\rm a}<t)\) that the first arrival time \(t_{\rm a}\) out of a cohort of N independent random walkers is less than \(t\) and shows that if \(P_{\rm a}(t_{\rm a}<t)\) can be approximated in the short time limit as  
\[P_{\rm a}(t_{\rm a}<t)\sim At^{q}e^{-\frac{B}{t}},\quad t\to 0^{+}, \tag{7}\]  
for a given \(q\), \(A\
eq 0\), and \(B>0\), then the first arrival time \(t_{\rm a}\) is distributed according to Gumbel distribution:  
\[P_{\rm a}(t_{\rm a})=\frac{1}{a_{N}}\exp\left[\frac{t_{\rm a}-b_{N}}{a_{N}}- \exp\left(\frac{t_{\rm a}-b_{N}}{a_{N}}\right)\right], \tag{8}\]  
where  
\[a_{N}=\frac{b_{N}}{q(1+W_{N})},\qquad b_{N}=\frac{B}{qW_{N}},\qquad W_{N}=W_{* }\left(\frac{B}{q}(AN)^{\frac{1}{q}}\right), \tag{9}\]  
and where \(W_{*}(z)\) is the principal branch of the LambertW function, defined as the inverse function of \(f(z)=ze^{z}\). The first two moments of the Gumbel distribution are  
\[\mathbb{E}[t_{\rm a}]=b_{N}-\gamma_{e}a_{N},\qquad\mathrm{Var}[t_{\rm a}]= \frac{\pi^{2}}{6}a_{N}^{2}, \tag{10}\]  
where \(\gamma_{e}\approx 0.5772\) is the Euler-Mascheroni constant.  
We can use these results by noting that, within our model, \(P_{\rm a}(t_{\rm a}<t)\) can be identified with the capture/mating probability \(C(t)\): if the first arrival time is less than \(t\), then at time \(t\), the moth will have reached the plume. In Equation (20) of Appendix B, we derive the short time behavior for \(C(t)\) and verify that it follows Equation (7) as \(t\to 0^{+}\). We find  
\[C(t)\sim At^{q}e^{-\frac{B}{t}},\quad t\to 0^{+}, \tag{11}\]  
with  
\[q=\frac{1}{2},\qquad A=\frac{2}{(1-R)r_{\rm mate}}\sqrt{\frac{DR}{\pi}},\qquad B= \frac{r_{0}^{2}(1-R)^{2}}{4D}, \tag{12}\]  
from which we conclude that the average first arrival time \(t_{\rm a}\) for \(N\to\infty\) competing male moths is given by  
\[\mathbb{E}[t_{\rm a}]\sim\frac{r_{0}^{2}(1-R)^{2}}{2 DW_{N}}\left[1-\frac{2 \gamma_{e}}{1+W_{N}}\right],\] (13a) where \[W_{N}=W_{*}\left(\frac{2N^{2}R}{\pi}\right)\sim\ln N. \tag{13b}\]  
The mean first arrival time \(\mathbb{E}[t_{\rm a}]\) is thus a decreasing function of \(D\) and a quadratically increasing function of \(r_{0}(1-R)=(x_{0}^{2}+y_{0}^{2})^{1/2}-r_{\rm mate}\), as can be expected from random searchers in two dimensions. Also note that, since \(W_{N}\sim\ln N\), in the \(N\to\infty\) limit, \(\mathbb{E}[t_{\rm a}]\) scales as \((\ln N)^{-1}\). Finally, note that these estimates assume that the time frame for male moths to reach the female is unbounded, so that all values of \(t_{\rm a}\), including large ones, contribute to \(\mathbb{E}[t_{\rm a}]\) as evaluated in Equation (13a). In practice, the female call may last for a limited period \(t_{\rm call}\), so that arrival times \(t_{\rm a}>t_{\rm call}\) would not be tallied.