## 3 Male Random Flight Model
We use absorbing boundary conditions in Equation (1b) to indicate that random flight ceases once the moth has reached the plume's edge. At this point, the male moth will navigate toward the female using well-known surging and casting behavior. The overall probability that the moth has not captured the plume at time \(t\) is thus given by\[P(t)=\int_{\mathbb{R}^{2}\setminus\Omega(t)}g(\mathbf{x},t)\,dxdy, \tag{2}\]  
while the overall probability that the moth has captured the plume at time \(t\) is defined as \(C(t)=1-P(t)\). Since we equate reaching the plume with mating success, \(C(t)\) can also be referred to as the moth mating probability. One quantity of interest is the arrival time distribution \(S(t)\) of a moth to the trapping region (Chou and D'Orsogna [65]), which is determined by calculating the total flux:  
\[S(t)=-\frac{dP}{dt}=-\frac{d}{dt}\left(\int_{\mathbb{R}^{2}\setminus\Omega(t) }g(\mathbf{x},t)\,dxdy\right). \tag{3}\]  
Using standard differentiation tools for the moving surface in Equation (1a) and the divergence theorem, we write the following:  
\[S(t)=-\left(D\int_{\partial\Omega(t)}\
abla g\cdot\mathbf{n}\,d\ell+\int_{ \partial\Omega(t)}(\mathbf{v}^{\mathbf{b}}\cdot\mathbf{n})\,g\,d\ell\right), \tag{4}\]  
where \(\mathbf{n}\) is the outward facing normal vector to the \(\partial\Omega(t)\) boundary contour, \(\mathbf{v}^{\mathbf{b}}\) is the velocity of the boundary contour element, and \(d\ell\) is the arc length along \(\partial\Omega(t)\). Equation (4) implies that the extent of trapping region \(\Omega(t)\) is time-dependent and that its changing boundaries define non-isotropic shapes. To follow the time evolution of \(\Omega(t)\) we need to specify the dynamics of \(c(\mathbf{x},t)\), which will be done in Section 5. Here, instead, we introduce some simplifications to Equation (4) which will allow for a more basic yet fundamental understanding of the mating process. We thus assume that \(\Omega(t)\) is static and circular, centered at the origin, and characterized by a radius \(r_{\text{mate}}\) so that the first term in Equation (4) is an integral over a circumference and the second term vanishes. We associate higher quality moths with a larger \(r_{\text{mate}}\) (and larger \(D\)) and lower quality moths with a smaller \(r_{\text{mate}}\) (and smaller \(D\)). In this scenario, closed form expressions for the quantities of \(P(t)\) and \(S(t)\) can be calculated ([PERSON] [66], [PERSON] and [PERSON] [67]). Detailed derivations are given in Appendix A. We find  
\[S(t) =\frac{2D}{\pi r_{\text{mate}}^{2}}\int_{\omega=0}^{\infty}\left[ \frac{J_{0}(\omega)Y_{0}(\omega R^{-1})-J_{0}(\omega R^{-1})Y_{0}(\omega)}{Y_ {0}^{2}(\omega)+J_{0}^{2}(\omega)}\right]\omega e^{-\omega^{2}Dt/r_{\text{mate }}^{2}}\,d\omega, \tag{5a}\] \[P(t) =\frac{2}{\pi}\int_{\omega=0}^{\infty}\left[\frac{J_{0}(\omega)Y_ {0}(\omega R^{-1})-J_{0}(\omega R^{-1})Y_{0}(\omega)}{Y_{0}^{2}(\omega)+J_{0} ^{2}(\omega)}\right]\frac{1}{\omega}e^{-\omega^{2}Dt/r_{\text{mate}}^{2}}d\omega, \tag{5b}\]  
where \(J_{0}(z)\) and \(Y_{0}(z)\) are Bessel functions of the first and second kinds, respectively, and the dimensionless \(R\) is defined as  
\[R=\frac{r_{\text{mate}}}{\sqrt{x_{0}^{2}+y_{0}^{2}}}=\frac{r_{\text{mate}}}{r_ {0}}, \tag{6}\]  
and where \(r_{0}\) is the male moth's initial distance from the female. If we assume that the male moth is initially outside the circular plume, \(r_{0}\geq r_{\text{mate}}\), Equation (6) implies that \(R\leq 1\). We illustrate some characteristic forms for \(P(t),C(t)\), and \(S(t)\) in Figure 2A and typical moth trajectories obtained through numerical simulations in Figure 2B.