## 4 Numerical Results for the Male Random Flight Model
### Single Female Calling Period  
In this section, we consider male moths responding to a single female calling period. Moths are initially placed in an equidistant manner on a circle of radius \(r_{0}=(x_{0}^{2}+y_{0}^{2})^{1/2}>r_{\text{mate}}\), so that \(R<1\), and undergo a random walk until the stationary, circular plume of radius \(r_{\text{mate}}\) is reached or the total simulation time \(t_{\text{sim}}\) elapses. We use an adaptive time stepping scheme to ensure that sufficiently small time steps (hence sufficiently small spatial steps) are taken near the plume circumference to reduce boundary errors in our simulation. If the time step is too large, the corresponding spatial step may take a free moth that starts at the exterior of the plume at time \(t\) across a short secant through the plume to an updated position that remains on the exterior of the plume at time \(t+\Delta t\). The plume flyover would incorrectly tally this moth as unsuccessful in reaching the female and would result in an underestimate of the mating probability. The position \(\mathbf{x}(t)\) of a given moth at time \(t\) is updated as  
\[\mathbf{x}(t+\Delta t)=\mathbf{x}(t)+\sqrt{4D\Delta t}\,r_{\text{s}}\hat{ \mathbf{r}}, \tag{14}\]  
where \(D\) is the diffusion constant, \(r_{\text{s}}\) is randomly drawn from the standard normal distribution, and \(\hat{\mathbf{r}}\) is a unit vector isotropically oriented. This corresponds to a position-jump process where the mean square displacement after time \(T\) is \(4 DT\), as expected for two-dimensional random movement with diffusion coefficient \(D\). As discussed above, the time step \(\Delta t\) is chosen in an adaptive manner depending on the moth's distance to the circular plume \(d(t)=(x(t)^{2}+y(t)^{2})^{1/2}-r_{\text{mate}}\). We pose  
\[\Delta t(d)=(\Delta t_{\text{max}}-\Delta t_{\text{min}})\tanh(kd)+\Delta t_{ \text{min}}, \tag{15}\]  
so that \(\Delta t(d)\) interpolates between \(\Delta t_{\text{min}}\) (the shortest time step) and \(\Delta t_{\text{max}}\) (the largest time step). In the vicinity of the plume, \(\Delta t(d\to 0)=\Delta t_{\text{min}}\), whereas at sufficiently large distances, \(\Delta t(d\to\infty)=\Delta t_{\text{max}}\). The parameter \(k\) controls the steepness of the transition between \(\Delta t_{\text{min}}\) and \(\Delta t_{\text{max}}\). We set \(k=1\) m\({}^{-1}\). A male moth is assumed to have captured the plume and to have successfully mated once \(d(t)<0\). We follow the trajectories of \(N=5000\) moths engaged in a random flight for a range of diffusivities \(D\) and mating radii \(r_{\text{mate}}\). All moths are initiated at an initial radius \(r_{0}=1\) m. Simulations are halted once the simulation time \(t_{\text{sim}}=100\) s has elapsed. Since the \(N\) male moths are independent, the fraction of moths to have reached the female by \(t_{\text{sim}}\) can reasonably approximate the capture and mating probability \(C(t_{\text{sim}})=1-P(t_{\text{sim}})\) as analytically evaluated in Equation (5b). We find that the analytical expression in Equation (5b) and our simulation results are in good agreement for all surveyed values of \(D\) and \(r_{\text{mate}}\), as shown in Figure 3A, for the specific choice \(t_{\text{sim}}=100\) s. The dimensionless \(0\leq R\leq 1\) is plotted on the horizontal axis of Figure 3A instead of \(r_{\text{mate}}\) for compactness. Larger values of \(R\to 1\) indicate moths that are initiated closer to the stationary plume, whereas smaller values of \(R\to 0\) indicate moths that are initiated further away from the female. A contour plot for the numerical capture and mating probability at time \(t_{\text{sim}}\) as a function of \(D\) and \(R\) is shown in Figure 3B. Increasing the simulation time \(t_{\text{sim}}\) yields capture and mating probabilities that approach one. As can be seen from Figure 3A, and as can be expected, male moths with very low diffusivity are not able to reach the female within \(t_{\text{sim}}\) unless they are initially very close to her and \(R\simeq 1\).