## 3 Male Random Flight Model  
We begin with modeling the male moth random search for the female-generated pheromone plume. This \"foraging\" stage of the mating process has often been considered independent of the surging/casting navigation processes, and few theoretical studies have focused on it ([PERSON] and [PERSON] [32], [PERSON] and [PERSON] [21], [PERSON] and [PERSON] [61], [PERSON] [62; 63]). The arrival time \(t_{a}\) to first reach the plume however is important, since males may spend a significant portion of the night in the foraging state and since shorter arrival times are indicative of moths that are better suited for mating. Although capturing the plume necessitates finding the female through surging/casting, for simplicity, we assume here that \(t_{a}\) is much larger than the time spent in the plume and study how the capture (and mating) probability depends on male traits relevant to random search, such as detection thresholds and diffusivity. We utilize a two-dimensional setting with female and male moths located on the same plane at a fixed altitude based on the observed lack of male vertical motion in experimental studies, as discussed in Section 2.2. We also model the concentration of female-emitted plume \(c(\mathbf{x},t)\) as a two-dimensional field. This quantity varies in space and time, and it is a function of wind currents and female emission activity. A male moth is assumed to be able to detect the plume if \(c(\mathbf{x},t)\) is larger than a given threshold \(C_{\text{tol}}\), a value that indicates male chemosensitivity to pheromone and is a proxy for the length and/or elaborateness a male's antenna, where longer/more elaborate antennae are able to sense smaller quantities of pheromone ([PERSON] et al. [64]). The spatial region where \(c(\mathbf{x},t)>C_{\text{tol}}\) defines the trapping region \(\Omega(t)\):  
\[\Omega(t)=\{\mathbf{x}:c(\mathbf{x},t)\geq C_{\text{tol}}\}.\]  
Once the plume is detected, the male moth will cease random flight and will surge upwind toward the female. Since lower detection thresholds \(C_{\text{tol}}\) can be used as a proxy for male moths that are able to detect smaller amounts of pheromone and, thus, are higher quality, the trapping region \(\Omega(t)\) is larger for higher quality males. Moths located outside the trapping region are not affected by the plume and are assumed to engage in a two-dimensional random flight with diffusion constant \(D\), which serves as a proxy for flight speed. Similar to \(C_{\text{tol}}\), \(D\) can be considered an indicator of male moth quality: moths with larger \(D\) are more diffusive and are able to locate the plume earlier than moths with lower \(D\). We investigate how \(C_{\text{tol}}\) and \(D\) influence \(t_{a}\), especially when \(C_{\text{tol}}\) and \(D\) are both relatively large or when both are relatively small. These are interesting biological scenarios, since moths with longer and/or more elaborate antennae (smaller thresholds \(C_{\text{tol}}\)) are associated to larger trapping regions but are also greater in body size and less motile (smaller diffusion constants \(D\)) ([PERSON] et al. [64], [PERSON] et al. [16], [PERSON] et al. [37]). The probability density function \(g(\mathbf{x},t)\) that a moth is at \(\mathbf{x}=(x,y)\) outside the trapping region \(\Omega(t)\) at time \(t\) and undergoing random motion satisfies the exterior parabolic problem:  
\[\frac{\partial g}{\partial t} =D\
abla^{2}g, \mathbf{x}\in\mathbb{R}^{2}\setminus\Omega(t),\quad t>0, \tag{1a}\] \[g =0, \mathbf{x}\in\partial\Omega(t),\quad t>0,\] (1b) \[g =\delta(x-x_{0})\delta(y-y_{0}), \mathbf{x}\in\mathbb{R}^{2}\setminus\Omega(t),\quad t=0, \tag{1c}\]  
where \(\mathbf{x}_{0}=(x_{0},y_{0})\) is the initial moth location, \(\Omega(t)\) is the dynamic trapping region defined by the time evolution of \(c(\mathbf{x},t)\), and \(\partial\Omega(t)\) is its boundary. Equation (1c) implies that the moth is initially located outside the trapping region, in \(\mathbb{R}^{2}\setminus\Omega(t)\).