# Moth Mating: Modeling Female Pheromone Calling and Male Navigational Strategies to Optimize Reproductive Success

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## Abstract

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## 1. Introduction

## 2. Moth Mating Mechanisms

#### 2.1. Female Calling Strategies

- calling duration: ranges from 0.5 to 8 h and
- emitted pheromone: ranges from 1 to 30 ng per night.

#### 2.2. Male Navigational Strategies

- male flight speed: ranges from 0.5 to 5 m/s
- male turning rate: ranges from 3 to 4 turns/s

^{2}/s, as also confirmed by other studies on Lepidoptera diffusivity estimated at $D\sim 1$ to ${10}^{-1}$ m

^{2}/s (Watanabe [54], Kareiva [55].) For transitions between flight modes, the time necessary for males to measure pheromone concentrations is less than 100 ms, while the time to realize that the pheromone plume is lost is estimated as 300–500 ms (Stengl [5], de Bruyne and Baker [56], Baker and Vogt [57]). Since the typical time spent in any flight mode varies between seconds and hours, several orders of magnitude larger than the above awareness/decision times, we neglect the time lag between transitions. Finally, studies on pheromone-specific receptor neurons reveal that male moth sensilla are able to detect pheromones at very low concentrations, allowing for males to detect the plume at large distances (Angioy et al. [58], Tabuchi et al. [59]). If we assume a typical plume length of 2 km and estimate male flight speed at 1 m/s, we can also give the rough estimate of 30 min as the time a male moth spends in the female-generated plume. Most nocturnal male moths begin flying in the evening and until the predawn hours, indicating that male activity normally occurs away from the plume or when searching for it (Nowinszky et al. [60]).

## 3. Male Random Flight Model

#### First Arrival and Mating Time

## 4. Numerical Results for the Male Random Flight Model

#### 4.1. Single Female Calling Period

^{−1}. A male moth is assumed to have captured the plume and to have successfully mated once $d\left(t\right)<0$.

#### 4.2. First Arrival Time

#### 4.3. Multiple Female Calling Periods

^{2}/s, resulting in four possibilities:

#### 4.4. Night-by-Night Mating Probabilities

## 5. Plume Navigation Model

#### 5.1. Male Navigation Algorithm

- Random flight: If the male moth does not detect pheromone ($c(\mathbf{x},t)<{C}_{\mathrm{tol}}$) and has never detected pheromone at earlier times ${t}^{\prime}<t$, the moth chooses a random direction $\theta $ drawn uniformly from $[0,2\pi )$. Note that this motion implies that, overall, the moth executes a random flight with effective diffusivity $D\simeq {v}^{2}\Delta t/4$. For the experimentally measured male flight speeds reported in Section 2.2, $v=0.5$ to 5 m/s, and for $\Delta t=1$ s, this corresponds to $D\simeq 1$ to ${10}^{-2}$ m
^{2}/s. - Surging: Upon detection of pheromone signals for $c(\mathbf{x},t)\ge {C}_{\mathrm{tol}}$, the moth will align with the upwind direction of airflow ($\theta =\pi $) with a margin of error $\u03f5$. The updated direction of moment $\theta $ will be selected uniformly from $(\pi -\u03f5,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\pi +\u03f5)$ at each timestep.
- Casting: Upon loss of contact with the pheromone plume for $c(\mathbf{x},t)<{C}_{\mathrm{tol}}$, the moth will search for the plume perpendicular to the direction of airflow ($\theta =\pm \pi /2$) with a margin of error $\u03f5$. The updated direction of movement $\theta $ will be selected uniformly in $(-\pi /2-\u03f5,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}-\pi /2+\u03f5)$ or in $(\pi /2-\u03f5,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\pi /2+\u03f5)$ at each timestep. If the male moth is unable to find the lost pheromone plume after casting for a given time ${t}_{\mathrm{cast}}$, it returns to random flight mode. Otherwise, it returns to surging. Typical total casting times are on the order of 10 s (Martinez et al. [20]), so we choose ${t}_{\mathrm{cast}}=10$ s.
- Mating: Once the male moth is within a radius ${r}_{\mathrm{cap}}$ of a female, the two successfully mate.

#### 5.2. Female-Generated Pheromone Plume Dynamics

#### 5.3. The Interplay of Calling Time and Pheromone Amount on Male Fitness

^{2}and a flight speed v uniformly sampled between 0.5 and 2 m/s.

## 6. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Derivation of Arrival Time to Circular Plumes

## Appendix B. Asymptotic of the Survival Probability at Short Times

## Appendix C. MothPy Simulations

- female pheromone puff release rate:$${f}_{\mathrm{r}}=\{50,66.66,83.33,100,116.66,133.33,150,166.66,183.33,200\}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{s}}^{-1}$$
- female pheromone puff molecular amount:$${m}_{\mathrm{p}}=\{0.1,0.68,1.26,1.84,2.42,3.00,3.58,4.16,4.74,5.32,6.48,8.80,13.44,22.72,41.28\}\phantom{\rule{3.33333pt}{0ex}}\begin{array}{c}\hfill \mathrm{pg}\end{array}$$
- male ground flight speed:$$v=\{50,140,230,320,410,500,590\}\phantom{\rule{3.33333pt}{0ex}}\mathrm{cm}/\mathrm{s}$$
- male pheromone detection threshold:$${C}_{\mathrm{tol}}=\{120,500,880,1260,1640,2020,2400\}\phantom{\rule{3.33333pt}{0ex}}\begin{array}{c}\hfill {\mathrm{pg}/\mathrm{cm}}^{2}\end{array}$$

**Figure A1.**Successful mating percentage using MothPy [75] as described in Appendix C. Velocities v are measured in cm/s; female pheromone amounts ${m}_{\mathrm{p}}$ are measured in pg; pheromone puff release rates ${f}_{\mathrm{r}}$ are measured is s${}^{-1}$; and pheromone detection thresholds ${C}_{\mathrm{tol}}$ are measured in pg/cm${}^{2}$.

**Figure A2.**First male arrival time to the female, normalized by the total simulation time ${t}_{\mathrm{sim}}=20$ s using MothPy [75] as described in Appendix C. White regions correspond to no male moths reaching the female within ${t}_{\mathrm{sim}}$, velocities v are measured in cm/s; female pheromone amounts ${m}_{\mathrm{p}}$ are measured in pg; pheromone puff release rates ${f}_{\mathrm{r}}$ are measured is s${}^{-1}$; and pheromone detection thresholds ${C}_{\mathrm{tol}}$ are measured in pg/cm${}^{2}$.

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**Figure 1.**Schematic of four female moth calling strategies, expressed as female calling effort over a typical seven night call. Calling effort is quantified as the time females spent calling males and/or the pheromone titer.

**Figure 2.**(

**A**) Typical profiles for $P\left(t\right),C\left(t\right)=1-P\left(t\right)$, and $S\left(t\right)$ as derived from Equation (5), for D = 1 m/s

^{2}, ${r}_{\mathrm{mate}}=1$ m, and ${r}_{0}=5$ m. The arrival time distribution $S\left(t\right)$ is plotted on an axis with a smaller scale for clarity (red axis, right). (

**B**) Example numerical simulation for 6 independent male moths (various colors; D = 1 m/s

^{2}), with ${r}_{\mathrm{mate}}=1$ m (solid circle) and ${r}_{0}=5$ m (dotted circle).

**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 $C\left(t\right)$ 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 $C\left(t\right)$ plotted via contours in $\{R,D\}$ space.

**Figure 4.**Male arrival time distributions evaluated in $\{R,D\}$ phase space, where $R={r}_{\mathrm{mate}}/{r}_{0}$. (

**A**) Mean first arrival times $\mathbb{E}\left[{t}_{\mathrm{a}}\right]$ calculated from Equation (13a) for $N=1000$ moths. (

**B**) Mean first arrival times $\mathbb{E}\left[{t}_{\mathrm{a}}\right]$ calculated from numerical simulations with $N=1000$ arrivals and ${t}_{\mathrm{call}}=1800$ s (Section 4.1). (

**C**) Mean arrival times $\mathbb{E}\left[t\right]$ calculated from numerical simulations. Mean first arrival times $\mathbb{E}\left[{t}_{\mathrm{a}}\right]$ are the lowest for large diffusivities D and for $R\to 1$, as can be expected. Note the nine orders of magnitude variation for $\mathbb{E}\left[{t}_{\mathrm{a}}\right]$ across the $\{R,D\}$ ranges considered. The mean arrival time $\mathbb{E}\left[t\right]$ is generally much larger than $\mathbb{E}\left[{t}_{\mathrm{a}}\right]$ and only spans three orders of magnitude in the same $\{R,D\}$ space.

**Figure 5.**Effectiveness of each of the four female calling strategies described in Section 2.1 and Figure 1. Capture and mating probability $C\left(t\right)$ for four representative moths with $\{R,D\}=\{0.75,{10}^{-1.5}$ m

^{2}/s} (blue curve), {$0.75,{10}^{-3.5}$ m

^{2}/s} (red curve), {$0.25,{10}^{-1.5}$ m

^{2}/s} (yellow curve), and {$0.25,{10}^{-3.5}$ m

^{2}/s} (purple curve). Calling times ${t}_{\mathrm{call}}\left(k\right)$ for $1\le k\le 7$ nights are specified in Section 4.3. Male moths of the highest quality display similar capture and mating probability across female strategies, with $C\left(t\right)\simeq 1$ in all cases. For intermediate quality moths $C\left(t\right)$ depends on the chosen female strategy.

**Figure 6.**Effectiveness (mating probability/time spend calling) of the four female calling strategies. Each dot corresponds to a night. The increase strategy is the most effective for small calling times, while all strategies have similar effectiveness for large calling times.

**Figure 7.**Night-by-night and total mating probabilities for the increase calling strategy. Note the different likelihood scales for each panel. On the first night, the mating likelihood ${\mathcal{M}}_{1}$ is highest for high-quality moths (large $\{R,D\}$), and lower for intermediate and low-quality moths (medium and low $\{R,D\}$). The latter reaches the female with relatively large likelihoods on the second and successive nights, as can be seen for ${\mathcal{M}}_{\ell}$ with $\ell >1$. The last panel displays the total capture and mating likelihood at the end of the one-week mating period.

**Figure 8.**Night-by-night and total mating probabilities under decrease calling. Note the different likelihood scales for each panel. Similar to Figure 7, intermediate-quality moths (medium $\{R,D\}$) display relatively large mating likelihoods ${\mathcal{M}}_{\ell}$ on the second and successive nights.

**Figure 9.**Night-by-night and total mating probabilities under constant calling. Note the different likelihood scales for each panel. Similar to Figure 7, intermediate-quality moths (medium $\{R,D\}$) display relatively large mating likelihoods ${\mathcal{M}}_{\ell}$ on the second and successive nights.

**Figure 10.**Night-by-night and total mating probabilities for the hat calling strategy. Note the different likelihood scales for each panel. Similar to Figure 7, intermediate-quality moths (medium $\{R,D\}$) display relatively large mating likelihoods ${\mathcal{M}}_{\ell}$ on the second and successive nights.

**Figure 11.**(

**A**) Schematic of the agent-based Plume Navigation Model as described in Section 5. A stationary female at the origin releases pheromone puffs while male moths fly upwind toward her. Note the capture radius ${r}_{\mathrm{cap}}$ and the presence of an advecting wind. (

**B**) Representative female pheromone plume and male flight trajectories in our Plume Navigation Model simulations as described in Section 5.

**Figure 12.**Simulation results from the agent-based Plume Navigation Model described in Section 5 for $N=\mathrm{10,000}$ moths. The range of the male quality trait ${C}_{\mathrm{tol}}$ is ${10}^{-10}$ to ${10}^{-6}$pg/m

^{2}; for v, it is $0.5$ to 2 m/s. (

**A**) Contour plot of mating probability as a function of female calling effort in $\{{t}_{\mathrm{call}},{m}_{\mathrm{p}}\}$ phase space, regardless of male quality: The panel tallies moths with all levels of ${C}_{\mathrm{tol}}$ and v that reach the female. (

**B**) Contour plot of the mean speed $\mathbb{E}[v]$ of males upon reaching the female in $\{{t}_{\mathrm{call}},{m}_{\mathrm{p}}\}$ phase space and for any level of male ${C}_{\mathrm{tol}}$. (

**C**) Contour plot of the mean detection threshold $\mathbb{E}\left[{C}_{\mathrm{tol}}\right]$ of males upon reaching the female in $\{{t}_{\mathrm{call}},{m}_{\mathrm{p}}\}$ phase space for any level of male v. Male arrival likelihood is highly dependent on ${t}_{\mathrm{call}}$: modulating ${t}_{\mathrm{call}}$ allows the female to select for male speed v, and modulating ${m}_{\mathrm{p}}$ allows the female to select for ${C}_{\mathrm{tol}}$.

**Figure 13.**Simulation results from the agent-based Plume Navigation Model described in Section 5 for $N=\mathrm{10,000}$ moths attempting to reach a female calling for ${t}_{\mathrm{call}}=8$ h via the continued release of ${m}_{\mathrm{p}}=1$ pg per second. (

**A**) Contour plot of mean first arrival time $\mathbb{E}[{t}_{\mathrm{a}}]$ of males in $\{{C}_{\mathrm{tol}},v\}$ phase space. (

**B**) Contour plot of the mean arrival time of males in $\{{C}_{\mathrm{tol}},v\}$ phase space. Both mean first arrival times and mean arrival times are measured in hours.

Symbol | Representation | Value/Units |
---|---|---|

$\mathbf{x}$ | Position of moth | m |

${t}_{\mathrm{a}}$ | First arrival and mating time | s |

Male Random Flight Model | ||

$\Delta {t}_{\mathrm{max}}$ | Largest simulation time step | ${10}^{-2}$ s |

$\Delta {t}_{\mathrm{min}}$ | Smallest simulation time step | ${10}^{-5}$ s |

${t}_{\mathrm{sim}}$ | Total simulation time | 100 s |

N | Number of male moths | 1000 to 5000 |

${r}_{\mathrm{mate}}$ | Mating radius | 0 to 1 m |

${r}_{0}={({x}_{0}^{2}+{y}_{0}^{2})}^{1/2}$ | Initial position of male moth | 1 m |

${r}_{\mathrm{mate}}$ | Mating radius | 0 to 1 m |

$R={r}_{0}/{r}_{\mathrm{mate}}$ | Male initial position to mating radius ratio | dimensionless |

D | Male moth diffusion constant | ${10}^{-1}$ to ${10}^{-4}$ m^{2}/s |

Plume Navigation Model | ||

$\Delta t$ | Time step | 1 s |

N | Number of male moths | 10,000 |

${\mathbf{x}}_{p}$ | Center of wind-advected plume | m |

v | Speed of male moths | 0.5 to 2 m/s |

${C}_{\mathrm{tol}}$ | Pheromone detection threshold | ${10}^{-10}$ to ${10}^{-6}$ pg/m^{2} |

$\u03f5$ | Casting and surging margin of error | $\pi /6$ |

${t}_{\mathrm{cast}}$ | Maximum time spent casting | 10 s |

${m}_{\mathrm{p}}$ | Pheromone amount per puff | 1 to 10 pg |

${f}_{\mathrm{r}}$ | Puff release rate | 1 puff/s |

${\sigma}_{0}$ | Puff growth parameter | 0.1 m/s |

U | Wind velocity | 5 m/s |

${r}_{\mathrm{cap}}$ | Capture distance | 5 m |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Stepien, T.L.; Zmurchok, C.; Hengenius, J.B.; Caja Rivera, R.M.; D’Orsogna, M.R.; Lindsay, A.E.
Moth Mating: Modeling Female Pheromone Calling and Male Navigational Strategies to Optimize Reproductive Success. *Appl. Sci.* **2020**, *10*, 6543.
https://doi.org/10.3390/app10186543

**AMA Style**

Stepien TL, Zmurchok C, Hengenius JB, Caja Rivera RM, D’Orsogna MR, Lindsay AE.
Moth Mating: Modeling Female Pheromone Calling and Male Navigational Strategies to Optimize Reproductive Success. *Applied Sciences*. 2020; 10(18):6543.
https://doi.org/10.3390/app10186543

**Chicago/Turabian Style**

Stepien, Tracy L., Cole Zmurchok, James B. Hengenius, Rocío Marilyn Caja Rivera, Maria R. D’Orsogna, and Alan E. Lindsay.
2020. "Moth Mating: Modeling Female Pheromone Calling and Male Navigational Strategies to Optimize Reproductive Success" *Applied Sciences* 10, no. 18: 6543.
https://doi.org/10.3390/app10186543