# Modeling Migratory Flight in the Spruce Budworm: Circadian Rhythm

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Mathematical Model

_{s}. This pattern can be shifted by ambient temperature conditions. We thus develop a time transformation τ around the central time t

_{c}

_{0}and t

_{m}, respectively, and the full likelihood distribution of moth liftoff described by

_{t}= 1 is the maximum at t = t

_{c}or τ = 0 (Figure 1).

_{0}and the time of peak likelihood t

_{c}is used to define Δ

_{0}(in hours):

_{m}can be used with the onset time t

_{0}to define the full duration of liftoff activity Δ

_{f}(in hours):

_{c}, by virtue of Equation (1). Expecting the peak of the moth liftoff period to occur after the time of local sunset, we can define the time difference Δ

_{s}(in hours):

_{s}depends on latitude and time of year.

_{0}to 1 at t

_{m}so that we can calculate the probability of moth liftoff in a short interval. This definite integral is given by:

_{t}remains strictly in the range (0,1) for τ (defined by Equation (1)) in the range (−1, 1). The probability of a moth taking off in an interval Δτ is therefore:

_{0}for Equation (3), Δ

_{f}for Equation (4), and Δ

_{s}for Equation (5).

#### 2.2. Observations and Model Calibration

_{0}, Δ

_{f}, and Δ

_{s}, respectively) must be determined from observations. For this purpose, we assembled 22 historical datasets of diel SBW moth flight observations. Two of these datasets were published previously by Greenbank et al. [10]. The other 20 datasets were obtained from similar observations made in the vicinity of Sault Ste. Marie and Black Sturgeon Lake, ON, by the late C.J. Sanders and his team in 1976 (two datasets), 1987 (eight datasets), and 1989 (10 datasets). In all cases, the number of moths ascending into emigration flight was observed over short periods of time (typically 5 min) from tall scaffolds above the canopy and recorded at intervals of 15 min from early evening (prior to sunset) to total darkness. On some nights, night-vision (infrared) devices were used after total darkness to observe late flying moths, primarily to ensure that those were not numerous. Air temperature was recorded at the scaffold location at the same frequency. Sunset times for each location and observation date were obtained from www.timeanddate.com/sun/canada/ (last accessed 1 October 2019). Because these canopy-level observations did not distinguish between males and females, and because Greenbank et al. [10] reported that the sex ratio of moths caught during emigration liftoff did not vary much with time of night, we made no attempt to model explicitly sex differences in the diel periodicity of SBW flight.

_{0}and Δ

_{s}from the time at which the first (t

_{0}) and the median (50%; t

_{c}) migrating moths were observed on each observation night. The value of Δ

_{f}(duration of the diel liftoff period) was determined from the time of first (t

_{0}) and last (t

_{m}) emigration. The values of Δ

_{s}were related by regression analysis to T

_{19h30}the ambient air temperature (°C) at 19h30 local DST, prior to the earliest observed emigration flights:

_{0}were related by regression analysis to the observed values of Δ

_{s}:

_{f}in hours from first to last moth seen emigrating) was related to the observed values of Δ

_{0}. Given pronounced heteroscedasticity in these observations, we expressed this relationship as

_{f}that, by multiplying with the value of p

_{5}, maximized the R

^{2}between observed and simulated proportions of moths taking flight (with a precision of ±5%). The potential duration of the flight period is then given by

#### 2.3. Simulations

## 3. Results

_{19h30}and the median timing of moth emigration Δ

_{s}(Figure 2a) as described by Equation (11). There was also a significant negative relationship between the timing of the onset of emigration Δ

_{0}and the value of Δ

_{s}(Figure 2b), as described by Equation (12). The average value of parameter p

_{5}in Equation (13) provided a good description of the relationship between observed flight duration Δ

_{f}and the time Δ

_{0}between first flight and peak flight (Figure 2c). Using the calibrated values for model parameters p

_{1}through p

_{5}(Table 1), the goodness-of-fit (R

^{2}) between observed and simulated cumulative liftoff patterns improved from 0.72 to an asymptotic maximum of 0.775 using k

_{f}= 1.35 (Figure 2d) in Equation (14). The resulting simulated patterns of liftoff timing corresponded well with field observations (Figure 3).

_{19h30}= 20 °C), nearly 0.5 h after sunset for a warmer evening (e.g., T

_{19h30}= 25 °C), and even later, nearly 1.5 h after sunset, for a relatively hot evening (e.g., T

_{19h30}= 30 °C). However, the relationships between the wingbeat frequency and temperature, body weight, and wing area also determine how many moths can lift off at any given time. The resulting migration pattern, and its amplitude, are the composite of these effects (Figure 4a). Interestingly, as a result of the liftoff wingbeat relationship (Equations (8) and (9)), the sex ratio (proportion females) of emigrating moths varies with temperature and time during the evening (Figure 4b). Females make up the larger portion of migrants on cooler evenings, and later in the evening, contradicting the observation reported by Greenbank et al. [10] that the sex ratio of moths caught during migration liftoff did not vary much with time of night. The fecundity carried away by emigrating females also varies as a function of temperature and time through the evening (Figure 4c). Heavier females (i.e., those with a greater proportion of their remaining fecundity) can only lift off at warmer temperatures and thus typically earlier in the evening, while lighter females emigrating in cooler air at the end of the evening tend to have lower gravidity.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Dingle, H. Migration: The Biology of Life on the Move, 2nd ed.; Oxford University Press: New York, NY, USA, 2014; 326p. [Google Scholar] [CrossRef]
- Régnière, J.; Delisle, J.; Sturtevant, B.R.; Garcia, M.; St-Amant, R. Modeling migratory flight in the spruce budworm: Temperature constraints. Forests
**2019**, 10, 802. [Google Scholar] [CrossRef] - Hu, G.; Lim, K.S.; Reynolds, D.R.; Reynolds, A.M.; Chapman, J.W. Wind-related orientation patterns in diurnal, crepuscular and nocturnal high-altitude insect migrants. Front. Behav. Neurosci.
**2016**, 10, 32. [Google Scholar] [CrossRef] [PubMed] - Kawahara, A.Y.; Plotkin, D.; Hamilton, C.A.; Gough, H.; St Laurent, R.; Owens, H.L.; Homziak, N.T.; Barber, J.R. Diel behavior in moths and butterflies: A synthesis of data illuminates the evolution of temporal activity. Org. Divers. Evol.
**2018**, 18, 13–27. [Google Scholar] [CrossRef] - Rund, S.; O’Donnell, A.; Gentile, J.; Reece, S. Daily rhythms in mosquitoes and their consequences for malaria transmission. Insects
**2016**, 7, 14. [Google Scholar] [CrossRef] - Broadhead, G.T.; Basu, T.; von Arx, M.; Raguso, R.A. Diel rhythms and sex differences in the locomotor activity of hawkmoths. J. Exp. Biol.
**2017**, 220, 1472–1480. [Google Scholar] [CrossRef] [PubMed][Green Version] - Wellington, W.G. The light reactions of the spruce budworm, Choristoneura fumiferana Clemens (Lepidoptera: Tortricidae). Can. Entomol.
**1948**, 80, 56–82. [Google Scholar] [CrossRef] - Henson, W.R. Mass flights of the spruce budworm. Can. Entomol.
**1951**, 83, 240. [Google Scholar] [CrossRef] - Schaefer, G.W. Radar observations of insect flight. Symp. R. Entomol. Soc. Lond.
**1976**, 7, 157–197. [Google Scholar] - Greenbank, D.O.; Schaefer, G.W.; Rainey, R.C. Spruce budworm (Lepidoptera: Tortricidae) moth flight and dispersal: New understanding from canopy observations, radar and aircraft. Mem. Entomol. Soc. Can.
**1980**, 112, 1–49. [Google Scholar] [CrossRef] - Kipp, L.R.; Lonergan, G.C.; Bell, W.J. Male periodicity and the timing of mating in the spruce budworm (Lepidoptera: Tortricidae): Influence of population density and temperature. Environ. Entomol.
**1995**, 24, 1150–1159. [Google Scholar] [CrossRef] - Cardé, R.T.; Roelofs, W.L. Temperature modification of male sex pheromone response and factors affecting female calling in Holomelina immaculata (Lepidoptera: Arctiidae). Can. Entomol.
**1973**, 105, 1505–1512. [Google Scholar] [CrossRef] - Cardé, R.T.; Comeau, A.; Baker, T.C.; Roelofs, W.L. Moth mating periodicity: Temperature regulates the circadian gate. Experientia
**1975**, 31, 46–48. [Google Scholar] [CrossRef] [PubMed] - Comeau, A.; Cardé, R.T.; Roelofs, W.L. Relationship of ambient temperatures to diel periodicities of sex attraction in six species of Lepidoptera. Can. Entomol.
**1976**, 108, 415–418. [Google Scholar] [CrossRef] - Alerstam, T.; Chapman, J.W.; Bäckman, J.; Smith, A.D.; Karlsson, H.; Nilsson, C.; Reynolds, D.R.; Klaassen, H.G.; Hill, J.K. Convergent patterns of long-distance nocturnal migration in noctuid moths and passerine birds. Proc. R. Soc. B Biol. Sci.
**2011**, 278, 3074–3080. [Google Scholar] [CrossRef] - Krauel, J.J.; Westbrook, J.K.; McCracken, G.F. Weather-driven dynamics in a dual-migrant system: Moths and bats. J. Anim. Ecol.
**2015**, 84, 604–614. [Google Scholar] [CrossRef] - Krauel, J.J.; Brown, V.A.; Westbrook, J.K.; McCracken, G.F. Predator–prey interaction reveals local effects of high-altitude insect migration. Oecologia
**2018**, 186, 49–58. [Google Scholar] [CrossRef] - Sanders, C.J. Daily activity patterns and sex pheromone specificity as sexual isolating mechanisms in two species of Choristoneura (Lepidoptera: Tortricidae). Can. Entomol.
**1971**, 103, 498–502. [Google Scholar] [CrossRef] - Sanders, C.J.; Lucuik, G.S. Effects of photoperiod and size on flight activity and oviposition in the eastern spruce budworm (Lepidoptera: Tortricidae). Can. Entomol.
**1975**, 107, 1289–1299. [Google Scholar] [CrossRef] - Simmons, G.A.; Chen, C.W. Application of harmonic analysis and polynomial regression to study flight activity of Choristoneura fumiferana (Clem.) (Lepidoptera: Tortricidae) in the field. Abstracts, Forty-Seventh Annual Meeting, Eastern Branch Entomological Society of America. J. N. Y. Entomol. Soc.
**1975**, 83, 266. [Google Scholar] - Régnière, J.; Powell, J.; Bentz, B.; Nealis, V. Effects of temperature on development, survival and reproduction of insects: Experimental design, data analysis and modeling. J. Insect Physiol.
**2012**, 58, 634–647. [Google Scholar] [CrossRef] - Wellington, W.G.; Henson, W.R. Notes on the effects of physical factors on the spruce budworm, Choristoneura fumiferana (Clem.). Can. Entomol.
**1947**, 79, 168–170. [Google Scholar] [CrossRef] - Dickison, R.B.B.; Haggis, M.J.; Rainey, R.C. Spruce budworm moth flight and storms: Case study of a cold front system. J. Clim. Appl. Meteorol.
**1983**, 22, 278–286. [Google Scholar] [CrossRef] - Dickison, R.B.B.; Haggis, M.J.; Rainey, R.C.; Burns, L.M.D. Spruce budworm moth flight and storms: Further studies using aircraft and radar. J. Clim. Appl. Meteorol.
**1986**, 25, 1600–1608. [Google Scholar] [CrossRef] - Pedgley, D.E.; Reynolds, D.R.; Riley, J.R.; Tucker, M.R. Flying insects reveal small-scale wind systems. Weather
**1982**, 37, 295–306. [Google Scholar] [CrossRef] - Drake, V.A. The vertical distribution of macro-insects migrating in the nocturnal boundary layer: A radar study. Bound. Layer Meteorol.
**1984**, 28, 353–374. [Google Scholar] [CrossRef] - Drake, V.A. Radar observations of moths migrating in a nocturnal low-level jet. Ecol. Entomol.
**1985**, 10, 259–265. [Google Scholar] [CrossRef] - Reynolds, D.R.; Chapman, J.W.; Edwards, A.S.; Smith, A.D.; Wood, C.R.; Barlow, J.F.; Woiwod, I.P. Radar studies of the vertical distribution of insects migrating over southern Britain: The influence of temperature inversions on nocturnal layer concentrations. Bull. Entomol. Res.
**2005**, 95, 259–274. [Google Scholar] [CrossRef] - Reynolds, D.R.; Smith, A.D.; Chapman, J.W. A radar study of emigratory flight and layer formation by insects at dawn over southern Britain. Bull. Entomol. Res.
**2008**, 98, 35–52. [Google Scholar] [CrossRef] [PubMed] - Chapman, J.W.; Drake, V.A.; Reynolds, D.R. Recent insights from radar studies of insect flight. Ann. Rev. Entomol.
**2011**, 56, 337–356. [Google Scholar] [CrossRef] - Westbrook, J.K.; Eyster, R.S.; Wolf, W.W. WSR-88D doppler radar detection of corn earworm moth migration. Int. J. Biometeorol.
**2014**, 58, 931–940. [Google Scholar] [CrossRef] - Boulanger, Y.; Fabry, F.; Kilambi, A.; Pureswaran, D.S.; Sturtevant, B.R.; Saint-Amant, R. The use of weather surveillance radar and high-resolution three-dimensional weather data to monitor a spruce budworm mass exodus flight. Agric. For. Meteorol.
**2017**, 234, 127–135. [Google Scholar] [CrossRef] - Westbrook, J.K.; Eyster, R.S. Doppler weather radar detects emigratory flights of noctuids during a major pest outbreak. Remote Sens. Appl. Soc. Environ.
**2017**, 8, 64–70. [Google Scholar] [CrossRef] - Sanders, J.C.; Wallace, D.R.; Luicuik, G.S. Flight activity of female eastern spruce budworm (Lepidoptera: Tortricidae) at constant temperatures in the laboratory. Can. Entomol.
**1978**, 110, 627–632. [Google Scholar] [CrossRef] - Régnière, J.; Cooke, B.; Béchard, A.; Dupont, A.; Therrien, P. Dynamics and management of rising outbreak spruce budworm populations. Forests
**2019**, 10, 748. [Google Scholar] [CrossRef] - Baklanov, A.A.; Grisogono, B.; Bornstein, R.; Zilitinkevich, S.S.; Taylor, P.; Larsen, S.E.; Rotach, M.W.; Fernando, H.J.S. The nature, theory, and modeling of atmospheric planetary boundary layers. Bull. Am. Meteorol. Soc.
**2011**, 92, 123–128. [Google Scholar] [CrossRef] - Angevine, W.M. Transitional, entraining, cloudy, and coastal boundary layers. Acta Geophysica
**2008**, 56, 2–20. [Google Scholar] [CrossRef] - Mahrt, L. Stably stratified atmospheric boundary layers. Ann. Rev. Fluid Mech.
**2014**, 46, 23–45. [Google Scholar] [CrossRef] - Mahrt, L. The early evening boundary layer transition. Quart. J. R. Meteorol. Soc.
**1981**, 107, 329–343. [Google Scholar] [CrossRef] - Mahrt, L. Nocturnal boundary-layer regimes. Bound. Layer Meteorol.
**1998**, 88, 255–278. [Google Scholar] [CrossRef] - Mahrt, L. The near-surface evening transition. Quart. J. R. Meteorol. Soc.
**2017**, 143, 2940–2948. [Google Scholar] [CrossRef] - Acevedo, O.C.; Fitzjarrald, D.R. The early evening surface-layer transition: Temporal and spatial variability. J. Atmos. Sci.
**2001**, 58, 2650–2667. [Google Scholar] [CrossRef] - Sastre, M.; Yagüe, C.; Román-Cascón, C.; Maqueda, G. Atmospheric boundary-layer evening transitions: A comparison between two different experimental sites. Bound. Layer Meteorol.
**2015**, 157, 375–399. [Google Scholar] [CrossRef] - Angevine, W.M.; Tjernström, M.; Žagar, M. Modeling of the coastal boundary layer and pollutant transport in New England. J. Appl. Meteorol. Climatol.
**2006**, 45, 137–154. [Google Scholar] [CrossRef] - Nieuwstadt, F.T.M. The turbulent structure of the stable, nocturnal boundary layer. J. Atmos. Sci.
**1984**, 41, 2202–2216. [Google Scholar] [CrossRef] - Acevedo, O.C.; Mahrt, L.; Puhales, F.S.; Costa, F.D.; Medeiros, L.E.; Degrazia, G.A. Contrasting structures between the decoupled and coupled states of the stable boundary layer. Quart. J. R. Meteorol. Soc.
**2016**, 142, 693–702. [Google Scholar] [CrossRef] - Mahrt, L. Microfronts in the nocturnal boundary layer. Quart. J. R. Meteorol. Soc.
**2019**, 145, 546–562. [Google Scholar] [CrossRef] - Reynolds, A.M.; Reynolds, D.R.; Smith, A.D.; Chapman, J.W. Orientation cues for high-flying nocturnal insect migrants: Do turbulence-induced temperature and velocity fluctuations indicate the mean wind flow? PLoS ONE
**2010**, 5, e15758. [Google Scholar] [CrossRef] - Riley, J.R.; Reynolds, D.R.; Rainey, R.C. Radar-based studies of the migratory flight of grasshoppers in the middle Niger area of Mali. Proc. R. Soc. Lond. Ser. B Biol. Sci.
**1979**, 204, 67–82. [Google Scholar] [CrossRef] - Rennie, S.J. Common orientation and layering of migrating insects in southeastern Australia observed with a Doppler weather radar. Meteorol. Appl.
**2014**, 21, 218–229. [Google Scholar] [CrossRef] - Feng, H.; Wu, X.; Wu, B.; Wu, K. Seasonal migration of Helicoverpa armigera (Lepidoptera: Noctuidae) over the Bohai Sea. J. Econ. Entomol.
**2009**, 102, 95–104. [Google Scholar] [CrossRef] - Fu, X.; Zhao, X.; Xie, B.; Ali, A.; Wu, K. Seasonal pattern of Spodoptera litura (Lepidoptera: Noctuidae) migration across the Bohai Strait in northern China. J. Econ. Entomol.
**2015**, 108, 525–538. [Google Scholar] [CrossRef] [PubMed] - Wood, C.R.; Chapman, J.W.; Reynolds, D.R.; Barlow, J.F.; Smith, A.D.; Woiwod, I.P. The influence of the atmospheric boundary layer on nocturnal layers of noctuids and other moths migrating over southern Britain. Int. J. Biometeorol.
**2006**, 50, 193–204. [Google Scholar] [CrossRef] [PubMed] - Wood, C.R.; Reynolds, D.R.; Wells, P.M.; Barlow, J.F.; Woiwod, I.P.; Chapman, J.W. Flight periodicity and the vertical distribution of high-altitude moth migration over southern Britain. Bull. Entomol. Res.
**2009**, 99, 525–535. [Google Scholar] [CrossRef] [PubMed][Green Version] - Wood, C.R.; Clark, S.J.; Barlow, J.F.; Chapman, J.W. Layers of nocturnal insect migrants at high-altitude: The influence of atmospheric conditions on their formation. Agric. For. Entomol.
**2010**, 12, 113–121. [Google Scholar] [CrossRef] - Dreyer, D.; El Jundi, B.; Kishkinev, D.; Suchentrunk, C.; Campostrini, L.; Frost, B.J.; Zechmeister, T.; Warrant, E.J. Evidence for a southward autumn migration of nocturnal noctuid moths in central Europe. J. Exp. Biol.
**2018**, 221, 179218. [Google Scholar] [CrossRef] - Wang, H.-H.; Grant, W.E.; Elliott, N.C.; Brewer, M.J.; Koralewski, T.E.; Westbrook, J.K.; Alves, T.M.; Sword, G.A. Integrated modelling of the life cycle and aeroecology of wind-borne pests in temporally-variable spatially-heterogeneous environment. Ecol. Model.
**2019**, 399, 23–38. [Google Scholar] [CrossRef] - Reynolds, A.M.; Reynolds, D.R.; Smith, A.D.; Chapman, J.W. A single wind-mediated mechanism explains high-altitude “non-goal oriented” headings and layering of nocturnally migrating insects. Proc. R. Soc. Lond. B Biol. Sci.
**2010**, 277, 765–772. [Google Scholar] [CrossRef] - Aralimarad, P.; Reynolds, A.M.; Lim, K.S.; Reynolds, D.R.; Chapman, J.W. Flight altitude selection increases orientation performance in high-flying nocturnal insect migrants. Anim. Behav.
**2011**, 82, 1221–1225. [Google Scholar] [CrossRef] - Riley, J.R. Collective orientation in night-flying insects. Nature
**1975**, 253, 113. [Google Scholar] [CrossRef] - Reynolds, A.M.; Reynolds, D.R.; Riley, J.R. Does a “turbophoretic” effect account for layer concentrations of insects migrating in the stable night-time atmosphere? J. R. Soc. Interface
**2009**, 6, 87–95. [Google Scholar] [CrossRef] - Holtslag, A.A.M.; Svensson, G.; Baas, P.; Basu, S.; Beare, B.; Beljaars, A.C.M.; Bosveld, F.C.; Cuxart, J.; Lindvall, J.; Steeneveld, G.J.; et al. Stable atmospheric boundary layers and diurnal cycles: Challenges for weather and climate models. Bull. Am. Meteorol. Soc.
**2013**, 94, 1691–1706. [Google Scholar] [CrossRef] - Davy, R. The climatology of the atmospheric boundary layer in contemporary global climate models. J. Clim.
**2018**, 31, 9151–9173. [Google Scholar] [CrossRef] - McNider, R.T.; Steeneveld, G.J.; Holtslag, A.A.M.; Pielke, R.A.; Mackaro, S.; Pour-Biazar, A.; Walters, J.; Nair, U.; Christy, J. Response and sensitivity of the nocturnal boundary layer over land to added longwave radiative forcing. J. Geophys. Res. Atmos.
**2012**, 117, D14106. [Google Scholar] [CrossRef] - Horvath, K.; Koracin, D.; Vellore, R.; Jiang, J.; Belu, R. Sub-kilometer dynamical downscaling of near-surface winds in complex terrain using WRF and MM5 mesoscale models. J. Geophys. Res. Atmos.
**2012**, 117, D11111. [Google Scholar] [CrossRef]

**Figure 1.**Circadian model of crepuscular migration liftoff activity (Equation (2)) illustrating terms used to define τ in Equation (1), with t

_{s}: Time of sunset; Δ

_{s}: Time between sunset t

_{s}and peak flight t

_{c}; Δ

_{0}: Time between first flight t

_{0}and peak flight t

_{c}; Δ

_{f}: Duration of the migration liftoff period; and t

_{m}: End of flight period.

**Figure 2.**(

**a**) Relationship between T

_{19h30}the temperature at 19h30 local DST and Δ

_{s}the shift between the time at which 50% of migrating moths were observed and the time of sunset t

_{s}. Line is Equation (11). (

**b**) Relationship between Δ

_{s}and the shift between the first migrating moth and sunset Δ

_{0}. Line is Equation (12). (

**c**) Relationship between Δ

_{0}and the observed duration of the moth migration period Δ

_{f}. Line is Equation (13). (

**d**) Optimization (maximum R

^{2}) of the value of parameter k

_{f}in Equation (14). The value k

_{f}=1.35 yielded R

^{2}= 0.775 between observed and simulated cumulative proportion of moths emigrating over 22 evenings (see Figure 3).

**Figure 3.**Observed (closed circles with dotted lines) and simulated (solid lines) cumulative moths flights with observed evening temperatures (dashed grey lines). (

**a**) and (

**b**) 9 and 16 July 1973 using data from [19]. (

**c**) and (

**d**) 2 and 5 July 1976, from observations by C.J. Sanders near Sault Ste. Marie, Ontario. (

**e**–

**v**): 18 individual nights in July 1987 and 1989, from observations by C.J. Sanders near Black Sturgeon Lake, Ontario.

**Figure 4.**(

**a**) Proportion of moths (black curves) emigrating under three temperature regimes (gray lines) on cool (solid), warm (dotted), and hot (dash-dotted) nights. (

**b**) Proportion female among the moths emigrating at each time step. (

**c**) Average proportion of initial fecundity carried away by emigrating females (Equation (10)).

Parameter | Calibration Value | Equation | Regression Statistics |
---|---|---|---|

p_{1} | −3.8 ± 0.7 h | (11) | F = 22.5; df = 1,20; R^{2} = 0.529; p < 0.001 |

p_{2} | 0.145 ± 0.031 h/°C | ||

p_{3} | −1.267 ± 0.146 h | (12) | F = 4.5; df = 1,20; R^{2} = 0.183; p < 0.047 |

p_{4} | −0.397 ± 0.187 | ||

p_{5} | −2.465 ± 0.152 | (13) | |

k_{f} | 1.35 ± 0.025 | (14) | R^{2} = 0.775 |

© 2019 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**

Régnière, J.; Garcia, M.; Saint-Amant, R. Modeling Migratory Flight in the Spruce Budworm: Circadian Rhythm. *Forests* **2019**, *10*, 877.
https://doi.org/10.3390/f10100877

**AMA Style**

Régnière J, Garcia M, Saint-Amant R. Modeling Migratory Flight in the Spruce Budworm: Circadian Rhythm. *Forests*. 2019; 10(10):877.
https://doi.org/10.3390/f10100877

**Chicago/Turabian Style**

Régnière, Jacques, Matthew Garcia, and Rémi Saint-Amant. 2019. "Modeling Migratory Flight in the Spruce Budworm: Circadian Rhythm" *Forests* 10, no. 10: 877.
https://doi.org/10.3390/f10100877