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# Effects of intense storm events on dolphin occurrence and foraging behavior

## Abstract

As storms become increasingly intense and frequent due to climate change, we must better understand how they alter environmental conditions and impact species. However, storms are ephemeral and provide logistical challenges that prevent visual surveys commonly used to understand marine mammal ecology. Thus, relatively little is known about top predators’ responses to such environmental disturbances. In this study, we utilized passive acoustic monitoring to characterize the response of bottlenose dolphins to intense storms offshore Maryland, USA between 2015 and 2017. During and following four autumnal storms, dolphins were detected less frequently and for shorter periods of time. However, dolphins spent a significantly higher percentage of their encounters feeding after the storm than they did before or during. This change in foraging may have resulted from altered distributions and behavior of their prey species, which are prone to responding to environmental changes, such as varied sea surface temperatures caused by storms. It is increasingly vital to determine how these intense storms alter oceanography, prey movements, and the behavior of top predators.

## Introduction

Coastal shelf marine ecosystems are both highly productive and particularly susceptible to disturbances through natural events and anthropogenic activities. Disturbances, such as storm events, often alter sound or nutrient levels in the environment1, increase turbulent mixing of the water column, and rapidly change ocean salinity and temperature gradients2,3. Such environmental changes can affect the distribution and behavior of a diverse array of shelf fauna4,5 and have long-lasting effects on inshore ecosystems such as coral reefs6 and other hard-bottom habitats. Extreme storm events are expected to become more frequent7,8 and intense8,9,10 as human-driven climate change continues to alter oceanic conditions. Determining how apex predators respond to storms is an important component in understanding how climate change mediates top-down effects in marine ecosystems.

Opportunistically recorded changes in several marine species’ behavior and distribution have been attributed to storms. Passive acoustic monitoring revealed that spotted seatrout11 (Cynoscion nebulosus) and sand seatrout4 (Cynoscion arenarius) altered their vocalizations and shifted the timing of their spawning activity both during and immediately after hurricanes. Temporary and permanent range shifts in several shark12 and fish species13,14 have also been observed following intense storm activity, although the response varied in duration and amongst species and age classes. More extreme impacts have included increased mortality or decreased abundance in macrobenthic organisms because of changes in oxygen concentrations15 and in juvenile fish13 and sessile invertebrates16 as a result of wave action. The environmental drivers of many of these organismal responses, however, are not fully understood.

As top predators, marine mammals may be directly affected by bottom-up changes, environmental changes caused by storm events or indirectly through changes in their prey. Storm events were considered the direct cause of bottlenose dolphin (Tursiops truncatus) movement into shallower waters17,18 and lower adult survival probabilities for manatees (Trichechus manatus latirostris) in years with intense hurricanes19. Following two hurricanes, Bahamian dolphin populations experienced 30% population loss, but it is not known whether this was due to emigration and/or mortality20. Hurricanes in the Mississippi Sound area had variable effects on short21 and long-term dolphin foraging activities22. In addition to altering salinity and temperature gradients, storms can alter upwelling patterns and lead to harmful algal blooms23,24 including lethal marine mammal domoic acid poisoning25. Increased sound levels from anthropogenic sources caused short-term changes in odontocete foraging26,27,28 and communication28,29,30. While we increasingly understand how anthropogenic noise impacts marine mammals, it is not yet clear how elevated sound levels from natural sources, such as those caused by high wind and wave action during storms, affects their behavior and ecology.

In this study, we analyzed passive acoustic detections of bottlenose dolphin calls within the US Mid-Atlantic Bight to determine whether dolphin occurrence and foraging behavior were affected by intense storm events (Table 1, Fig. 1b). Passive acoustic monitoring is advantageous to traditional visual monitoring techniques, especially during storm events, because it does not require favorable viewing conditions for marine mammal detection and dramatically extends the observation period. The US Mid-Atlantic Bight is a large shelf region in the Northwest Atlantic Ocean spanning subtropical and temperate zones that are susceptible to rapid changes in ocean conditions caused by autumnal storm events. During these intense storm events, flow conditions change, environmental noise is increased, and shelf waters rapidly destratify causing large temperature changes14,31. Common dolphins (Delphinus delphis), spotted dolphins (Stenella frontalis), and bottlenose dolphins are present in this region32. During the summer and fall, bottlenose dolphins are the primary species sighted32,33,34,35, and they feed on a variety of fish and squid species36,37. While bottlenose dolphins and other marine mammals are highly mobile, adaptable, and potentially less directly impacted by changes in water temperature than ectothermic species, they could be indirectly affected by alterations in prey behavior and distribution. We hypothesized that dolphin detections and foraging behavior in the Mid-Atlantic Bight would consequently be reduced during and following intense storm events such as tropical storms due to altered prey distribution14 and avoidance of the storm. In the future, marine ecosystems may be increasingly exposed to disturbances such as storm events, which are expected to occur with higher frequency and intensity as the climate changes8.

## Results

Here we present results based on the mixed-effect estimates, $${\widehat{b}}_{j}$$ (j = 0, 1, 2), from model (1) for each metric. There were significantly fewer daily dolphin encounters during (mean = 2.80, n = 17, p < 0.01) and after (mean = 5.26, n = 65, p = 0.02) storms compared to before storms (mean = 6.77, n = 97; Table 2, Fig. 2a). When dolphins did occur during and following storms, encounters were significantly shorter than they were before the storms [mean = 2.55 log(min), n = 11, p = 0.04 ; mean = 2.76  log(min), n = 65, p = 0.02, respectively; Table 2, Fig. 2b]. However, the percentage of time dolphins spent foraging during an encounter increased after the storms (mean = 32.75, n = 64, p = 0.02; Table 2, Fig. 2c). Ambient sound levels also increased significantly during storms (mean = 119.74 SPL rms, n = 17, p < 0.01; Table 2, Fig. 2d), and how the storm affected SSTa varied (Fig. 2e).

For all models, ANOVA F-tests confirmed that the means of each variable were statistically different by period (the p-values for the individual models were adjusted for the number of tests using the Benjamini–Hochberg procedure38). Mean values were obtained from dolphin occurrence and behavior metrics across periods, with p-values accounting for the random or autocorrelation effects structure in the final models (Table 2). Residual diagnostics plots were used to check homogeneity, possible covariance, and normality of model residuals. One of the models (for encounter duration) required a logarithmic transformation of the response to achieve normality of the residuals; however, both the transformed and untransformed models indicated the high statistical significance of the storm effects.

For the 4 days in 2017 on which the C-POD detected dolphins but did not indicate foraging, concurrent sound files from the SM3M acoustic recorder were examined to verify that no detectable foraging occurred. The C-POD, as a conservative classifier of dolphin clicks, missed two feeding buzzes. On the loudest days in 2016 and 2017, dolphin presence and foraging were detected despite elevated ambient sound levels during the storms.

## Discussion

Our study found that storms in the US Mid-Atlantic Bight caused sea surface temperature anomalies to vary, as was expected based on the mixing of the stratified water column with cooler bottom water, thus lowering the surface temperature, as found in other studies14. While lower water temperatures are unlikely to have directly affected the dolphins’ behavior, their ectothermic prey species are influenced by these changes14,39. The resulting changes in prey distribution may have indirectly influenced dolphin movements and foraging. The significantly lower dolphin occurrence and shorter encounters indicate that dolphins had to search more widely for prey during the storms or alter their foraging strategies, but were able to increase the percentage of time spent foraging after the storms compared to before storms (Table 2, Fig. 2c). They could also be compensating for reduced foraging during the storm, taking advantage of the changed behavior and disorientation of fish following the disturbance and changed conditions caused by the storm.

Bottlenose dolphins are highly mobile animals that can have large ranges of 500 km or more40 and are generalist carnivores36,37. Their diet is composed of cephalopods, crustaceans, and large and small fish species36,37,41. The movements and concentrations of their prey are influenced by a variety of environmental factors including water temperature14, salinity15, oxygen concentration15,42, barometric pressure, and wind speeds12, all of which are altered by storms. In California, bottlenose dolphins extended their range in response to prey shifts during El Niño43. Similar short-term changes in dolphin presence, as assumed from fewer recorded dolphin vocalizations, were also shown in our study. However, this reduction in recorded vocalizations may have also resulted from changes in group size or vocal behavior. Further research on the foraging techniques, prey choices, and specific prey responses to storms would improve our understanding of the cause of top predators’ responses to these disturbances.

Fish species exhibit highly variable responses to storms. While environmental changes elicited permanent range shifts in some prey species14, short-term distributional changes (on the scale of days) were found in others12,44. Vertical and horizontal range shifts in fish to avoid hypoxic zones caused by storm events have also been observed41. During Tropical Storms Jose and Maria, intense wave action caused gray triggerfish (Balistes capriscus) in North Carolina to increase their emigration rates44. Concurrent research in our study region found that black sea bass (Centropristis striata) evacuated the region as a result of Tropical Storm Hermine in 201614. Water column mixing and changing bottom temperatures initiated a migration of 40% of tagged black sea bass and caused reduced activity rates by those that remained14. When environmental changes, including those from storm events, result in fish that are stunned, less active, or concentrated in a region, it potentially increases their catchability by predators. This prey behavior is a possible cause of increased dolphin foraging following storm events. After two intense hurricanes in Mississippi, bottlenose dolphin foraging increased 15% over pre-hurricane levels and remained heightened for 2 years, potentially due to increased biodiversity, decreased vessel traffic, or less fishing as a result of the storm22.

Although storms may create unique foraging opportunities for dolphins, severe wave action from storms has displaced dolphins17,18,45 and caused natural ambient sound levels to peak during storms. Exposure to elevated sound levels from anthropogenic sources can result in abandonment of important habitat46 or a diminished ability to compensate for short-term changes in activities such as feeding47. Should dolphins experience range shifts due to direct displacement or in search of prey, they may be exposed to changes in pollutants, pathogens45, and predators48. Interspecific dynamics20 and biodiversity have also been affected by range shifts in these top predators48. Bottlenose dolphin and spotted dolphin (Stenella frontalis) populations in the Bahamas decreased in size by 30% following two hurricanes, likely due to death and/or emigration20. These dramatic effects are increasingly likely as dolphins are forced to cope with the cumulative stress of altered prey distributions, physical and acoustic disturbances, and changing environmental conditions49.

While dolphins were detected throughout our study period, including on days with high ambient sound levels (up to 123 dB re 1 µPa rms), elevated ambient sound levels during the storms likely masked dolphin clicks50,51 or decreased the detection range of the C-POD52. The C-POD’s KERNO classifier is also a conservative classifier of dolphin clicks53. While no dolphins were detected during the 2 loudest days of our study period (average sound levels of up to 126 dB re 1 µPa rms in 2015), on the loudest days in 2016 and 2017, dolphin presence and foraging were detected despite elevated ambient sound levels during the storm. A previous study in this region showed that the C-POD’s ability to detect dolphins (based on click trains between 20 and 160 kHz) was not significantly affected by broadband (2 Hz–24 kHz, up to 130 dB re 1 µPa rms) ambient sound levels53. Storm noise, which has most of its energy in lower frequencies, did not appear to greatly reduce detections by the C-POD. Our review of concurrent sound files from the SM3M acoustic recorder similarly indicated decreased occurrence of feeding buzzes rather than masking by the storm noise. The C-POD’s passive acoustic click data allowed us to measure the number of dolphin encounters per day, but we were unable to determine the number or identity of dolphins in the area at any time. As a result, we could not measure the effects of storms on dolphin abundance. Dolphin signature whistles recorded with devices like the SM3M, however, allow identification of individuals54 and could offer future insight into how many dolphins were impacted by storm events and whether they permanently evacuated or returned.

Understanding the effects of storms on top predators such as marine mammals can serve as an indicator for the overall resiliency of the ecosystem they inhabit. Yet the key to understanding trophic level responses is rooted in understanding the changes in environmental conditions. As storms become increasingly frequent and intense due to climate change, it is vital to determine how storms alter oceanography and prey movements. This will provide a more complete understanding of storms as drivers of behavioral changes in dolphins and other top predators.

## Methods

### Study area

Our study area was located in the US Mid-Atlantic Bight between 12 and 63 km east of Ocean City, Maryland, USA (Fig. 1a). At the four sites in this area (Site 1: nearest inshore—Site 4: farthest offshore, Fig. 1a), water depths ranged from approximately 20–45 m, and acoustic recording instruments were deployed with the hydrophone located approximately 1.5 m above the ocean floor using bottom-anchored moorings (see53,55 for mooring configurations). Automatic dolphin click detectors (C-PODs; Chelonia Ltd., UK) were deployed at all four sites from 2015 through 2017, a Marine Autonomous Recording Unit (MARU; Cornell Lab of Ornithology Bioacoustics Research Program, Ithaca, NY) continuously monitored sound levels at a sampling rate of 2 kHz [15 Hz–1 kHz] at Site 2 from 2015 through 2017, and an SM3M acoustic recorder (Wildlife Acoustics, Maynard, MA, USA) sampling at 48 kHz [2 Hz–24 kHz] was deployed at Site 2 from May of 2016 through 2017 to verify dolphin detections (Fig. 1a).

### Storm and environmental data

One strong extratropical storm (2015) and three classified tropical storms (2016 and 2017) that occurred during the late summer and autumn near the study area were tracked using data obtained from the U.S. National Oceanic and Atmospheric Administration’s (NOAA’s) Tropical Cyclone Reports (www.nhc.noaa.gov/data/#tcr) and wind speeds were obtained from NOAA’s National Buoy Data Center (www.ndbc.noaa.gov; Station OCIM2; Table 1). Late summer and autumn storms (16 August–15 October) were chosen for analysis because dolphins are most common in this coastal region during that period33,34,56. In 2015, a strong storm event took place within the study area on 1–4 October with wind speeds up to 13 m/s and gusts up to 22 m/s (Table 1). In 2016, Tropical Storm Hermine passed through the study area on 3–8 September with wind speeds up to 11 m/s and gusts up to 19 m/s (Fig. 1b; Table 1). In 2017, Tropical Storms Jose and Maria moved through the region on 19–23 and 27–28 September, respectively, with wind speeds up to 12 m/s and 11 m/s and gusts up to 19 m/s and 15 m/s, respectively (Fig. 1b; Table 1). The days from 16 August to the date of storm arrival were categorized as the period before the storms (Table 1). The days on which the storms’ winds affected conditions in the study area (1–4 October 2015, 3–8 September 2016, and 19–23 and 27–28 September 2017) were categorized as the period during the storms, and the days since the storms’ departure to 15 October were categorized as the periods after the storms. The period of 24–26 September 2017 was omitted to ensure the before and after periods were at least 10 days long (Table 1).

Sea surface temperature anomaly data (°C, SSTa) were downloaded from NOAA’s ERDDAP database57. These data were provided by the Group for High Resolution Sea Surface Temperature (GHRSST), and SSTa were calculated as a retrospective dataset, comparing daily temperatures with a 1 km spatial resolution to historically averaged climatology between 2003 and 201458.

### Acoustic data analysis

#### Dolphin detections

C-PODs were utilized to determine the occurrence of bottlenose dolphins. C-PODs continuously monitor for dolphin click trains (five or more clicks) between 20 and 160 kHz, detecting dolphins up to 1.8 km away52. Only click trains classified as being of high and moderate qualities by the C-POD’s KERNO cetacean classifier were included in the analysis. C-PODs were deployed at each of the four sites during the 2015 through 2017 study period (Fig. 1a). To determine whether storms affected the C-POD’s dolphin detection ability, an SM3M acoustic recorder sampling at 48 kHz ([2 Hz–24 kHz]; hydrophone sensitivity: 165 dB re 1 V/µPa, gain 12 dB re 1 µPa) was also deployed on the mooring at Site 2 in 2016. Spectrograms from the SM3M recordings were manually reviewed in Raven Pro 1.5 (Cornell Lab of Ornithology Bioacoustics Research Program, Ithaca, NY) to determine whether dolphin foraging occurred during storm events in 2016 and 2017. The SM3M was not yet gathering data when the 2015 storm occurred.

#### Dolphin occurrence and behavior metrics

To determine how dolphin behaviors changed as a result of storms, behavior metrics were calculated from the C-POD data. These metrics were the total number of encounters per day, average daily encounter length, and daily average proportion of minutes per encounter spent foraging. Because C-PODs are conservative indicators of dolphin presence53, an encounter was defined as a sequence of detections with a new encounter being assigned when there were more than 37 min between detections (as described in56). Foraging behavior was defined as the occurrence of an inter-click interval (ICI) of 9.9 ms or less56. Any minute in which this short ICI, a feeding buzz26,56,59, occurred was labeled a foraging minute. Feeding buzzes are associated with the majority of prey interceptions in bottlenose dolphins60. The percentage of minutes spent foraging during an encounter was calculated for each encounter and these percentages were averaged at the daily scale for analysis.

#### Ambient sound levels

Underwater sounds were analyzed from the MARU, which recorded continuously at a 2 kHz sampling rate [15 Hz–1 kHz]. Because sampling was performed continuously, a very low sampling rate was selected to maximize battery life. While dolphins do not echolocate in such low frequencies, they can hear within this range61, and much of the ambient sound energy is within this bandwidth62. The MARU was anchored in the vicinity of the C-POD at Site 2. Acoustic data were processed using the Raven-X toolbox63 in MATLAB (Mathworks, Inc.). The sound pressure level was used to calculate the root-mean-square (rms) pressure within one-hour time bins to represent the ambient sound levels. These binned values were averaged to obtain the mean daily ambient sound levels. As the relationship between ocean ambient sound levels and wind speed, especially during storms, has been well established64,65, sound level was used as an indication of storm intensity.

### Statistical analysis

Linear mixed-effects models66 were used to determine whether any change in dolphin behavior was associated with storms. The model for each metric included all 3 years, and the year was treated as a random effect to account for inter-annual differences. The models determined the effect of the period (before, during, and after the storm) on each environmental (sound level and SSTa) and dolphin (number of dolphin encounters, encounter length, percentage of minutes per encounter spent foraging) metric. The linear mixed-effects model took the form:

$$Y_{i} = b_{0} + b_{j} + \alpha_{Year} + \varepsilon_{ij} ,$$
(1)

where Yi is the dolphin encounter or environmental metric, i is the date of the observation, b0 is the grand intercept or the reference level during the period before the storm, bj is the effect of storm period relative to the period before the storm (j = 1, 2 for during and after the storm), αYear is the random intercept varying by year to account for differences in mean responses simply due to the use of different years of data. The residuals εij were assumed to be uncorrelated, with a mean of 0 and variance σ2 (Var(εij) = σ2), or with heterogeneous variance by period (Var(εij) = $${\sigma }_{j}^{2}$$), or autocorrelated and following an autoregressive process of first order, AR(1) (Cov(εij, εi−1,j) = φσ2, where |φ|< 1 is the autoregressive coefficient).

Models were fit in the statistical software R (Version 3.5.2; R Core Team, 2018) using the package lme467. The final model for each response metric was chosen based on the smallest value of Akaike information criterion (AIC) from the alternative model specifications (presence or absence of the random intercept and different structure of the residual covariance).

## References

1. 1.

Harley, C. D. G. et al. The impacts of climate change in coastal marine systems. Ecol. Lett. 9, 228–241 (2006).

2. 2.

Gong, D., Kohut, J. T. & Glenn, S. M. Seasonal climatology of wind-driven circulation on the New Jersey Shelf. J. Geophys. Res. Oceans 115(C4) (2010).

3. 3.

Glenn, S. M. et al. Stratified coastal ocean interactions with tropical cyclones. Nat. Commun. 7, 10887 (2016).

4. 4.

Locascio, J. V. & Mann, D. A. Effects of Hurricane Charley on fish chorusing. Biol. Let. 1, 362–365 (2005).

5. 5.

Fiedler, P. C. et al. Effects of a tropical cyclone on a pelagic ecosystem from the physical environment to top predators. Mar. Ecol. Prog. Ser. 484, 1–16 (2013).

6. 6.

Harmelin-Vivien, M. L. The effects of storms and cyclones on coral reefs: A review. J. Coastal Res. 211–231 (1994).

7. 7.

Easterling, D. R. et al. Climate extremes: Observations, modeling, and impacts. Science 289, 2068–2074 (2000).

8. 8.

International Panel on Climate Change (IPCC). Climate Change 2007: Impacts, Adaptation and Vulnerability: Contribution of Working Group II to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change (Cambridge University Press, 2007).

9. 9.

Jentsch, A., Kreyling, J. & Beierkuhnlein, C. A new generation of climate change experiments: Events, not trends. Front. Ecol. Environ. 5, 365–374 (2007).

10. 10.

Knutson, T. R. et al. Tropical cyclones and climate change. Nat. Geosci. 3, 157–163 (2010).

11. 11.

Biggs, C. R., Lowerre-Barbieri, S. K. & Erisman, B. Reproductive resilience of an estuarine fish in the eye of a hurricane. Biol. Let. 14, 20180579 (2018).

12. 12.

Udyawer, V., Chin, A., Knip, D. M., Simpfendorfer, C. A. & Heupel, M. R. Variable response of coastal sharks to severe tropical storms: Environmental cues and changes in space use. Mar. Ecol. Prog. Ser. 480, 171–183 (2013).

13. 13.

Lassig, B. R. The effects of a cyclonic storm on coral reef fish assemblages. Environ. Biol. Fishes 9, 55–63 (1983).

14. 14.

Secor, D. H., Zhang, F., O’Brien, M. H. & Li, M. Ocean destratification and fish evacuation caused by a Mid-Atlantic tropical storm. ICES J. Mar. Sci. 76, 573–584 (2019).

15. 15.

Boesch, D. F., Diaz, R. J. & Virnstein, R. W. Effects of Tropical Storm Agnes on soft-bottom macrobenthic communities of the James and York estuaries and the lower Chesapeake Bay. Chesapeake Sci. 17, 246–259 (1976).

16. 16.

Walker, S. J., Degnan, B. M., Hooper, J. N. A. & Skilleter, G. A. Will increased storm disturbance affect the biodiversity of intertidal, nonscleractinian sessile fauna on coral reefs?. Glob. Change Biol. 14, 2755–2770 (2008).

17. 17.

Rosel, P. E. & Watts, H. Hurricane Impacts on Bottlenose Dolphins in the Northern Gulf of Mexico. Gulf Mexico Sci. 25, 7 (2008).

18. 18.

Mullin, K. D., et al. Common bottlenose dolphins (Tursiops truncatus) in Lake Pontchartrain, Louisiana: 2007 to mid-2014. Prepared for the U.S. Department of Commerce, NOAA Technical Memorandum NMFS-SEFSC-673 43 (2015).

19. 19.

Langtimm, C. A. & Beck, C. A. Lower survival probabilities for adult Florida manatees in years with intense coastal storms. Ecol. Appl. 13, 257–268 (2003).

20. 20.

Elliser, C. R. & Herzing, D. L. Changes in interspecies association patterns of Atlantic bottlenose dolphins, Tursiops truncatus, and Atlantic spotted dolphins, Stenella frontalis, after demographic changes related to environmental disturbance. Mar. Mammal Sci. 32, 602–618 (2016).

21. 21.

Sinclair, C. Comparison of group size, abundance estimates and movement patterns of common bottlenose dolphins (Tursiops truncatus) in Mississippi Sound, Mississippi. MS Thesis, Louisiana State University, Baton Rouge. (2016).

22. 22.

Smith, C. E. et al. Hurricane impacts on the foraging patterns of bottlenose dolphins Tursiops truncatus in Mississippi Sound. Mar. Ecol. Prog. Ser. 487, 231–244 (2013).

23. 23.

Trainer, V. L., Hickey, B. M. & Horner, R. A. Biological and physical dynamics of domoic acid production off the Washington coast. Limnol. Oceanogr. 47, 1438–1446 (2002).

24. 24.

Bassos-Hull, K. et al. Long-term site fidelity and seasonal abundance estimates of common bottlenose dolphins (Tursiops truncatus) along the southwest coast of Florida and responses to natural perturbations. J. Cetacean Res. Manag. 13, 19–30 (2013).

25. 25.

Scholin, C. A. et al. Mortality of sea lions along the central California coast linked to a toxic diatom bloom. Nature 403, 80 (2000).

26. 26.

Pirotta, E., Brooks, K. L., Graham, I. M. & Thompson, P. M. Variation in harbour porpoise activity in response to seismic survey noise. Biol. Let. 10, 20131090. https://doi.org/10.1098/rsbl.2013.1090 (2014).

27. 27.

Wisniewska, D. M. et al. High rates of vessel noise disrupt foraging in wild harbour porpoises (Phocoena phocoena). Proc. R Soc. B Biol. Sci. 285(1872), 20172314 (2018).

28. 28.

Gordon, C. Anthropogenic Noise and Cetacean Interactions in the 21st Century: A Contemporary Review of the Impacts of Environmental Noise Pollution on Cetacean Ecologies. University Honors Theses. Paper 625 (2018).

29. 29.

Bittencourt, L. et al. Underwater noise in an impacted environment can affect Guiana dolphin communication. Mar. Pollut. Bull. 114, 1130–1134 (2017).

30. 30.

Fouda, L. et al. Dolphins simplify their vocal calls in response to increased ambient noise. Biol. Lett. 14, 20180484 (2018).

31. 31.

Seroka, G. et al. Rapid shelf-wide cooling response of a stratified coastal ocean to hurricanes. J. Geophys. Res. Oceans 122, 4845–4867 (2017).

32. 32.

Barco, S., Burt, L., DePerte, A., & Digiovanni Jr, R. Marine mammal and sea turtle sightings in the vicinity of the maryland wind energy area July 2013–June 2015. Prepared for Maryland Department of Natural Resources, VAQF Scientific Report # 2015-06 (2015).

33. 33.

Barco, S. G., Swingle, W. M., Mlellan, W. A., Harris, R. N. & Pabst, D. A. Local abundance and distribution of bottlenose dolphins (Tursiops truncatus) in the nearshore waters of Virginia Beach, Virginia. Mar. Mammal Sci. 15, 394–408 (1999).

34. 34.

Toth, J. L., Hohn, A. A., Able, K. W. & Gorgone, A. M. Patterns of seasonal occurrence, distribution, and site fidelity of coastal bottlenose dolphins (Tursiops truncatus) in southern New Jersey, USA. Mar. Mammal Sci. 27, 94–110 (2011).

35. 35.

Roberts, J. J. et al. Habitat-based cetacean density models for the US Atlantic and Gulf of Mexico. Sci. Rep. 6, 22615 (2016).

36. 36.

Gannon, D. P. & Waples, D. M. Diets of coastal bottlenose dolphins from the US Mid-Atlantic coast differ by habitat. Mar. Mammal Sci. 20, 527–545 (2004).

37. 37.

dos Santos, M. E., Coniglione, C. & Louro, S. Feeding behaviour of the bottlenose dolphin, Tursiops truncatus (Montagu, 1821) in the Sado estuary, Portugal, and a review of its prey species. Revista Brasileira de Zoociências 9, 31–39 (2007).

38. 38.

Benjamini, Y. & Hochberg, Y. Controlling the false discovery rate: A practical and powerful approach to multiple testing. J. Roy. Stat. Soc. Ser. B (Methodol.) 57, 289–300 (1995).

39. 39.

Bailey, H. & Secor, D. H. Coastal evacuations by fish during extreme weather events. Sci. Rep. 6, 30280 (2016).

40. 40.

Defran, R. H., Weller, D. W., Kelly, D. L. & Espinosa, M. A. Range characteristics of Pacific coast bottlenose dolphins (Tursiops truncatus) in the Southern California Bight. Mar. Mammal Sci. 15, 381–393 (1999).

41. 41.

Spitz, J., Rousseau, Y. & Ridoux, V. Diet overlap between harbour porpoise and bottlenose dolphin: An argument in favour of interference competition for food?. Estuar. Coast. Shelf Sci. 70, 259–270 (2006).

42. 42.

Craig, J. K. Aggregation on the edge: effects of hypoxia avoidance on the spatial distribution of brown shrimp and demersal fishes in the Northern Gulf of Mexico. Mar. Ecol. Prog. Ser. 445, 75–95 (2012).

43. 43.

Wells, R. S., et al. Northward extension of the range of bottlenose dolphins along the California coast. in The Bottlenose Dolphin 421–432. (Academic Press, 1990).

44. 44.

Bacheler, N. M., Shertzer, K. W., Cheshire, R. T. & MacMahan, J. H. Tropical storms influence the movement behavior of a demersal oceanic fish species. Sci. Rep. 9, 1481 (2019).

45. 45.

Schumann, N., Gales, N. J., Harcourt, R. G. & Arnould, J. P. Impacts of climate change on Australian marine mammals. Aust. J. Zool. 61, 146–159 (2013).

46. 46.

Weilgart, L. S. The impacts of anthropogenic ocean noise on cetaceans and implications for management. Can. J. Zool. 85(11), 1091–1116 (2007).

47. 47.

New, L. F. et al. Modelling the biological significance of behavioural change in coastal bottlenose dolphins in response to disturbance. Funct. Ecol. 27, 314–322 (2013).

48. 48.

Heithaus, M. R., Frid, A., Wirsing, A. J. & Worm, B. Predicting ecological consequences of marine top predator declines. Trends Ecol. Evol. 23, 202–210 (2008).

49. 49.

National Academies of Sciences Engineering and Medicine. Approaches to Understanding the Cumulative Effects of Stressors on Marine Mammals (The National Academies Press, Washington, 2017). https://doi.org/10.17226/23479

50. 50.

Clark, C. W. et al. Acoustic masking in marine ecosystems: intuitions, analysis, and implication. Mar. Ecol. Prog. Ser. 395, 201–222 (2009).

51. 51.

Clausen, K. T., Tougaard, J., Carstensen, J., Delefosse, M. & Teilmann, J. Noise affects porpoise click detections—The magnitude of the effect depends on logger type and detection filter settings. Bioacoustics 28(5), 443–458 (2019).

52. 52.

Nuuttila, H. K. et al. Acoustic detection probability of bottlenose dolphins, Tursiops truncatus, with static acoustic dataloggers in Cardigan Bay, Wales. J. Acoust. Soc. Am. 134, 2596–2609 (2013).

53. 53.

Garrod, A. et al. Validating automated click detector dolphin detection rates and investigating factors affecting performance. J. Acoust. Soc. Am. 144, 931–939 (2018).

54. 54.

Janik, V. M. & Sayigh, L. S. Communication in bottlenose dolphins: 50 years of signature whistle research. J. Comp. Physiol. A. 199, 479–489 (2013).

55. 55.

Wingfield, J. E. et al. Year-round spatiotemporal distribution of harbour porpoises within and around the Maryland wind energy area. PLoS ONE 12, e0176653 (2017).

56. 56.

Bailey, H. et al. Empirical evidence that large marine predator foraging behavior is consistent with area-restricted search theory. Ecology 100, e02743 (2019).

57. 57.

Simons, R. A. ERDDAP. NOAA/NMFS/SWFSC/ERD https://coastwatch.pfeg.noaa.gov/erddap (2018).

58. 58.

JPL MUR MEaSUREs Project. GHRSST Level 4 MUR global foundation sea surface temperature analysis (v4.1). PO.DAAC. https://doi.org/10.5067/GHGMR-4FJ04 (2015).

59. 59.

Carlström, J. Diel variation in echolocation behavior of wild harbor porpoises. Mar. Mammal Sci. 21, 1–12 (2005).

60. 60.

Wisniewska, D. M., Johnson, M., Nachtigall, P. E. & Madsen, P. T. Buzzing during biosonar-based interception of prey in the delphinids Tursiops truncatus and Pseudorca crassidens. J. Exp. Biol. 217(24), 4279–4282 (2014).

61. 61.

Au, W. W. L. The Sonar of Dolphins (Springer-Verlag, Berlin, 1993).

62. 62.

Hildebrand, J. A. Anthropogenic and natural sources of ambient noise in the ocean. Mar. Ecol. Prog. Ser. 395, 5–20 (2009).

63. 63.

Dugan, P. J., Klinck, H., Roch, M. A. & Helble, T. A. RAVEN X: High performance data mining toolbox for bioacoustic data analysis. arxiv:1610.03772. 2016.

64. 64.

Wenz, G. M. Acoustic ambient noise in the ocean: Spectra and sources. J. Acoust. Soc. Am. 34, 1936–1956 (1962).

65. 65.

Perrone, A. J. Ambient-noise-spectrum levels as a function of water depth. J. Acoust. Soc. Am. 48, 362–370 (1970).

66. 66.

Bates, D. M. & Pinheiro, J. C. Linear and nonlinear mixed-effects models. Abstract submitted to the 1998 Kansas State University Annual Conference on Applied Statistics in Agriculture (1998).

67. 67.

Bates, D., Mächler, M., Bolker, B. & Walker, S. Fitting linear mixed-effects models using lme4. J. Stat. Softw. 67(1), 1–48 (2015).

## Acknowledgements

Thanks to the many volunteers from the Chesapeake Biological Laboratory and the crew from Cornell University for field assistance. The Maryland Department of Natural Resources secured funding for this project from the Maryland Energy Administration’s Offshore Wind Development Fund and the US Department of Interior’s Bureau of Ocean Energy Management, Environmental Studies Program (Contract numbers 14-14-1916 BOEM/BOEM Award #M14AC00018 and 14-17-2241 BOEM). The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the opinions or policies of the US Government, the Maryland Department of Natural Resources, or the Maryland Energy Administration. Mention of trade names or commercial products does not constitute their endorsement by the US Government or the state.

## Author information

Authors

### Contributions

The manuscript has been reviewed and approved by all co-authors. A.D.F. conducted the analysis and wrote the manuscript with H.B. A.G., A.L.H., J.E.W., A.R., K.B.H., and A.D.F. conducted the fieldwork, and assistance with the analysis was received from V.L., D.H.S., and A.G.

### Corresponding author

Correspondence to Amber D. Fandel.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

### Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Fandel, A.D., Garrod, A., Hoover, A.L. et al. Effects of intense storm events on dolphin occurrence and foraging behavior. Sci Rep 10, 19247 (2020). https://doi.org/10.1038/s41598-020-76077-3

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# A helical cycloid?

While combing around my notes looking for other possible examples for this question, I chanced upon another one of my unsolved problems:

Cycloidal curves are curves generated by a circle rolling upon a plane or space curve. It's not too hard to derive the required parametric equations if the fixed curve is a plane curve, but I've had some trouble deriving the corresponding expression for space curves.

More specifically, here is the particular problem I was concerned with: consider a (cylindrical) helix:

\begin{align*}x&=a\cos\;t\\y&=a\sin\;t\\z&=ct\end{align*}

and imagine a circle of radius $r$ whose plane is always perpendicular to the x-y plane rolling on the helix, starting at the point $(a,0,0)$ ($t=0$). Imagine a point in the plane of the circle at a distance $hr$ from the center. What are the parametric equations for the locus of the point?

The two obvious pieces of information I have are that the center of the circle also traces a helix, whose parametric equation differs from the original by a vertical shift of $r$ (per Tony, that was an erroneous assumption), and that the expression for the arclength of the helix, $s=\sqrt{a^2+c^2}t$, should figure into the final parametric equations. Otherwise, I'm not sure how to start.

How does one derive parametric equations for the "helical cycloid"?

The physical model I had in mind was a screw ramp winding around a cylinder. Supposing that there was a car that needed to go to the top of the cylinder by driving on the ramp, and supposing that a spot is placed on one of the car's wheels, what are the equations for the locus of the spot?

• Where does the point start? – Jonas Meyer Dec 24 '10 at 5:06
• @Jonas: I was assuming that in the $h=1$ case, the point of tangency of the circle and the helix is $(a,0,0)$, and that the first cusp of the curve should start from there. – J. M. is a poor mathematician Dec 24 '10 at 7:50
• I think you are wrong that the centre of the circle lies directly above a point on the helix. The radius from the centre of the circle to the point of contact is skew, and (except for the point of contact itself) lies outside the bounding cylinder of the helix. – TonyK Dec 24 '10 at 8:23
• Aha @Tony, I think that's why my attempts at deriving it looked weird... how would this be fixed? – J. M. is a poor mathematician Dec 24 '10 at 8:41
• M.: I think I fixed my answer. – Jonas Meyer Dec 24 '10 at 9:04

I like to think of helices on cylinders as images of lines under planar curling, so here's an approach incorporating that idea.

Imagine the circle rolling up a line drawn in the plane, with $P$ the point of tangency along the line and $Q$ a distinguished point on the circumference of the circle. (So, $Q$ traces out a cycloid in the plane.) Now, imagine the plane is made of thin paper, but the circle is made of stiff cardboard. If we curl the plane into a cylinder but the circle remains flat, the plane of the circle will be tangent to the cylinder along the (vertical) line passing through the point $P$. The point $P$ will follow a helix, and the point $Q$ --which lies in the plane of the circle-- traces out the helical cycloid in space. While the path that $P$ (and $Q$) takes is decidedly different after curling than before, the displacement vector between $P$ and $Q$ in the plane of the circle is the same throughout.

Before curling, if the circle of radius $r$ rolls along a horizontal track in the $uv$-plane, then we have $P=(r t, 0)$, and the point $Q$ traces out the standard cycloid:

\begin{align} u &= r ( t - \sin t ) \\ v &= r ( 1 - \cos t ) \end{align}

The displacement vector from $P$ to $Q$ at time $t$ is given by

$$d := PQ = [m, n] = r[-\sin t, 1 - \cos t ] = - 2 r \sin\frac{t}{2} \; [\cos\frac{t}{2},-\sin\frac{t}{2}]$$

Rotating the plane through an angle, say, $\theta := \rm{atan\frac{c}{a}}$, and writing $b$ for $\sqrt{a^2+c^2}$, the circle's point of tangency along the tilted track is given by ...

$$P_2 = (r t \cos\theta,r t \sin\theta) = \left( \frac{r t a}{b}, \frac{r t c}{b}\right)$$

... and the displacement vector, $d_2$, to the point $Q_2$ tracing out the tilted cycloid is given by ...

\begin{align} d_2 :&= [m \cos\theta - n \sin \theta, m \sin\theta + n \cos \theta ] \\ &= \frac{1}{b} [a m - c n, c m + a n] \\ &= -\frac{2r}{b}\sin\frac{t}{2}\left[\cos\left(\theta-\frac{t}{2}\right), \sin\left(\theta - \frac{t}{2} \right) \right] \\ \end{align}

Now the fun part: Curl the $(u,v)$-plane around a cylinder of radius $a$, such that the $uv$-origin aligns with the $xyz$-point $(a,0,0)$ and the $v$-axis runs parallel to the $z$ axis. The curled, tilted track will coincide with a helix. The horizontal distance travelled by $P_2$ in the plane becomes the length of horizontal circular arc travelled by $P_3$ around the cylinder; upon dividing by the radius, $a$, of the cylinder, this length becomes the angular "distance" --$s := \frac{rt}{b}$-- travelled by $P_3$; the vertical distances match. Therefore:

$$P_3 = \left(a\cos s, a\sin s, \frac{rtc}{b} \right)=(a\cos s,a\sin s,c s)$$

As for the displacement vector: The $u$ direction of the tangent plane coincides with the horizontal vector tangent to the cylinder at $P_3$; the $v$ direction coincides with the $z$ direction. Thus, the transformation from $uv$-coordinates to $xyz$-coordinates is given by

$$[1,0]\to[-\sin s, \cos s, 0]\hspace{0.5in}[0,1]\to[0,0,1]$$

The image, $d_3$, of the displacement vector $d_2$, then, is

$$d_3 = -\frac{2r}{b}\sin\frac{t}{2}\left[-\sin s \cos\left(\theta-\frac{t}{2}\right), \cos s \cos\left(\theta-\frac{t}{2}\right), \sin\left(\theta - \frac{t}{2} \right) \right]$$

and the path of $Q_3$, which traces the helical cycloid, is given by

Q_3 = P_3 + d_3 = \left\{ \begin{align} x &= a \cos s &+ \frac{2 r}{b} \sin s \sin\frac{bs}{2r} \cos\left(\theta-\frac{bs}{2r}\right)\\ y &= a \sin s &- \frac{2 r}{b} \cos s \sin\frac{bs}{2r} \cos\left(\theta-\frac{bs}{2r}\right)\\ z &= c s &- \frac{2 r}{b} \sin\frac{b s}{2r}\sin\left(\theta-\frac{bs}{2r}\right) \end{align} \right.

Here's a picture with $a=c=1$ and $r=1/2$:

Note: The above does not simply curl the planar cycloid around the cylinder. Since

$$x^2 + y^2 = a^2 + \frac{4r^2}{b^2} \sin^2\frac{bs}{2r} \cos^2\left(\theta-\frac{bs}{2r}\right) \ge a^2$$

we see that most of the helical cycloid lies outside the surface of the cylinder.

The tangent to the helix at $P = (a \cos t, a \sin t, ct)$ is

$T = (-a \sin t, a \cos t, c)$.

The vector

$H = (\cos t, \sin t, 0)$

is horizontal, and perpendicular to $T$. The radius vector $R$ from $P$ to the centre $C$ of the circle is perpendicular to both of these, so it is proportional to

$H \wedge T = (c \sin t, -c \cos t, a)$,

which has length $\sqrt{a^2 + c^2}$. So the centre of the circle is at $P + h H \wedge T$, with $h = r/\sqrt{a^2 + c^2}$.

• Should that be $-c\cos t$ in the middle component of $H\wedge T$? – Jonas Meyer Dec 24 '10 at 9:10
• Yes, you're right. I've fixed it. – TonyK Dec 24 '10 at 9:22

I'm adjusting my answer, which as TonyK points out was based on an incorrect assumption. Hopefully it's correct now.

The center of the circle is not directly above the point of contact with the helix as my incorrect answer had asserted. To see where it is, first picture the starting point at $(a,0,0)$, where the circle is parallel to the $y$-$z$ plane. The tangent vector to the helix is in the direction of $(0,a,c)$, so the radius is perpendicular to this in the direction of $(0,-c,a)$. So you have to add $\frac{r}{\sqrt{a^2+c^2}}(0,-c,a)$ to get to the center. As the circle rolls along, the vector you add to get to the center will rotate around the $z$ axis to $\frac{r}{\sqrt{a^2+c^2}}(c\sin(t),-c\cos(t),a)$. So the center of the circle is parametrized by $(a\cos(t),a\sin(t),ct)+\frac{r}{\sqrt{a^2+c^2}}(c\sin(t),-c\cos(t),a)$.

The arclength formula allows you to write down the rotation angle of the point about the center of the circle in terms of $t$ as $\frac{\sqrt{a^2+c^2}}{r}t$. Let $b=\frac{\sqrt{a^2+c^2}}{r}$ from now on.

If we pretend for a minute that the point was just going to rotate about the center of the circle at this rate while the circle remains fixed, and simplify the problem for now by assuming that the point starts above the center of the circle, then the vector we would add to the center of the circle to parametrize the point would be $(0,hr\sin(bt),hr\cos(bt))$. When looking toward the $y$-$z$ plane from the positive $x$ axis, this motion would appear clockwise, as it should because the circle will be starting off to the right from this perspective.

But the vector we actually add to the center will also be rotated by $t$ about the $z$-axis, so putting it all together (with minimal simplification) we get

\begin{align*} x&=a\cos(t)+\frac{c}{b}\sin(t)-hr\sin(t)\sin(bt)\\ y&=a\sin(t)-\frac{c}{b}\cos(t)+hr\cos(t)\sin(bt)\\ z&=ct+\frac{a}{b}+hr\cos(bt). \end{align*}

Now to adjust the starting point, you can change $bt$ to $b(t-t_0)$ for suitable $t_0$.

• See my comment to the OP's question ("I think you are wrong that the centre of the circle lies above a point on the helix"). – TonyK Dec 24 '10 at 8:25
• @TonyK: You're right, thank you. I'll try to fix it. – Jonas Meyer Dec 24 '10 at 8:29


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Article Contents
Article Contents

# Pullback attractors for stochastic heat equations in materials with memory

• We study the asymptotic behaviour of a non-autonomous stochastic reaction-diffusion equation with memory. In fact, we prove the existence of a random pullback attractor for our stochastic parabolic PDE with memory. The randomness enters in our model as an additive Hilbert valued noise. We first prove that the equation generates a random dynamical system (RDS) in an appropriate phase space. Due to the fact that the memory term takes into account the whole past history of the phenomenon, we are not able to prove compactness of the generated RDS, but its asymptotic compactness, ensuring thus the existence of the random pullback attractor.
Mathematics Subject Classification: Primary: 60H15, 35B25, 35B40, 37L30.

 Citation:


text:
zbMATH — the first resource for mathematics

On the size of algebraic integers. (Russian) Zbl 0732.11052
For an algebraic integer $$\alpha$$ denote by $$\alpha '$$ the maximum of the absolute values of conjugates of $$\alpha$$. It follows from a result of E. Dobrowolski [Bull. Acad. Pol. Sci., Ser. Sci. Math. Astron. Phys. 26, 291-292 (1978; Zbl 0393.12003)] that if $$f(n)=\exp (\frac{\log n}{6n^ 2})$$ and $$\alpha '<f(n)$$ where n denotes the degree of $$\alpha$$ then $$\alpha$$ is a root of unity. A later result of E. Dobrowolski [Acta Arith. 34, 391-401 (1979; Zbl 0416.12001)] shows that one can replace here f(n) by $$\exp ((1-\epsilon)\frac{(\log \log n \log n)^ 3}{n})$$ for $$n>n_ 0(\epsilon)$$. The author improves slightly the first of these results and points out that although his result is weaker than the second it is effective.
Note of reviewer: It is stated in the second paper of Dobrowolski that if one replaces in his theorem 1-$$\epsilon$$ by 1/1200 then its assertion holds for all n, a much stronger result than that in the paper reviewed.

MSC:
 11R04 Algebraic numbers; rings of algebraic integers 11R09 Polynomials (irreducibility, etc.)


text:
• 能力建设 •

### 中国科技期刊国际影响力提升计划D类项目资助期刊的办刊成效——被SCIE/SSCI收录的情况

1. 沈阳药科大学《亚洲药物制剂科学》编辑部,辽宁省沈阳市文化路103号 110016
• 收稿日期:2019-08-06 修回日期:2019-10-25 出版日期:2019-12-15 发布日期:2019-12-30
• 作者简介:凌桂霞(ORID:0000-0003-4577-7342),博士,编辑,E-mail: ajps2006@163.com|张 宣,学士,编辑|任 珊,硕士,编辑|赵春杰,博士,教授,编辑部主任。

### Effectiveness of journals funded by Project for Enhancing International Impact of China STM Journals category D: Situation of journals indexed by SCIE/SSCI

LING Guixia,ZHANG Xuan,REN Shan,ZHAO Chunjie

1. Editorial Office of Asian Journal of Pharmaceutical Sciences, Shenyang Pharmaceutical University, 103 Wenhua Road, Shenyang 110016, China
• Received:2019-08-06 Revised:2019-10-25 Online:2019-12-15 Published:2019-12-30

【目的】 研究中国科技期刊国际影响力提升计划D类项目2013—2018年资助期刊被Web of Science (WoS) 数据库收录情况,考察D类项目资助期刊的发展成效,为世界一流期刊的建设提供参考。【方法】 利用WoS数据库,统计分析SCIE/SSCI收录D类项目资助期刊的学科分布和文献计量指标,评价D类项目资助期刊对SCI空白学科和优势学科的填补作用。【结果】 D类项目资助期刊中有21种期刊被SCIE/SSCI收录,47.6%的收录期刊进入Q1区,5种期刊填补了SCI学科空白,16种期刊有效增补了优势/前沿学科。【结论】 D类项目资助培育了一批优质期刊,成果显著,21种已被SCIE/SSCI收录的期刊代表了优势/前沿学科,或填补了学科空白,对世界一流期刊建设起到了推动作用。

Abstract:

[Purposes] This paper aims to study the status of the journals funded by Project for Enhancing International Impact of China STM Journals category D (PIIJ D) in 2013—2018, which are indexed by Web of Science (WoS) database, and provide references for the construction of world-class journals in China. [Methods] The journals indexed by SCIE/SSCI were selected from the PIIJ D-supported journals through the WoS database, and their qualitative indicators and subject distribution/contribution were investigated and searched. The effect of PIIJ D on filling the blank of the SCI subjects was also evaluated. [Findings] A total of 90 journals have been selected by PIJJ D with high academic influence, among which 21 journals stand out and are indexed by SCIE/SSCI, and 47.6% of the indexed journals rank in Q1 region. In the 21 journals indexed by SCIE/SSCI, 5 journals have filled in the gaps in SCI disciplines, and 16 journals have effectively supplemented the frontier disciplines. [Conclusions] The PIIJ D has cultivated a number of high-quality journals with remarkable performance. The 21 journals have been indexed by SCIE/SSCI representing the frontier disciplines, or filling the gaps, which play a role in promoting the construction of world-class journals.


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• ### To Do Lessons

Mathematics » Introducing Fractions » Add and Subtract Mixed Numbers

This is a lesson from the tutorial, Introducing Fractions and we encourage you to log in or register before you continue, so that you can track your progress.

Modeling with fraction circles helps illustrate the process for adding mixed numbers: We add the whole numbers and add the fractions, and then we simplify the result, if possible.

### Adding Mixed Numbers With a Common Denominator

Step 1. Add the whole numbers.

Step 3. Simplify, if possible.

## Example

Add: $$3\frac{4}{9}+2\frac{2}{9}.$$

### Solution

 $$3\frac{4}{9}+2\frac{2}{9}$$ Add the whole numbers. Add the fractions. Simplify the fraction.

In the example above, the sum of the fractions was a proper fraction. Now we will work through an example where the sum is an improper fraction.

## Example

Find the sum: $$9\frac{5}{9}+5\frac{7}{9}.$$

### Solution

 $$9\frac{5}{9}+5\frac{7}{9}$$ Add the whole numbers and then add the fractions. $$\begin{array}{}\phantom{\rule{0.6em}{0ex}}9\frac{5}{9}\hfill \\ +5\frac{7}{9}\hfill \\ \text{________}\hfill \\ \phantom{\rule{0.8em}{0ex}}14\frac{12}{9}\hfill \end{array}$$ Rewrite $$\frac{12}{9}$$ as an improper fraction. $$14+1\frac{3}{9}$$ Add. $$15\frac{3}{9}$$ Simplify. $$15\frac{1}{3}$$

An alternate method for adding mixed numbers is to convert the mixed numbers to improper fractions and then add the improper fractions. This method is usually written horizontally.

## Example

Add by converting the mixed numbers to improper fractions: $$3\frac{7}{8}+4\frac{3}{8}.$$

### Solution

 $$3\frac{7}{8}+4\frac{3}{8}$$ Convert to improper fractions. $$\frac{31}{8}+\frac{35}{8}$$ Add the fractions. $$\frac{31+35}{8}$$ Simplify the numerator. $$\frac{66}{8}$$ Rewrite as a mixed number. $$8\frac{2}{8}$$ Simplify the fraction. $$8\frac{1}{4}$$

Since the problem was given in mixed number form, we will write the sum as a mixed number.

The table below compares the two methods of addition, using the expression $$3\frac{2}{5}+6\frac{4}{5}$$ as an example. Which way do you prefer?

Mixed NumbersImproper Fractions
$$\begin{array}{} \hfill \phantom{\rule{0.8em}{0ex}}3\frac{2}{5}\hfill \\ \hfill \frac{+6\frac{4}{5}}{\phantom{\rule{0.6em}{0ex}}9\frac{6}{5}}\hfill \\ \hfill 9+\frac{6}{5}\hfill \\ \hfill 9+1\frac{1}{5}\hfill \\ \hfill 10\frac{1}{5}\hfill \end{array}$$$$\begin{array}{} \hfill 3\frac{2}{5}+6\frac{4}{5}\hfill \\ \hfill \frac{17}{5}+\frac{34}{5}\hfill \\ \hfill \frac{51}{5}\hfill \\ \hfill 10\frac{1}{5}\hfill \end{array}$$


text:
## fog

https://doi.org/10.1351/goldbook.F02470
A general term applied to a @S06198@ of droplets in a gas. In meteorology, it refers to a @S06198@ of water droplets resulting in a visibility of less than $$1\ \text{km}$$.
Sources:
PAC, 1972, 31, 577. (Manual of Symbols and Terminology for Physicochemical Quantities and Units, Appendix II: Definitions, Terminology and Symbols in Colloid and Surface Chemistry) on page 606 [Terms] [Paper]
PAC, 1990, 62, 2167. (Glossary of atmospheric chemistry terms (Recommendations 1990)) on page 2190 [Terms] [Paper]


text:
For height-balanced Binary trees, the height is defined to be logarithmic $$O(\log n)$$ in the number $$n$$ of items.

## References#

“Self-Balancing Binary Search Tree.” 2022. Wikipedia, June. https://en.wikipedia.org/w/index.php?title=Self-balancing_binary_search_tree&oldid=1092928206.


text:
Today I try to delete a directory with 485,000+ files on Linux server. It shows the error message below.

If you’re trying to delete a very large number of files in a single directory at one time, you will probably run into this error. The problem is that when you type something like rm -f /data/*. So how do we working with directories with a large number of files? Let talk about it below.

## Show first files

To show first n files in the directory instead of to show all files.

## Remove files

### Using Find

Using find -delete. (2000 files/second, it is about three time faster rm)

Using find and xargs. To find every file and pass them one-by-one to the “rm” command (not recommend, xargs is dangerous*)


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At a certain upscale restaurant, there just two kinds of food items: e

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At a certain upscale restaurant, there just two kinds of food items: e  [#permalink]

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At a certain upscale restaurant, there just two kinds of food items: entrees and appetizers. Each entrée item costs $30, and each appetizer item costs$12. Last year, it had a total of 15 food items on the menu, and the average price of a food item on its menu was $18. This year, it added more appetizer items, and the average price of a food item on its menu dropped to$15. How many appetizer items did it add this year?

A. 3
B. 6
C. 9
D. 12
E. 15

Kudos for a correct solution.

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Re: At a certain upscale restaurant, there just two kinds of food items: e  [#permalink]

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17 Mar 2015, 07:00
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It is just about resolving some equations :

X is the number of entrees
Y is the number of appetizers

1) X+Y = 15
2) (30X+12Y)/15=18

1 and 2 give X=5 and Y = 10

Then next year,
Z is the number of added appetizers
3) (30*5+12*(10+Z))/(15+Z)=15

I found Z = 15
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Re: At a certain upscale restaurant, there just two kinds of food items: e  [#permalink]

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17 Mar 2015, 23:17
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Let the number of entrees = x, then number of appetizers = 15-x

Given that $$\frac{30x + 12(15-x)}{15} = 18$$

Solving we get x = number of entrees = 5 & number of appetizers = 10

Say number of appetizers added = y

New equation $$= \frac{30*5 + 12(10+y)}{15+y} = 15$$

y = 15
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Re: At a certain upscale restaurant, there just two kinds of food items: e  [#permalink]

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18 Mar 2015, 04:13
PareshGmat wrote:

Let the number of entrees = x, then number of appetizers = 15-x

Given that $$\frac{30x + 12(15-x)}{15} = 18$$

Solving we get x = number of entrees = 5 & number of appetizers = 10

Say number of appetizers added = y

New equation $$= \frac{30*5 + 12(10+y)}{15+y} = 15$$

y = 15

Hi Paresh,

One question :
The current price is as follows: Each entrée item costs $30, and each appetizer item costs$12.
Last year the breakup was --> Last year, it had a total of 15 food items on the menu, and the average price of a food item on its menu was $18. Then why are we setting up a equation which takes current prices and average prices from last year ? $$\frac{30x + 12(15-x)}{15} = 18$$ Thanks Lucky Current Student Joined: 09 Feb 2014 Posts: 39 Concentration: General Management, Operations Schools: Darden '19 (M$)
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Re: At a certain upscale restaurant, there just two kinds of food items: e  [#permalink]

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18 Mar 2015, 08:29
PareshGmat wrote:

Let the number of entrees = x, then number of appetizers = 15-x

Given that $$\frac{30x + 12(15-x)}{15} = 18$$

Solving we get x = number of entrees = 5 & number of appetizers = 10

Say number of appetizers added = y

New equation $$= \frac{30*5 + 12(10+y)}{15+y} = 15$$

y = 15

This is pretty straightforward and I took the same approach, but it still took me over 3 minutes to work out the equations and solve.

1. Is this too long to take for this type of question?
2. Any tips to increase speed?
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At a certain upscale restaurant, there just two kinds of food items: e  [#permalink]

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18 Mar 2015, 09:26
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1
mr7183 wrote:
This is pretty straightforward and I took the same approach, but it still took me over 3 minutes to work out the equations and solve.

1. Is this too long to take for this type of question?
2. Any tips to increase speed?

hi,
1. 3 minutes is slightly more and specially when you feel it is straightforward..
2. ofcourse you need to tackle different type of Q in different ways to lower the time..
- few Q are best by choosing/putting appropriate value for variable and then check the answer
- few are best solved by putting the value given in choices
- few through proper mathematical way...

in this Q, best method is weighted average but you require to practice these..

At a certain upscale restaurant, there just two kinds of food items: entrees and appetizers. Each entrée item costs $30, and each appetizer item costs$12. Last year, it had a total of 15 food items on the menu, and the average price of a food item on its menu was $18. This year, it added more appetizer items, and the average price of a food item on its menu dropped to$15. How many appetizer items did it add this year?

A. 3
B. 6
C. 9
D. 12
E. 15

first lets find initial ratio of E:A... there average values are 12 and 30 and combined average brings it to 18... so A:E=30-18:18-12=12:6=2:1 or A=10 and E=5 as total is 15..
let y appetizers be added so number of A=10+y.... and now combined average is 15 so A:E=30-15:15-12=5:1=10+y:5.....
$$\frac{5}{1}=\frac{(10+y)}{5}$$
10+y=25... so y=15...
ans E..
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Re: At a certain upscale restaurant, there just two kinds of food items: e  [#permalink]

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18 Mar 2015, 10:10
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The total cost of appetizers + entrees last year = $18 * 15 Let there be 'x' appetizers added this year, then total cost of food items =$15 * (15+x)

Since, cost of each appetizer is $12, we have 18*15 + 12x = 15*15 + 15x 3x = 3*15 Therefore, no. of appetizers added this year, x = 15 Answer (E) Math Expert Joined: 02 Sep 2009 Posts: 58320 Re: At a certain upscale restaurant, there just two kinds of food items: e [#permalink] Show Tags 23 Mar 2015, 04:10 Bunuel wrote: At a certain upscale restaurant, there just two kinds of food items: entrees and appetizers. Each entrée item costs$30, and each appetizer item costs $12. Last year, it had a total of 15 food items on the menu, and the average price of a food item on its menu was$18. This year, it added more appetizer items, and the average price of a food item on its menu dropped to $15. How many appetizer items did it add this year? A. 3 B. 6 C. 9 D. 12 E. 15 Kudos for a correct solution. MAGOOSH OFFICIAL SOLUTION: Method I: using sums First, last year. Let x be the number of entrees. Then (15 – x) is the number of appetizers. The sums are: entrees = 30x appetizers = (15 – x)*12 = 12*15 – 12x = 6*30 – 12x = 180 – 12x total = 15*18 = 30*9 = 270 Notice the use of the Doubling and Halving trick in the second and third lines. The two individual sums should add up to the total sum. 30x + 180 – 12x = 270 18x = 90 x = 5 They start out with 5 entrees and 10 appetizers. Let N be the number of appetizers added, so now there are 5 entrees and (10 + N) appetizers. We need to solve for N. Again, the sums: entrees = 5*30 = 150 appetizers = (10 + N)*12 = 120 + 12N total = (15 + N)*15 = 225 + 15N Again, the two individual sums should add up to the total sum. 150 + 120 + 12N = 225 + 15N 270 = 225 + 3N 45 = 3N 15 = N They added 15 more appetizers. Answer = (E) Method I was do-able, but we had to solve for many values. Method II: proportional placement of the total average Originally, the entrée price was 30 – 18 = 12 from the total average, and the appetizer price was 18 – 12 = 6. This means there must have been twice as many appetizers as entrees. Therefore , with 15 items, there must have been 10 appetizers and 5 entrees. The number of entrees doesn’t change. The average drops to$15, so the distance from the entrée price is now 30 – 15 = 15, and the distance from the appetizer price is now 15 – 12 = 3. That’s a 5-to-1 ratio, which means there must be 5x as many appetizers as entrees. Since there still are 5 entrees, there must now be 25 appetizers, so 15 have been added.

If you know how to employ this method, it is much more elegant.
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Re: At a certain upscale restaurant, there just two kinds of food items: e  [#permalink]

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21 Dec 2015, 02:35
I directly solved for X the number of appetizers added.
We know that [(18 x 15) + 12 x X ] / (15 + X) = 15
We can solved directly for X without caring what was the number of appetizers last year.
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Re: At a certain upscale restaurant, there just two kinds of food items: e  [#permalink]

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21 Dec 2015, 04:15
Total of last year = 15*18 = 270

this year -> (270 + 12x)/(15+x) = 15

x= 15
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