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Fitch, W. T., Neubauer, J., and Herzel, H. (2002). "Calls out of chaos: the adaptive significance of nonlinear phenomena in mammalian vocal production," Anim. Behav. 63, Hegger, R., Kantz, H., and Schreiber, T. (1999). "Practical implementation of nonlinear time series methods: The TISEAN package," Chaos 9(2), Johnson, M. P. and Tyack, P. L. (2003). "A digital acoustic recording tag for measuring the response of wild marine mammals to sound," IEEE J. Ocean Eng. 28, Kantz, H. and Schreiber, T. (2004). Nonlinear Time Series Analysis, Second ed. (Cambridge University Press, Cambridge), 369 pp. Miller, P.J. and Tyack, P.L. (1998). "A small towed beamforming array to identify vocalizing resident killer whales (Orcinus orca) concurrent with focal behavioral observations, " Deep Sea Res. II 45, The presence of nonlinear phenomena in the analyzed vocalizations of both right and killer whales suggests that these features are important and could play an integral role in the communication of these species. Fitch et al. (2002) hypothesized that these features function to increase auditory impact on listeners by providing cues as to signaler fitness, mate quality, and overall health and may assist in communicating individual identification, animal size, and urgency. The lack of correlating detailed behavioral information in this study and the lack of information on the individual whales that produced each vocalization particularly for the right whales did not allow us to address any of the proposed functional hypotheses but the prevalence of features found indicates that they are likely more than an artifact of production and may serve communicative functions. Nonlinear phenomena in the vocalizations of North Atlantic right whales (Eubalaena glacialis) and killer whales (Orcinus orca)* Tyson, Reny B. 1,2 ; Nowacek, Douglas P. 1, and Miller, Patrick, J. O. 3 1 Department of Oceanography, Florida State University, Tallahassee, Florida 32306, 2 Department of Psychology, Florida State University, Tallahassee, FL 32306, and 3 School of Biology, University of St. Andrews, St. Andrews, Fife, KY16 9TS, United Kingdom * Adapted from: Tyson, R. B. et al. (2007). J. Acoustic. Soc. Am. 122, Nonlinear phenomena or nonlinearities in animal vocalizations include features such as subharmonics, deterministic chaos, biphonation, and frequency jumps that until recently were generally ignored in acoustic analyses. Recent documentation of these phenomena in several species suggests that they may play a communication role, though the exact function is still under investigation. Here, qualitative descriptions and quantitative analyses of nonlinearities in the vocalizations of killer whales (Orcinus orca) and North Atlantic right whales (Eubalaena glacialis) are provided. All four nonlinear features were present in both species, with at least one feature occurring in 92.4% of killer and 65.7% of right whale vocalizations analyzed. Occurrence of biphonation varied the most between species, being present in 89.0% of killer whale vocalizations and only 20.4% of right whale vocalizations. Because deterministic chaos is qualitatively and quantitatively different than random or Gaussian noise, we used a program (TISEAN © ) designed specifically to identify deterministic chaos to confirm the presence of this nonlinearity. All segments tested in this software indicate that both species do indeed exhibit deterministic chaos. The results of this study provide confirmation that such features are common in the vocalizations of cetacean species and lay the groundwork for future studies. 2. METHODS 4. ANALYSIS OF CHAOTIC SEGMENTS 7. ACKNOWLEDGMENTS 6. LITERATURE CITED 3. NONLINEAR PHENOMENA1. ABSTRACT FIG 4. Spectrograms of representative right whale vocalizations exhibiting nonlinear phenomena. Labels are defined in TABLE I for nonlinear features (SB is biphonation in the form of sidebands). 5. DISCUSSION David Mann, Michael Owren, Anna Nousek, Susan Parks, Rainer Hegger, Mike Kaschack & Frank Johnson. Right whale recordings in 2001 and 2002 were made under a NOAA National Marine Fisheries permit no issued to Scott Kraus & Canadian Department of Fisheries & Oceans permits & Right whale recordings in 2005 were made under a Canadian Department of Fisheries & Oceans permit, MAR-SA Killer whale recordings were also made under research permits from the Canadian Department of Fisheries & Oceans following Canadian law. TABLE I. Frequency of occurrence of nonlinear phenomena in the analyzed right whale (RW) and killer whale (KW) vocalizations: frequency jump (FJ), subharmonics (SH), biphonation (BP), and deterministic chaos (DC). FIG 5. Spectrograms of representative killer whale vocalizations exhibiting nonlinear phenomena. Labels are defined in TABLE I for nonlinear features (SB is biphonation in the form of sidebands, F o is the fundamental frequency for the low frequency component, and G o is the fundamental frequency for the high frequency component). TABLE II. Analysis of four signals exhibiting deterministic chaos and the generated harmonic and random signals. DTAG FIG 3. Towed array of 16 hydrophones that recorded northern resident killer whale vocalizations in Johnstone Strait, British Columbia in 1998 and 1999 (Fs = 48 kHz) (Schematic figure from Miller and Tyack 1998) FIG 2. Multi-sensor digital acoustic recording tags (‘DTAGs’; Johnson and Tyack 2003) that recorded right whale vocalizations in the Bay of Fundy, Canada in 2001 (Fs = 16 kHz), 2002 (Fs = 32 kHz), and 2005 (Fs = 96 kHz). D. Nowacek FIG 1. Schematic narrowband spectrogram illustrating the four nonlinear features analyzed in the vocalizations used in this study (a stable limit cycle is considered normal phonation and is not included as a nonlinear feature; modified after Fig. 1 from Riede et al. 2004). These features are produced by nonlinearities in the vocal production system, where rather simple commands to the system can result in highly complex and individually variable acoustic signals (Fitch et al. 2002). Spectrograms were isolated and inspected for the presence of these nonlinearities using Adobe Audition ® software. SubharmonicsBiphonationDeterministic Chaos NonlinearitiesStable Limit Cycle Frequency Jump Because deterministic chaos can resemble Gaussian noise, we used TISEAN © software to determine if segments we suspected were chaotic were not random noise (FIG. 6, Table II; Hegger et al. 1999). Two segments from each species were cut from the original vocalizations and analyzed. We also analyzed a generated harmonic and a generated random signal for comparison. We implemented the method of surrogate data, the method of delays, a false nearest-neighbors analysis, and created the spectrum of Lyapunov exponents to reconstruct the attractor of the underlying dynamics of each signal in the phase space. The Lyapunov exponents give a measure of the local stability properties of a trajectory and quantify the strength of chaos (Kantz and Schreiber 2004). All signals suspected of being chaotic exhibited one positive exponent indicating they are indeed non-Gaussian signals FIG 6 (left). Spectrum of Lyapunov Exponents. A trajectory was considered chaotic if it had at least one positive exponent in the Lyapunov Spectrum since chaotic segments are identified as having maximal Lyapunov exponent (MLE) between zero and infinity (Kantz and Schreiber 2004). The exponent for the harmonic signal is the maximum exponent found; an additional exponent was found at
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