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Three months journeying of a Hawaiian monk seal

David R. Brillinger, Brent S. Stewart, and Charles L. Littnan

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Abstract

Hawaiian monk seals (Monachus schauinslandi) are endemic to the Hawaiian Islands and are the most endangered species of marine mammal that lives entirely within the jurisdiction of the United States. The species numbers around 1300 and has been declining owing, among other things, to poor juvenile survival which is evidently related to poor foraging success. Consequently, data have been collected recently on the foraging habitats, movements, and behaviors of monk seals throughout the Northwestern and main Hawaiian Islands.

Our work here is directed to exploring a data set located in a relatively shallow offshore submerged bank (Penguin Bank) in our search of a model for a seal’s journey. The work ends by fitting a stochastic differential equation (SDE) that mimics some aspects of the behavior of seals by working with location data collected for one seal. The SDE is found by developing a time varying potential function with two points of attraction. The times of location are irregularly spaced and not close together geographically, leading to some difficulties of interpretation. Synthetic plots generated using the model are employed to assess its reasonableness spatially and temporally. One aspect is that the animal stays mainly southwest of Molokai. The work led to the estimation of the lengths and locations of the seal’s foraging trips.

Chapter information

Source
Deborah Nolan and Terry Speed, eds., Probability and Statistics: Essays in Honor of David A. Freedman (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2008), 246-264

Dates
First available in Project Euclid: 7 April 2008

Permanent link to this document
https://projecteuclid.org/euclid.imsc/1207580087

Digital Object Identifier
doi:10.1214/193940307000000473

Zentralblatt MATH identifier
1171.62073

Subjects
Primary: 60J60: Diffusion processes [See also 58J65] 62G08: Nonparametric regression 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 70F99: None of the above, but in this section

Keywords
bagplot boundary GPS locations Molokai Monachus schauinslandi Hawaiian Monk seal moving bagplot potential function robust methods simulation spatial locations stochastic differential equation synthetic plot UTM coordinates

Rights
Copyright © 2008, Institute of Mathematical Statistics

Citation

Brillinger, David R.; Stewart, Brent S.; Littnan, Charles L. Three months journeying of a Hawaiian monk seal. Probability and Statistics: Essays in Honor of David A. Freedman, 246--264, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2008. doi:10.1214/193940307000000473. https://projecteuclid.org/euclid.imsc/1207580087


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References

  • [1] Aldous, D. (1989). Probability Approximations via the Poisson Clumping Heuristic. Springer, New York.
  • [2] Baker, J. D. and Johanos, T. C. (2004). Abundance of the Hawaiian monk seal in the main Hawaiian Islands. Biological Conservation 116 103–110.
  • [3] Brillinger, D. R. (1997) A particle migrating randomly on a sphere. J. Theoret. Probab. 10 429–443.
  • [4] Brillinger, D. R. (2003). Simulating constrained animal motion using stochastic differential equations. Probability, Statistics and Their Applications. Lecture Notes Monogr. Ser. 41 35–48. Institute of Mathematical Statistics, Hayward.
  • [5] Brillinger, D. R. (2007). Learning a potential function from a trajectory. Signal Processing Letters 14 867–870.
  • [6] Brillinger, D. R. (2007) A potential function approach to the flow of play in soccer. J. Quantitative Analysis in Sports 3 Issue 1.
  • [7] Brillinger, D. R. and Stewart, B. S. (1998). Elephant-seal movements: Modelling migration. Canad. J. Statist. 26 431–443.
  • [8] Brillinger, D. R., Preisler, H. K., Ager, A. A. and Kie, J. G. (2001). The use of potential functions in modeling animal movement. In Data Analysis from Statistical Foundations (A. K. Md. E. Saleh, ed.) 369–386. Nova Science, New York.
  • [9] Brillinger, D. R., Preisler, H. K., Ager, A. A., Kie, J. G. and Stewart, B. S. (2002). Employing stochastic differential equations to model wildlife motion. Bull. Brazilian Math. Soc. 33 385–408.
  • [10] DeLong, R. L., Kooyman, G. L., Gilmartin, W. G. and Loughlin, T. R. (1984). Hawaiian monk seal diving behavior. Acta Zoologica Fennica 172 129–131.
  • [11] Flemming, J. E., Field, C. A., James, M. C., Jonsen, I. D. and Myers, R. A. (2005). How well can animals navigate? Estimating the circle of confusion over tracking data. Environmetrics 16 1–12.
  • [12] Freedman, D. A. (2005). Statistical Models, Theory and Practice. Cambridge Univ. Press.
  • [13] Gilmartin, W. G. and Eberhardt, L. L. (1995). Status of the Hawaiian monk seal (Monachus schauinslandi) population. Canad. J. Zoology 73 1185–1190.
  • [14] Gutenkunst, R., Newlands, N., Lutcavage, M. and Edelstein-Keshet, L. (2005). Inferring resource distributions from Atlantic bluefin tuna movements: An analysis based on net displacement and length of track. J. Theoret. Biology 245 243–257.
  • [15] Heyde, C. C. (1994). A quasi-likelihood approach to estimating parameters in diffusion-type processes. J. Applied Probab. 31 283–290.
  • [16] Ihaka, R. and Gentleman, R. (1996). R: A language for data analysis and graphics. J. Graph. Computat. Statist. 5 299–314.
  • [17] Jonsen, I. D., Flemming, J. M. and Myers, R. A. (2005). Robust state-space modeling of animal movement data. Ecology 86 2874–2880.
  • [18] Kendall, D. G. (1974). Pole-seeking Brownian motion and bird navigation. J. Roy. Statist. Soc. Ser. B 36 365–417.
  • [19] Kloeden, P. E. and Platen, P. (1995). Numerical Solution of Stochastic Diferential Equations. Springer, New York.
  • [20] Littnan, C. L., Stewart, B. S., Braun, R. C. and Yochem, P. K. (2005). Risk of pathogen exposure of the endangered Hawaiian monk seal ranging in the main Hawaiian Islands. EcoHealth. In review.
  • [21] Littnan, C. L., Stewart, B. S., Yochem, P. K. and Braun, R. C. (2006). Survey for selected pathogens and evaluation of disease risk factors for endangered Hawaiian monk seals in the Main Hawaiian Islands. EcoHealth 3 232–244.
  • [22] Lépingle, D. (1995). Euler scheme for reflected stochastic differential equations. Math. Comput. Simulation 38 119–126.
  • [23] Nelson, E. (1967). Dynamical Theories of Brownian Motion. Princeton Univ. Press.
  • [24] Neyman, J. and Scott, E. L.S. (1956). The distribution of galaxies. Scientific American (September) 187–200.
  • [25] Neyman J., Scott, E. L.S. and Shane, C. D. (1952). On the spatial distribution of galaxies, a specific model. Astrophysical J. 117 92–133.
  • [26] Oksendal, B. (1998). Stochastic Differential Equations, 5th ed. Springer, New York.
  • [27] Parrish, F. A., Abernathy, K., Marshall, G. J. and Buhleier, B. M. (2002). Hawaiian monk seals (Monachus schauinslandi) foraging in deep water coral beds. Marine Mammal Science 18 244–258.
  • [28] Parrish, F. A., Marshall, G. J., Littnan, C. L., Heithaus, M., Canja, S., Becker, B., Braun, R. and Antonelis, G. A. (2005). Foraging of juvenile monk seals at French Frigate Shoals, Hawaii. Marine Mammal Science 21 93–107.
  • [29] Pettersson, R. (1995). Approximations for stochastic differential equations with reflecting convex boundaries. Stochastic Process. Appl. 59 295–308.
  • [30] Rousseuw, P. J., Ruts, I. and Tukey, J. W. (1999). The bagplot: A bivariate boxplot. The American Statistician 53 382–387.
  • [31] Schruben, L. W. (1980). Establishing the credibility of simulations. Simulation 34 101–105.
  • [32] Sorensen, M. (1997). Estimating functions for discretely observed diffusions: A review. Selected Proceedings of the Symposium on Estimating Functions. Lecture Notes Monogr. Ser. 32 305–325. Institute of Mathematical Statistics, Hayward.
  • [33] Spring, J. (2006). On faraway shores, researchers struggle to save the seals. New York Times October 31.
  • [34] Stewart, B. S. (2004). Foraging biogeography of Hawaiian monk seals (Monachus schauinslandi) in the Northwestern Hawaiian Islands (NWHI): Relevance to considerations of marine zones for conservation and management in the NWHI Coral Reef Ecosystem Reserve. HSWRI Technical Report 2004-354 1–91.
  • [35] Stewart, B. S., Antonelis, G. A., Yochem, P. K. and Baker, J. D. (2006). Foraging biogeography of Hawaiian monk seals in the northwestern Hawaiian Islands. Atoll Research Bulletin 543 131–145.
  • [36] Venables, W. N. and Ripley, B. D. (2002). Modern Applied Statistics with S, 4th ed. Springer, New York.
  • [37] Wolf, H. P. (2005). A rough R impementation (sic) of the bagplot. Available at http://cran.r-project.org.