Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-19T04:27:44.119Z Has data issue: false hasContentIssue false
  • English
  • Français

THE PHILLIP ISLAND PENGUIN PARADE (A MATHEMATICAL TREATMENT)

Part of: Applications - Dynamical systems and ergodic theory Mathematical biology in general

Published online by Cambridge University Press:  08 August 2018

SERENA DIPIERRO Open the ORCID record for SERENA DIPIERRO [Opens in a new window] ,
LUCA LOMBARDINI Open the ORCID record for LUCA LOMBARDINI [Opens in a new window] ,
PIETRO MIRAGLIO Open the ORCID record for PIETRO MIRAGLIO [Opens in a new window]  and
ENRICO VALDINOCI Open the ORCID record for ENRICO VALDINOCI [Opens in a new window]
SERENA DIPIERRO*
Affiliation:
Dipartimento di Matematica, Università degli Studi di Milano, 20133 Milan, Italy email [email protected] School of Mathematics and Statistics, University of Western Australia, Crawley, WA 6009, Australia
LUCA LOMBARDINI
Affiliation:
Dipartimento di Matematica, Università degli Studi di Milano, 20133 Milan, Italy email [email protected] Faculté des Sciences, Université de Picardie Jules Verne, 80039 Amiens CEDEX 1, France email [email protected]
PIETRO MIRAGLIO
Affiliation:
Dipartimento di Matematica, Università degli Studi di Milano, 20133 Milan, Italy email [email protected] Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, 08028 Barcelona, Spain email [email protected]
ENRICO VALDINOCI
Affiliation:
Dipartimento di Matematica, Università degli Studi di Milano, 20133 Milan, Italy email [email protected] Istituto di Matematica Applicata e Tecnologie Informatiche, 27100 Pavia, Italy email [email protected] School of Mathematics and Statistics, University of Melbourne, Parkville, Vic. 3010, Australia School of Mathematics and Statistics, University of Western Australia, Crawley, WA 6009, Australia
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Penguins are flightless, so they are forced to walk while on land. In particular, they show rather specific behaviours in their homecoming, which are interesting to observe and to describe analytically. We observed that penguins have the tendency to waddle back and forth on the shore to create a sufficiently large group, and then walk home compactly together. The mathematical framework that we introduce describes this phenomenon, by taking into account “natural parameters”, such as the eyesight of the penguins and their cruising speed. The model that we propose favours the formation of conglomerates of penguins that gather together, but, on the other hand, it also allows the possibility of isolated and exposed individuals.

The model that we propose is based on a set of ordinary differential equations. Due to the discontinuous behaviour of the speed of the penguins, the mathematical treatment (to get existence and uniqueness of the solution) is based on a “stop-and-go” procedure. We use this setting to provide rigorous examples in which at least some penguins manage to safely return home (there are also cases in which some penguins remain isolated). To facilitate the intuition of the model, we also present some simple numerical simulations that can be compared with the actual movement of the penguin parade.

Keywords

population dynamicsEudyptula minorPhillip Islandmathematical models

MSC classification

Primary: 92B05: General biology and biomathematics
Secondary: 92B25: Biological rhythms and synchronization 37N25: Dynamical systems in biology
Type
Research Article
Information
The ANZIAM Journal , Volume 60 , Issue 1 , July 2018 , pp. 27 - 54
Copyright
© 2018 Australian Mathematical Society 

References

Ballerini, M., Cabibbo, N., Candelier, R., Cavagna, A., Cisbani, E., Giardina, I., Lecomte, V., Orlandi, A., Parisi, G., Procaccini, A., Viale, M. and Zdravkovic, V., “Interaction ruling animal collective behaviour depends on topological rather than metric distance: Evidence from a field study”, Proc. Natl. Acad. Sci. USA 105 (2008) 1232–1237; doi:10.1073/pnas.0711437105.Google Scholar
Barbu, V., Differential equations, Springer Undergraduate Mathematics Series (Springer, Cham, 2016); Translated from the 1985 Romanian original by Liviu Nicolaescu. MR 3585801.Google Scholar
Berdahl, A., Torney, C. J., Ioannou, C. C., Faria, J. and Couzin, I. D., “Emergent sensing of complex environments by mobile animal groups”, Science 339 (2013) 574–576;doi:10.1126/science.1225883.Google Scholar
Bertozzi, A. L., Rosado, J., Short, M. B. and Wang, L., “Contagion shocks in one dimension?”, J. Stat. Phys. 158 (2015) 647–664; doi:10.1007/s10955-014-1019-6.Google Scholar
Chiaradia, A., McBride, J., Murray, T. and Dann, P., “Effect of fog on the arrival time of little penguins Eudyptula minor: a clue for visual orientation?”, J. Ornithol. 148 (2007) 229–233;doi:10.1007/s10336-007-0125-5.Google Scholar
Cristiani, E., Piccoli, B. and Tosin, A., “Modeling self-organization in pedestrians and animal groups from macroscopic and microscopic viewpoints”, in: Mathematical Modeling of Collective Behaviour in Socio-Economic and Life Sciences, Model. Simul. Sci. Eng. Technol. (Birkhauser Boston, Boston, MA, 2010) 337–364; doi:10.1007/978-0-8176-4946-3 13.Google Scholar
Daniel, T., Chiaradia, A., Logan, M., Quinn, G. and Reina, R., “Synchronized group association in little penguins, Eudyptula Minor”, Anim. Behav. 74 (2007) 1241–1248;doi:10.1016/j.anbehav.2007.01.029.Google Scholar
DiPerna, R. J. and Lions, P.-L., “Ordinary differential equations, transport theory and Sobolev spaces”, Invent. Math. 98 (1989) 511–547; doi:10.1007/BF01393835.Google Scholar
Du, N., Fan, J., Wu, H., Chen, S. and Liu, Y., “An improved model of heat transfer through penguin feathers and down”, J. Theoret. Biol. 248 (2007) 727–735; doi:10.1016/j.jtbi.2007.06.020.Google Scholar
Farine, D. R., Montiglio, P. O. and Spiegel, O., “From individuals to groups and back: the evolutionary implications of group phenotypic composition”, Trends Ecol. Evolut. 30 (2015) 609–621; doi:10.1016/j.tree.2015.07.005.Google Scholar
Filippov, A. F., Differential equations with discontinuous righthand sides, Mathematics and its Applications (Soviet Series) (Kluwer Academic Publishers Group, Dordrecht, 1988); Translated from the Russian. MR 1028776.Google Scholar
Gerum, R. C., Fabry, B., Metzner, C., Beaulieu, M., Ancel, A. and Zitterbart, D. P., “The origin of traveling waves in an emperor penguin huddle”, New J. Phys. 15 (2013) 125022; http://iopscience.iop.org/article/10.1088/1367-2630/15/12/125022/pdf.Google Scholar
Gheraibia, Y. and Moussaoui, A., “Penguins Search Optimization Algorithm (PeSOA)”, in: Recent Trends in Applied Artificial Intelligence: 26th International Conference on Industrial, Engineering and Other Applications of Applied Intelligent Systems, IEA/AIE 2013, Amsterdam, The Netherlands, June 17–21, 2013 (Springer, Berlin, 2013).Google Scholar
Giling, D., Reina, R. D. and Hogg, Z., “Anthropogenic influence on an urban colony of the little penguin Eudyptula minor ”, Marine Freshwater Res. 59 (2008) 647–651; doi:10.1071/MF08003.Google Scholar
Hegselmann, R. and Krause, U., “Opinion dynamics and bounded confidence: models, analysis and simulation”, J. Artif. Soc. Soc. Simul. 5 (2002) 1–33; http://jasss.soc.surrey.ac.uk/5/3/2/2.pdf.Google Scholar
Kong, Q., A Short Course in Ordinary Differential Equations (Universitext, Springer, Cham, 2014).Google Scholar
Laaksonen, S. M., Chiaradia, A. and Reina, R. D., “Behavioural plasticity of a multihabitat animal, the little penguin, Eudyptula minor, in response to tidal oscillations on its interhabitat transitions”, Preprint (2016).Google Scholar
Macintosh, A. J. J., Pelletier, L., Chiaradia, A., Kato, A. and Ropert-Coudert, Y., “Temporal fractals in seabird foraging behaviour: diving through the scales of time”, Sci. Rep. 3 (2013) 1–10;doi:10.1038/srep01884.Google Scholar
Miller, N., Garnier, S., Hartnett, A. T. and Couzin, I. D., “Both information and social cohesion determine collective decisions in animal groups”, Proc. Natl Acad. Sci. USA 110 (2013) 5263–5268; doi:10.1073/pnas.1217513110.Google Scholar
Ragonnet, R., Jumentier, R. and Beckermann, B., “La marche de l’empereur”, Matapli 102 (2013) 71–82; (French). MR 3235841.Google Scholar
Reynolds, A. M., Ropert-Coudert, Y., Kato, A., Chiaradia, A. and MacIntosh, A. J. J., “A priority-based queuing process explanation for scale-free foraging behaviours”, Anim. Behav. 108 (2015) 67–71; doi:10.1016/j.anbehav.2015.07.022.Google Scholar
Rodríguez, A., Chiaradia, A., Wasiak, P., Renwick, L. and Dann, P., “Waddling on the dark side: ambient light affects attendance behavior of little penguins”, J. Biol. Rhythms 31 (2016) 194–204; doi:10.1177/0748730415626010.Google Scholar
Sidhu, L. A., Catchpole, E. A. and Dann, P., “Modelling banding effect and tag loss for Little Penguins Eudyptula minor ”, ANZIAM J. Electron. Suppl. 52 (2010) C206–C221;https://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/viewFile/3941/1431.Google Scholar
Skewgar, E. A., “Behavior of Magellanic penguins at sea”, Ph.D. Thesis, University of Washington, ProQuest LLC, Ann Arbor, MI, 2009, MR 2718048.Google Scholar