Skip to main content
SpringerLink
Log in

Kinetic Models for Pattern Formation in Animal Aggregations: A Symmetry and Bifurcation Approach

  • Chapter
  • First Online:
Active Particles, Volume 2

  • 797 Accesses

Abstract

In this study we start by reviewing a class of 1D hyperbolic/kinetic models (with two velocities) used to investigate the collective behaviour of cells, bacteria or animals. We then focus on a restricted class of nonlocal models that incorporate various inter-individual communication mechanisms, and discuss how the symmetries of these models impact the various types of spatially heterogeneous and spatially homogeneous equilibria exhibited by these nonlocal models. In particular, we characterise a new type of equilibria that was not discussed before for this class of models, namely a relative equilibria. Then we simulate numerically these models and show a variety of spatio-temporal patterns (including classic equilibria and relative equilibria) exhibited by these models. We conclude by introducing a continuation algorithm (which takes into account the models symmetries) that allows us to track the solutions bifurcating from these different equilibria. Finally, we apply this algorithm to identify a D3-symmetric steady-state solution.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. E. L. Allgower and K. Georg. Introduction to Numerical Continuation Methods. SIAM, 2003.

    MATH  Google Scholar 

  2. I. Aoki. A simulation study on the schooling mechanism in fish. Bull. Jpn. Soc. Sci. Fish, pages 1081–1088, 1982.

    Google Scholar 

  3. S. Arganda, A. Pérez-Escudero, and G.G. de Polavieja. A common rule for decision making in animal collectives across species. Proc. Matl. Acad. Sci., 109:20508–20513, 2012.

    Google Scholar 

  4. P. Ashwin and I. Melbourne. Noncompact drift for relative equilibria and relative periodic orbits. Nonlinearity, 10:595, 1997.

    MathSciNet  MATH  Google Scholar 

  5. P. J. Aston, A. Spence, and W. Wu. Bifurcation to rotating waves in equations with O(2)–symmetry. SIAM J. Appl. Math., 52:792–809, 1992.

    MathSciNet  MATH  Google Scholar 

  6. D. Avitable and K.C.A. Wedgwood. Macroscopic coherent structures in a stochastic neural network: from interface dynamics to coarse-grained bifurcation analysis. J. Math. Biol., 75(4):885–928, 2017.

    MathSciNet  MATH  Google Scholar 

  7. E. Barbera, G. Consolo, and G. Valenti. A two or three compartments hyperbolic reaction-diffusion model for the aquatic food chain. Math. Biosci. Eng., 12(3):451–472, 2015.

    MathSciNet  MATH  Google Scholar 

  8. E. Barbera, C. Currò, and G. Valenti. Wave features of a hyperbolic prey-predator model. Math. Methods Appl. Sci., 33(12):1504–1515, 2010.

    MathSciNet  MATH  Google Scholar 

  9. N. Bellomo, A. Bellouquid, and M. Delitala. From the mathematical kinetic theory of active particles to multiscale modelling of complex biological systems. Math. Comput. Model., 47(7-8):687–698, 2008.

    MathSciNet  MATH  Google Scholar 

  10. A. Berdhal, C.J. Torney, C.C. Ioannou, J.J. Faria, and I.D. Couzin. Emergent sensing of complex environments by mobile animal groups. Science, 339(6119):574–576, 2013.

    Google Scholar 

  11. P-L. Buono and R. Eftimie. Analysis of Hopf/Hopf bifurcations in nonlocal hyperbolic models for self-organised aggregations. Math. Models Methods Appl. Sci., 24(2):327–357, 2014.

    MathSciNet  MATH  Google Scholar 

  12. P-L. Buono and R. Eftimie. Codimension-two bifurcations in animal aggregation models with symmetry. SIAM J. Appl. Dyn. Syst., 13(4):1542–1582, 2014.

    MathSciNet  MATH  Google Scholar 

  13. P.-L. Buono and R. Eftimie. Lyapunov-Schmidt and centre-manifold reduction methods for nonlocal PDEs modelling animal aggregations. In B. Tony, editor, Mathematical Sciences with Multidisciplinary Applications. Springer Proceedings in Mathematics & Statistics, volume 157, pages 29–59. Springer, Cham, 2016.

    Google Scholar 

  14. D. Burini, L. Gibelli, and N. Outada. A kinetic theory approach to the modeling of complex living systems. In N. Bellomo, P. Degond, and E. Tadmor, editors, Active Particles, volume 1, pages 229–258. Birkhäuser, Basel, 2017.

    Google Scholar 

  15. D.S. Calovi, U. Lopez, S. Ngo, C. Sire, H. Chaté, and G. Theraulaz. Swarming, schooling, milling: phase diagram of data-driven fish school model. New Journal of Physics, 16:015026, 2014.

    Google Scholar 

  16. H. Chaté, F. Ginelli, G. Grégoire, F. Peruani, and F. Raynaud. Modeling collective motion: variations on the Vicsek model. The European Physics Journal B, 64(3-4):451–456, 2008.

    Google Scholar 

  17. S.-H. Choi and Y.-J. Kim. A discrete velocity kinetic model with food metric: chemotaxis travelling waves. Bull. Math. Biol., 79(2):277–302, 2017.

    MathSciNet  MATH  Google Scholar 

  18. R.M. Colombo and E. Rossi. Hyperbolic predators vs. parabolic prey. Communications in Mathematical Sciences, 13(2):369–400, 2015.

    Google Scholar 

  19. I.D. Couzin, J. Krause, R. James, G.D. Ruxton, and N.R. Franks. Collective memory and spatial sorting in animal groups. J. Theor. Biol., 218:1–11, 2002.

    MathSciNet  Google Scholar 

  20. P. Degond, A. Frouvelle, S. Merino-Aceituno, and A. Trescases. Quaternions in collective dynamics. Multiscale Model. Simul., 16(1):28–77, 2018.

    MathSciNet  MATH  Google Scholar 

  21. R. Eftimie. Hyperbolic and kinetic models for self-organised biological aggregations and movement: a brief review. J. Math. Biol., 65(1):35–75, 2012.

    MathSciNet  MATH  Google Scholar 

  22. R. Eftimie. Simultaneous use of different communication mechanisms leads to spatial sorting and unexpected collective behaviours in animal groups. J. Theor. Biol., 337:42–53, 2013.

    MathSciNet  MATH  Google Scholar 

  23. R. Eftimie, G. de Vries, and M.A. Lewis. Complex spatial group patterns result from different animal communication mechanisms. Proc. Natl. Acad. Sci., 104(17):6974–6979, 2007.

    MathSciNet  MATH  Google Scholar 

  24. R. Eftimie, G. de Vries, M.A. Lewis, and F. Lutscher. Modeling group formation and activity patterns in self-organising collectives of individuals. Bull. Math. Biol., 69(5):1537–1566, 2007.

    MathSciNet  MATH  Google Scholar 

  25. R. Fetecau. Collective behaviour of biological aggregations in two dimensions: a nonlocal kinetic model. Math. Models Methods Appl. Sci., 21:1539–1569, 2011.

    MathSciNet  MATH  Google Scholar 

  26. B. Fiedler, S. Björn, A. Scheel, and C. Wulff. Bifurcation form relative equilibria of nonimpact group actions: Skew products, meanders, and drifts. Documenta Mathematica, 1:479–505, 1996.

    MathSciNet  MATH  Google Scholar 

  27. F. Filbet, P. Laurencot, and B. Perthame. Derivation of hyperbolic models for chemosensitive movement. J. Math. Biol., 50(2):189–207, 2005.

    MathSciNet  MATH  Google Scholar 

  28. A. Filella, F. Nadal, C. Sire, E. Kanso, and C. Eloy. Model of collective fish behavior with hydrodynamic interactions. Phys. Rev. Lett., 120:198101, 2018.

    Google Scholar 

  29. M. Golubitsky, I. Stewart, and D.G. Schaeffer. Singularities and Groups in Bifurcation Theory. Volume 2. Springer-Verlag New York Inc., 1988.

    Google Scholar 

  30. K.P. Hadeler. Reaction transport equations in biological modeling. Mathematical and Computer Modelling, 31(4-5):75–81, 2000. Proceedings of the Conference on Dynamical Systems in Biology and Medicine.

    Google Scholar 

  31. M. Haragus and G. Iooss. Local bifurcations, centre manifolds, and normal forms in infinite-dimensional systems. Springer, 2010.

    MATH  Google Scholar 

  32. T. Hillen. Invariance principles for hyperbolic random walk systems. J. Math. Anal. Appl., 210(1):360–374, 1997.

    MathSciNet  MATH  Google Scholar 

  33. T. Hillen. Hyperbolic models for chemosensitive movement. Mathematical Models and Methods in Applied Sciences, 12(07):1007–1034, 2002.

    MathSciNet  MATH  Google Scholar 

  34. T. Hillen. Existence theory for correlated random walks on bounded domains. Canad. Appl. Math. Quart, 18(1):1–40, 2010.

    MathSciNet  MATH  Google Scholar 

  35. T. Hillen and K.P. Hadeler. Hyperbolic systems and transport equations in mathematical biology. In Gerald Warnecke, editor, Analysis and Numerics for Conservation Laws, pages 257–279. Springer Berlin Heidelberg, 2005.

    Google Scholar 

  36. A. Huth and C. Wissel. The simulation of fish schools in comparison with experimental data. Ecol. Model, 75/76:135–145, 1994.

    Google Scholar 

  37. H. Inaba. Threshold and stability results for an age-structured epidemic model. J. Math. Biol., 28:411–434, 1990.

    MathSciNet  MATH  Google Scholar 

  38. B.L. Keyfitz and N. Keyfitz. The McKendrick partial differential equation and its uses in epidemiology and population study. Math. Comput. Modelling, 26(6):1–9, 1997.

    MathSciNet  MATH  Google Scholar 

  39. I. Kmit. Fredholm solvability of a periodic Neumann problem for a linear telegraph equation. Ukrainian Mathematical Journal, 65(3), 2013.

    Google Scholar 

  40. I. Kmit and L. Recke. Hopf bifurcation for semilinear dissipative hyperbolic systems. J. Differential Equations, 257:264–309, 2014.

    MathSciNet  MATH  Google Scholar 

  41. M. Kovacic. On matrix-free pseudo-arclength continuation methods applied to a nonlocal PDE in 1+1d with pseudo-spectral time-stepping. Master’s thesis, University of Ontario Institute of Technology, 2013.

    Google Scholar 

  42. M. Krupa. Bifurcations of relative equilibria. SIAM J. Math. Anal., 21(6):1453–1486, 1990.

    MathSciNet  MATH  Google Scholar 

  43. F. Lutscher. Modeling alignment and movement of animals and cells. J. Math. Biol., 45:234–260, 2002.

    MathSciNet  MATH  Google Scholar 

  44. P. Magal and S. Ruan. On integrated semigroups and age structured models in Lp spaces. Differential Integral Equations, 20(2):197–239, 2007.

    MathSciNet  MATH  Google Scholar 

  45. J.K. Parrish and L. Edelstein-Keshet. Complexity, pattern, and evolutionary trade-offs in animal aggregations. Science, 284(2):99–101, 1999.

    Google Scholar 

  46. J.K. Parrish, S.V. Viscido, and D. Grünbaum. Self-organised fish schools: an examination of emergent properties. Biol. Bull., 202:296–305, 2002.

    Google Scholar 

  47. B. Pfistner. A one dimensional model for the swarming behaviour of Myxobacteria. In G. Hoffmann W. Alt, editor, Biological Motion. Lecture Notes on Biomathematics, pages 556–563. Springer, Berlin, 1990.

    Google Scholar 

  48. M. Pineda, C.J. Weijer, and R. Eftimie. Modelling cell movement, cell differentiation, cell sorting and proportion regulation in Dictyostelium discoideum aggregations. J. Theor. Biol., 370:135–150, 2015.

    MathSciNet  MATH  Google Scholar 

  49. W. Pönisch, C.A. Weber, G. Juckeland, N. Biais, and V. Zaburdaev. Multiscale modeling of bacterial colonies: how pili mediate the dynamics of single cells and cellular aggregates. New Journal of Physics, 19:015003, 2017.

    MATH  Google Scholar 

  50. W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery. Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, 2007.

    MATH  Google Scholar 

  51. J. Rankin, D. Avitabile, J. Baladron, G. Faye, and D. J. B. Lloyd. Continuation of localized coherent structures in nonlocal neural field equations. SIAM J. Sci. Comp., 36:B70–B93, 2014.

    MathSciNet  MATH  Google Scholar 

  52. J. Sánchez Umbría and M. Net. Numerical continuation methods for large–scale dissipative dynamical systems. Eur. Phys. J. – Spec. Top., 225:2465–2486, 2016.

    Google Scholar 

  53. A.P. Solon, H. Chaté, and J. Tailleur. From phase to microphase separation in flocking models: the essential role of nonequilibrium fluctuations. Phys. Rev. Lett., 114(6):068101, 2015.

    Google Scholar 

  54. Pliny the Elder. The Natural History. H.G. Bohn, London, 1855. (Translated by John Bostock M.D. and F.R.S. Henry T. Riley Esq.).

    Google Scholar 

  55. C.M. Topaz and A.L. Bertozzi. Swarming patterns in a two-dimensional kinematic model for biological groups. SIAM J. Appl. Math, 65(1):152–174, 2004.

    MathSciNet  MATH  Google Scholar 

  56. C.M. Topaz, A.L. Bertozzi, and M.A. Lewis. A nonlocal continuum model for biological aggregation. Bull. Math. Biol., 68:1601–1623, 2006.

    MathSciNet  MATH  Google Scholar 

  57. L. N. Trefethen. Spectral methods in MATLAB. SIAM, 2000.

    MATH  Google Scholar 

  58. T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet. Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett., 75:1226, 1995.

    MathSciNet  Google Scholar 

  59. M. Witten, editor. Hyperbolic Partial Differential Equations. Populations, reactors, tides and waves: theory and applications. Pergamon, 1983.

    Google Scholar 

  60. D.J. Wollkind. Applications of linear hyperbolic partial equations: predator-prey systems and gravitational instability of nebulae. Mathematical Modelling, 7:413–428, 1986.

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

PLB and LvV acknowledge the financial support from NSERC in the form of a Discovery Grant.

Author information

Authors and Affiliations

  1. University of Ontario Institute of Technology, Oshawa, ON, Canada

    Pietro-Luciano Buono & Lennaert van Veen

  2. Present address: Université du Québec à Rimouski, Rimouski, Qué, Canada

    Pietro-Luciano Buono

  3. University of Dundee, Dundee, UK

    Raluca Eftimie

  4. Simon Fraser University, Burnaby, BC, Canada

    Mitchell Kovacic

Authors
  1. Pietro-Luciano Buono
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Raluca Eftimie
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mitchell Kovacic
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Lennaert van Veen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Raluca Eftimie .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, Politecnico di Torino, Torino, Italy

    Nicola Bellomo

  2. Department of Mathematics, Imperial College London, London, UK

    Pierre Degond

  3. CSCAMM, CSIC Bldg 406, University of Maryland, College Park, USA

    Eitan Tadmor

Appendix

Appendix

In Table 2 we summarise the parameters that appear in the nonlocal models (5).

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Buono, PL., Eftimie, R., Kovacic, M., van Veen, L. (2019). Kinetic Models for Pattern Formation in Animal Aggregations: A Symmetry and Bifurcation Approach. In: Bellomo, N., Degond, P., Tadmor, E. (eds) Active Particles, Volume 2. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-20297-2_2

Download citation

Publish with us

Policies and ethics

Search

Navigation