Nonlocal Hyperbolic Models in 1D

  • Raluca Eftimie
Part of the Lecture Notes in Mathematics book series (LNM, volume 2232)


More and more experimental studies show that nonlocal interactions play a role in the majority of biological aggregations. In this chapter we describe a few classes of hyperbolic models that include nonlocal interactions among cells/bacteria/animals, which can influence (1) their turning behaviour, (2) their speeding behaviour, or (3) both turning and speeding behaviours. In addition to emphasising the complexity of the numerical patterns that can be exhibited by these nonlocal models, we also present in more detail some analytical approaches, such as bifurcation theory or parabolic limits, that are being used to classify the various patterns exhibited by these nonlocal models. We conclude by discussing the patterns exhibited by a class of nonlocal models that incorporate explicitly stochastic environmental effects that could impact animal movement.


  1. 1.
    R. Larkin, R. Szafoni, Integr. Comp. Biol. 48(1), 40 (2008)CrossRefGoogle Scholar
  2. 2.
    A. Farnsworth, Auk 122(3), 733 (2005)CrossRefGoogle Scholar
  3. 3.
    M.C. Hager, G.S. Helfman, Behav. Ecol. Sociobiol. 29, 271 (1991)CrossRefGoogle Scholar
  4. 4.
    K. Zuberbühler, R. Noë, R. Seyfarth, Anim. Behav. 53(3), 589 (1997)CrossRefGoogle Scholar
  5. 5.
    S. Wich, C. Nunn, Behav. Ecol. Sociobiol. 52(6), 474 (2002)CrossRefGoogle Scholar
  6. 6.
    T. Das, K. Safferling, S. Rauch, N. Grabe, H. Boehm, J. Spatz, Nat. Cell Biol. 17, 276 (2015)CrossRefGoogle Scholar
  7. 7.
    R. Eftimie, Modelling group formation and activity patterns in self-organising communities of organisms. Ph.D. thesis, University of Alberta (2008)Google Scholar
  8. 8.
    A. Mogilner, L. Edelstein-Keshet, J. Math. Biol. 38, 534 (1999)MathSciNetCrossRefGoogle Scholar
  9. 9.
    L. Edelstein-Keshet, J. Watmough, D. Grünbaum, J. Math. Biol. 36(6), 515 (1998)MathSciNetCrossRefGoogle Scholar
  10. 10.
    N. Armstrong, K. Painter, J. Sherratt, J. Theor. Biol. 243(1), 98 (2006)CrossRefGoogle Scholar
  11. 11.
    R. Seyfarth, D. Cheney, P. Marler, Science 201, 801 (1980)CrossRefGoogle Scholar
  12. 12.
    I. Barber, OIKOS 101, 331 (2003)CrossRefGoogle Scholar
  13. 13.
    J.B.J. rgensen, Trends Ecol. Evol. 25, 292 (2010)Google Scholar
  14. 14.
    A. Leverentz, C. Topaz, A. Bernoff, SIAM J. Appl. Dyn. Syst. 8(3), 880 (2009)MathSciNetCrossRefGoogle Scholar
  15. 15.
    C.M. Topaz, A.L. Bertozzi, SIAM J. Appl. Math. 65, 152 (2004)MathSciNetCrossRefGoogle Scholar
  16. 16.
    A. Mogilner, L. Edelstein-Keshet, L. Bent, A. Spiros, J. Math. Biol. 47, 353 (2003)MathSciNetCrossRefGoogle Scholar
  17. 17.
    R. Fetecau, R. Eftimie, J. Math. Biol. 61(4), 545 (2010)MathSciNetCrossRefGoogle Scholar
  18. 18.
    R. Eftimie, M. Perez, P.L. Buono, Math. Biosci. 289, 96 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    R. Eftimie, G. de Vries, M.A. Lewis, F. Lutscher, Bull. Math. Biol. 69(5), 1537 (2007)MathSciNetCrossRefGoogle Scholar
  20. 20.
    R. Eftimie, G. de Vries, M.A. Lewis, Proc. Natl. Acad. Sci. USA 104(17), 6974 (2007)MathSciNetCrossRefGoogle Scholar
  21. 21.
    R. Eftimie, G. de Vries, M. Lewis, J. Math. Biol. 59, 37 (2009)MathSciNetCrossRefGoogle Scholar
  22. 22.
    X. Zhang, A. Chen, D.D. Leon, H. Li, E. Noiri, V. Moy, M. Goligorsky, Am. J. Physiol. Heart Circ. Physiol. 286(1), H359 (2004)CrossRefGoogle Scholar
  23. 23.
    D. Morale, V. Capasso, K. Oelschläger, J. Math. Biol. 50, 49 (2005)MathSciNetCrossRefGoogle Scholar
  24. 24.
    K. Fellner, G. Raoul, Math. Comput. Modell. 53, 1436 (2011)CrossRefGoogle Scholar
  25. 25.
    G. Raoul, Differ. Integr. Equ. 25(5/6), 417 (2012)MathSciNetGoogle Scholar
  26. 26.
    M. Bodnar, J. Velazquez, J. Differ. Equ. 222(2), 341 (2006)CrossRefGoogle Scholar
  27. 27.
    M. Kac, Rocky Mt. J. Math. 4, 497 (1974)CrossRefGoogle Scholar
  28. 28.
    H.G. Othmer, S.R. Dunbar, W. Alt, J. Math. Biol. 26, 263 (1988)MathSciNetCrossRefGoogle Scholar
  29. 29.
    D. Grünbaum, A. Okubo, in Frontiers in Mathematical Biology. Lecture Notes in Biomathematics, vol. 100, ed. by S.A. Levin (Springer, Berlin Heidelberg, 1994), pp. 296–325Google Scholar
  30. 30.
    J. Hutchinson, P. Waser, Biol. Rev. 82, 335–359 (2007)CrossRefGoogle Scholar
  31. 31.
    P.L. Buono, R. Eftimie, J. Math. Biol. 71(4), 847 (2014)CrossRefGoogle Scholar
  32. 32.
    J. Carrillo, R. Eftimie, F. Hoffmann, Kinet. Relat. Models 8(3), 413 (2015)MathSciNetCrossRefGoogle Scholar
  33. 33.
    B. Pfistner, in Biological Motion. Lecture Notes on Biomathematics, vol. 89, ed. by W. Alt, G. Hoffmann (Springer, Berlin, 1990), pp. 556–563Google Scholar
  34. 34.
    B. Pfistner, J. Biol. Syst. 3, 579 (1995)CrossRefGoogle Scholar
  35. 35.
    M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale, V. Zdravkovic, Proc. Natl. Acad. Sci. USA 105(5), 1232 (2008)CrossRefGoogle Scholar
  36. 36.
    S. Partan, in Handbook of Multi Sensory Processes, ed. by C. Calvert, B. Stein (The MIT Press, Cambridge, 2004), pp. 225–240Google Scholar
  37. 37.
    F. Lutscher, A. Stevens, J. Nonlinear Sci. 12, 619 (2002)MathSciNetCrossRefGoogle Scholar
  38. 38.
    K. Kang, A. Scheel, A. Stevens (2018). arXivGoogle Scholar
  39. 39.
    O. Igoshin, A. Mogilner, R. Welch, D. Kaiser, G. Oster, Proc. Natl. Acad. Sci. USA 98, 14913 (2001)CrossRefGoogle Scholar
  40. 40.
    H. Pomeroy, F. Heppner, Auk 109, 256 (1992)CrossRefGoogle Scholar
  41. 41.
    C. Zmurchok, G. de Vries, PLoS One 13(6), e0198550 (2018)CrossRefGoogle Scholar
  42. 42.
    W. Allee, Q. Rev. Biol. 2(3), 367 (1927)CrossRefGoogle Scholar
  43. 43.
    G. Harris, S. Thirgood, J. Hopcraft, J. Cromsigt, J. Berger, Endanger. Species Res. 7, 55 (2009)CrossRefGoogle Scholar
  44. 44.
    F. Siegert, C. Weijer, Curr. Biol. 5(8), 937 (1995)CrossRefGoogle Scholar
  45. 45.
    P.L. Buono, R. Eftimie, Math. Models Methods Appl. Sci. 24(2), 327–357 (2014)MathSciNetCrossRefGoogle Scholar
  46. 46.
    P.L. Buono, R. Eftimie, in Mathematical Sciences with Multidisciplinary Applications. Springer Proceedings in Mathematics & Statistics, vol. 157 (Springer, Berlin, 2016), pp. 29–59Google Scholar
  47. 47.
    P.L. Buono, R. Eftimie, SIAM J. Appl. Dyn. Syst. 13(4), 1542 (2014)MathSciNetCrossRefGoogle Scholar
  48. 48.
    M. Beekman, D.J.T. Sumpter, F.L.W. Ratnieks, Proc. Natl. Acad. Sci. USA 98(17), 9703 (2001)CrossRefGoogle Scholar
  49. 49.
    J. Buhl, D.J.T. Sumpter, I.D. Couzin, J.J. Hale, E. Despland, E.R. Miller, S.J. Simpson, Science 312, 1402 (2006)CrossRefGoogle Scholar
  50. 50.
    M. Golubitsky, I. Stewart, D.G. Schaeffer, Singularities and Groups in Bifurcation Theory, vol. II (Springer, New York, 1988)zbMATHCrossRefGoogle Scholar
  51. 51.
    M. Golubitsky, I. Stewart, The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space (Birkhäuser, Basel, 2002)Google Scholar
  52. 52.
    I.D. Couzin, J. Krause, R. James, G. Ruxton, N.R. Franks, J. Theor. Biol. 218, 1 (2002)CrossRefGoogle Scholar
  53. 53.
    R. Eftimie, J. Theor. Biol. 337, 42 (2013)MathSciNetCrossRefGoogle Scholar
  54. 54.
    W. Guo, X. Wang, X. Zheng, Physica A 432, 87 (2015)CrossRefGoogle Scholar
  55. 55.
    V. Curtis, Trends Immunol. 35(10), 457 (2014)CrossRefGoogle Scholar
  56. 56.
    C. Sarabian, V. Curtis, R. McMullan, Philos. Trans. R. Soc. Lond. B Biol. Sci. 373(1751), 20170256 (2018)CrossRefGoogle Scholar
  57. 57.
    M. Kavaliers, E. Choleris, Philos. Trans. R. Soc. Lond. B Biol. Sci. 373(1751), 20170206 (2018)CrossRefGoogle Scholar
  58. 58.
    G. Rubin, R. Amlôt, L. Page, S. Wessely, Br. Med. J. 339, 1 (2009)Google Scholar
  59. 59.
    L. Leaver, L. Hopewell, C. Caldwell, L. Mallarky, Anim. Cogn. 10(1), 23 (2007)CrossRefGoogle Scholar
  60. 60.
    T. Pitcher, C. Wyche, in Predators and Prey in Fishes. Developments in Environmental Biology of Fishes, vol. 2, ed. by D. Noakes, D. Lindquist, G. Helfman, J. Ward (Springer, Dordrecht, 1983), pp. 193–204Google Scholar
  61. 61.
    R. Eftimie, A. Coulier, Math. Model Nat. Phenom. 10(2), 27 (2015)MathSciNetCrossRefGoogle Scholar
  62. 62.
    D. Álvarez, A. Nicieza, J. Fish Biol. 63(6), 1565 (2003)CrossRefGoogle Scholar
  63. 63.
    S. Gueron, S.A. Levin, D.I. Rubenstein, J. Theor. Biol. 182, 85 (1996)CrossRefGoogle Scholar
  64. 64.
    B. Partridge, Sci. Am. 246(6), 114 (1982)CrossRefGoogle Scholar
  65. 65.
    D. Helbing, P. Molnar, Phys. Rev. E 51(5), 4282 (1995)CrossRefGoogle Scholar
  66. 66.
    W. Takken, Invertebr. Reprod. Dev. 36(1–3), 67 (1999)CrossRefGoogle Scholar
  67. 67.
    K. Lika, T. Hallam, J. Math. Biol. 38, 346 (1999)MathSciNetCrossRefGoogle Scholar
  68. 68.
    R. Mickens, SIAM Rev. 30(4), 629 (1988)MathSciNetCrossRefGoogle Scholar
  69. 69.
    D. Helbing, A. Hennecke, V. Shvetsov, M. Treiber, Math. Comput. Model. 35(5–6), 517 (2002)CrossRefGoogle Scholar
  70. 70.
    J.K. Parrish, L.E. Keshet, Science 284, 99 (1999)CrossRefGoogle Scholar
  71. 71.
    H. Brumm (ed.), Animal Communication and Noise (Springer, Berlin Heidelberg, 2013)Google Scholar
  72. 72.
    L. Rabin, B. McCowan, S. Hooper, D. Owings, Int. J. Comp. Psychol. 16, 172 (2003)Google Scholar
  73. 73.
    T. Ord, J. Stamps, Proc. Natl. Acad. Sci. USA 105(48), 18830 (2008)CrossRefGoogle Scholar
  74. 74.
    A. Codarin, L. Wysocki, F. Ladich, M. Picciulin, Mar. Pollut. Bull. 58, 1880 (2009)CrossRefGoogle Scholar
  75. 75.
    F. VanDyke, R. Broke, H. Shaw, B. Ackerman, T. Hemker, F. Lindzey, J. Wildl. Manage. 50(1), 95 (1986)CrossRefGoogle Scholar
  76. 76.
    M. Tucker, K. Böhning-Gaese, W. Fagan, J. Fryxell, B.V. Moorter, S. Alberts, A. Ali, A. Allen, N. Attias, T. Avgar, H. Bartlam-Brooks, B. Bayarbaatar, J. Belant, A. Bertassoni, D. Beyer, L. Bidner, F. van Beest, S. Blake, N. Blaum, C. Bracis, D. Brown, P. de Bruyn, F. Cagnacci, J.M. Calabrese, C. Camilo-Alves, S. Chamaillé-Jammes, A. Chiaradia, S. Davidson, T. Dennis, S. DeStefano, D. Diefenbach, I. Douglas-Hamilton, J. Fennessy, C. Fichtel, W. Fiedler, C. Fischer, I. Fischhoff, C. Fleming, A. Ford, S. Fritz, B. Gehr, J. Goheen, E. Gurarie, M. Hebblewhite, M. Heurich, A. Hewison, C. Hof, E. Hurme, L. Isbell, R. Janssen, F. Jeltsch, P. Kaczensky, A. Kane, P. Kappeler, M. Kauffman, R. Kays, D. Kimuyu, F. Koch, B. Kranstauber, S. LaPoint, P. Leimgruber, J. Linnell, P. López-López, A. Markham, J. Mattisson, E.P. Medici, U. Mellone, E. Merrill, G. de Miranda Mourào, R. Morato, N. Morellet, T. Morrison, S. Diaz-Munoz, A. Mysterud, D. Nandintsetseg, R. Nathan, A. Niamir, J. Odden, R. O’Hara, L. Oliveira-Santos, K. Olson, B. Patterson, R. de Paula, L. Pedrotti, B. Reineking, M. Rimmler, T. Rogers, C. Rolandsen, C. Rosenberry, D. Rubenstein, K. Safi, S. Saïd, N. Sapir, H. Sawyer, N. Schmidt, N. Selva, A. Sergiel, E. Shiilegdamba, J. Silva, N. Singh, E. Solberg, O. Spiegel, O. Strand, S. Sundaresan, W. Ullmann, U. Voigt, J. Wall, D. Wattles, M. Wikelski, C. Wilmers, J. Wilson, G. Wittemyer, F. Zieba, T. Zwijacz-Kozica, T. Mueller, Science 358(6374), 466 (2018)CrossRefGoogle Scholar
  77. 77.
    P. Marler, M. Konishi, A. Lutjen, M. Waser, Proc. Natl. Acad. Sci. USA 70, 1393 (1973)CrossRefGoogle Scholar
  78. 78.
    E. Chang, M. Merzenich, Science 300, 498 (2003)CrossRefGoogle Scholar
  79. 79.
    S. Sobrian, V. Vaughn, W. Ashe, B. Markovic, V. Djuric, B. Jankovic, Environ. Res. 73, 227 (1997)CrossRefGoogle Scholar
  80. 80.
    J. Hochel, R. Pirow, M. Nichelmann, Comp. Biochem. Physiol. 131, 805 (2002)CrossRefGoogle Scholar
  81. 81.
    H. Ising, B. Kruppa, Noise Health 6, 5 (2004)Google Scholar
  82. 82.
    I. Croy, M. Smith, K. Waye, Br. Med. J. Open 3, e002655 (2013)Google Scholar
  83. 83.
    C. Francis, C. Ortega, A. Cruz, Curr. Biol. 45, 1415 (2009)CrossRefGoogle Scholar
  84. 84.
    R. Eftimie, in BIOMATH (2018)Google Scholar
  85. 85.
    D. proppe, C. Sturdy, C.S. Clair, PLoS One 6, e25413 (2011)CrossRefGoogle Scholar
  86. 86.
    F. Ladich, in Animal Communication and Noise, ed. by H. Brumm (Springer, Berlin, 2013), pp. 65–90Google Scholar
  87. 87.
    A.J. Lotka, Elements of Physical Biology (Williams and Wilkins, Baltimore, 1924)zbMATHGoogle Scholar
  88. 88.
    T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, O. Shochet, Phys. Rev. Lett. 75(6), 1226 (1995)MathSciNetCrossRefGoogle Scholar
  89. 89.
    T. Vicsek, A. Czirok, I.J. Farkas, D. Helbing, Physica A 274, 182 (1999)CrossRefGoogle Scholar
  90. 90.
    T. Vicsek, A. Zafeiris, Phys. Rep. 517(3–4), 71 (2010)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Raluca Eftimie
    • 1
  1. 1.Division of MathematicsUniversity of DundeeDundeeUK

Personalised recommendations