Evolutionary Ecology

, Volume 12, Issue 5, pp 503–522 | Cite as

Schooling as a strategy for taxis in a noisy environment

  • Daniel GrÜnbaum


Many aquatic animals face a fundamental problem during foraging and migratory movements: while their resources commonly vary at large spatial scales, they can only sample and assess their environment at relatively small, local spatial scales. Thus, they are unable to choose movement directions by directly sampling distant parts of their environment. A common strategy to overcome this problem is taxis, a behaviour in which an animal performs a biased random walk by changing direction more rapidly when local conditions are getting worse. Such an animal spends more time moving in right directions than wrong ones, and eventually gets to a favourable area. Taxis is inefficient, however, when environmental gradients are weak or overlain by ‘noisy’ small-scale fluctuations. In this paper, I show that schooling behaviour can improve the ability of animals performing taxis to climb gradients, even under conditions when asocial taxis would be ineffective. Schooling is a social behaviour incorporating tendencies to remain close to and align with fellow members of a group. It enhances taxis because the alignment tendency produces tight angular distributions within groups, and dampens the stochastic effects of individual sampling errors. As a result, more school members orient up-gradient than in the comparable asocial case. However, overly strong schooling behaviour makes the school slow in responding to changing gradient directions. This trade-off suggests an optimal level of schooling behaviour for given spatio-temporal scales of environmental variations. Social taxis may enhance the selective value of schooling in pelagic grazers such as herrings, anchovies and Antarctic krill. Furthermore, the degree of aggregation in a population of schooling animals may affect directly the rate and direction of migration and foraging movements.

aggregation optimal foraging resource distributions schooling search strategies social behaviour taxis 


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  1. Alt, W. (1980) Biased random walk models for chemotaxis and related diffusion approximations. J. Math. Biol. 9, 147–177.Google Scholar
  2. Alt, W. (1985) Degenerate diffusion equations with drift functionals modelling aggregation. Nonlinear Analysis, Theory, Methods & Applications 9, 811–836.Google Scholar
  3. Aoki, I. (1982) A simulation study on the schooling mechanism in fish. Bull. Jap. Soc. Sci. Fish. 48, 1081–1088.Google Scholar
  4. Atema, J. (1988) Distribution of chemical stimuli. In Sensory Biology of Aquatic Animals (J. Atema, R.R. Fay, A.N. Popper and W.N. Tavolga, eds), pp. 29–57. Springer-Verlag, Berlin.Google Scholar
  5. Baker, P.S., Gewecke, M. and Cooter, R.J. (1984) Flight orientation of swarming Locusta migratoria. Pysiol. Ent. 9, 247–252.Google Scholar
  6. Clark, C.W. and Dukas, R. (1994) Balancing foraging and antipredator demands: An advantage of sociality. Am. Nat. 144, 542–548.Google Scholar
  7. Daly, K.L. and Macaulay, M.C. (1991) Influence of physical and biological mesoscale dynamics on the seasonal distribution and behavior of Euphausia superba in the Antarctic marginal ice zone. Mar. Ecol. Prog. Ser. 79, 37–66.Google Scholar
  8. Davis, C.S., Flierl, G.R., Wiebe, P.H. and Franks, P.J.S. (1991) Micropatchiness, turbulence, and recruitment in plankton. J. Mar. Res. 49, 1–43.Google Scholar
  9. Domenici, P. and Batty, R.S. (1994) Escape manoeuvres of schooling Clupea harengus. J. Fish Biol. 45 (suppl. A), 97–110.Google Scholar
  10. Edelstein-Keshet, L. (1988) Mathematical Models in Biology. Random House, New York.Google Scholar
  11. Ezoe, H., Iwasa, Y. and Umeda, T. (1994) Aggregation by chemotactic random walk: Drifting clusters and chemotactic friction. J. Theor. Biol. 168, 259–267.Google Scholar
  12. Foster, S.A. (1987) Acquisition of a defended resource: A benefit of group foraging for the neotropical wrasse, Thalassoma lucasanum. Env. Biol. Fish. 3, 215–222.Google Scholar
  13. Grünbaum, D. (1994) Translating stochastic density-dependent individual behavior with sensory constraints to a continuum model of animal swarming. J. Math. Biol. 33, 139–161.Google Scholar
  14. Grünbaum, D. and Okubo, A. (1994) Modelling social animal aggregations. In Frontiers in Theoretical Biology (S.A. Levin, ed.), pp. 296–325. Lecture Notes in Biomathematics Vol. 100. Springer-Verlag, Berlin.Google Scholar
  15. Gueron, S. and Levin, S.A. (1995) The dynamics of group formation. Math. Biosci. 128, 243–264.Google Scholar
  16. Hamner, W.M. (1984) Aspects of schooling in Euphausia superba. J. Crust. Biol. 4 (Spec. No. 1), 67–74.Google Scholar
  17. Hamner, W.M., Hamner, P.P., Strand, S.W. and Gilmer, R.W. (1983) Behaviour of Antarctic krill, Euphausia superba: Chemoreception, feeding, schooling, and molting. Science 220, 433–435.Google Scholar
  18. Haney, J.C., Fristrup, K.M. and Lee, D.S. (1992) Geometry of visual recruitment by seabirds to ephemeral forgaging flocks. Ornis Scand. 23, 49–62.Google Scholar
  19. Heppner, F. and Grenander, U. (1990) A stochastic nonlinear model for coordinated bird flocks. In The Ubiquity of Chaos (S. Krusna, ed.), pp. 233–238. AAAS Publications, Washington, DC.Google Scholar
  20. Huth, A. and Wissel, C. (1990) The movement of fish schools: A simulation model. In Bilogical Motion (W. Alt and G. Hoffmann, eds), pp. 577–590. Lecture Notes in Biomathematics, Vol. 89. Springer-Verlag, Berlin.Google Scholar
  21. Huth, A. and Wissel, C. (1992) The simulation of the movement of fish schools. J. Theor. Biol. 156, 365–385.Google Scholar
  22. Inagaki, T., Sakamoto, W. and Kuroki, T. (1976) Studies on the schooling behaviour of fish. II: Mathematical modelling of schooling form depending on the intensity of mutual force between individuals. Bull. Jap. Soc. Sci. Fish. 42, 265–270.Google Scholar
  23. Kareiva, P. and Odell, G. (1987) Swarms of predators exhibit ‘preytaxis’ if individual predators use area-restricted search. Am. Nat. 130, 207–228.Google Scholar
  24. Kawasaki, K. (1978) Diffusion and the formation of spatial distribution. Math. Sci. 16, 47–52.Google Scholar
  25. Keller, E.F. and Segel, L.A. (1971) Model for chemotaxis. J. Theor. Biol. 30, 225–234.Google Scholar
  26. Kennedy, J.S. (1951). The migration of the desert locust (Schistocerca gregaria Forsk.). Phil. Trans. R. Soc. Lond. B 235, 163–290Google Scholar
  27. Kils, U. (1981) The Swimming Behaviour, Swimming Performance and Energy Balance of Antarctic Krill, Euphausia superba. BIOMASS Scientific Series No. 3. SCAR/SCOR, Scott Polar Research Institute, Cambridge.Google Scholar
  28. Kshatriya, M. and Blake, R.W. (1992) Theoretical model of the optimum flock size of birds flying in for-mation. J. Theor. Biol. 157, 135–174.Google Scholar
  29. Levin, S.A., Morin, A. and Powell, T.M. (1989) Pattern and processes in the distribution and dynamics of Antarctic krill. Scientific Report VII/BG 20, 281–296. Report for the Commission for the Conservation of Antarctic Marine Living Resources (CCAMLR).Google Scholar
  30. Matuda, K. and Sannomiya, N. (1980) Computer simulation of fish behaviour in relation to fishing gear. I: Mathematical model of fish behaviour. Bull. Jap. Soc. Sci. Fish. 46, 689–697.Google Scholar
  31. Matuda, K. and Sannomiya, N. (1985) Computer simulation of fish behaviour in relation to a trap model. Bull. Jap. Soc. Sci. Fish. 51, 33–39.Google Scholar
  32. Miller, D.G.M. and Hampton, I. (1989) Biology and Ecology of the Antarctic Krill (Euphausia superba Dana): A review. BIOMASS Scientific Series No. 9. SCAR/SCOR, Scott Polar Research Institute, Cambridge.Google Scholar
  33. Monin, A.S. and Ozmidov, R.V. (1985) Turbulence in the Ocean. D. Reidel, Dordrecht.Google Scholar
  34. Morgan, M.J. and Colgan, P.W. (1987) The effects of predator presence and shoal size on foraging in bluntnose minnows, Pimephales notatus. Env. Biol. Fish. 20, 105–111.Google Scholar
  35. Mullen, A.J. (1989) Aggregation of fish through variable diffusivity. Fish. Bull. USA 87, 353–362.Google Scholar
  36. Murray, J.D. (1989) Mathematical Biology. Springer-Verlag, Berlin.Google Scholar
  37. Nihoul, J.C.J. (1981) Marine hydrodynamics at ecological scales. In Ecohydrodynamics (J.C.J. Nihoul, ed.), pp. 1–12. Elsevier, Amsterdam.Google Scholar
  38. O'Brien, D.P. (1989) Analysis of the internal arrangement of individuals within crustacean aggregations (Euphausiacea, Mysidacea). J. Exp. Mar. Biol. Ecol. 128, 1–30.Google Scholar
  39. Okubo, A. (1980) Diffusion and Ecological Problems: Mathematical Models. Biomathematics Vol. 10. Springer-Verlag, Berlin.Google Scholar
  40. Okubo, A. (1986) Dynamical aspects of animal grouping: Swarms, schools, flocks and herds. Adv. Biophys. 22, 1–94.Google Scholar
  41. Othmer, H.G., Dunbar, S.R. and Alt, W. (1988) Models of dispersal in biological systems. J. Math. Biol. 26, 263–298.Google Scholar
  42. Parrish, J.K. (1992) Levels of diurnal predation on a school of flat-iron herring, Harengula thrissina. Env. Biol. Fish. 34, 257–263.Google Scholar
  43. Pattiaratchi, C.B., Micallef, S., Aiken, J., Osborne, M.J., Collins, M.B. and Williams, R. (1989) Chlorophyll variation at Ocean Weather Station Lima (57 degrees N, 20 degrees W). In Reproduction, Genetics, and Distributions of Marine Organisms (J.S. Ryland and P.A. Tyler, eds), pp. 423–429. Olsen and Olsen, Fredensborg, Denmark.Google Scholar
  44. Pfistner, B. and Alt, W. (1990) A two dimensional random walk model for swarming behavior. In Biological Motion (W. Alt and G. Hoffmann, eds), pp. 584–565. Lecture Notes in Biomathematics Vol. 89. Springer-Verlag, Berlin.Google Scholar
  45. Pitcher, T. (1983) Heuristic definitions of shoaling behaviour. Anim. Behav. 31, 611–613.Google Scholar
  46. Pitcher, T.J. (1986) Functions of shoaling behaviour in teleosts. In Behaviour of Teleost Fishes (T.J. Pitcher, ed.), pp. 294–337. Chapman & Hall, London.Google Scholar
  47. Pitcher, T.J. and House, A.C. (1987) Foraging rules for group feeders: Area copying depends upon food density in shoaling goldfish. Ethology 76, 161–167.Google Scholar
  48. Pitcher, T.J. and Parrish, J.K. (1993) Functions of shoaling behaviour in teleosts. In Behaviour of Teleost Fishes (T.J. Pitcher, ed.), pp. 363–469. Chapman & Hall, London.Google Scholar
  49. Pitcher, T.J., Magurran, A.E. and Winfield, T.J. (1982) Fish in larger schools find food faster. Behav. Ecol. Sociobiol. 10, 149–151.Google Scholar
  50. Porter, J.M. and Sealy, S.G. (1982) Dynamics of seabirds multisepecies feeding flocks: Age-related feeding behaviour. Behaviour 81, 91–109.Google Scholar
  51. Powell, T.M. (1989) Physical and biological scales of variability in lakes, estuaries, and the coastal ocean. In Perspectives in Ecological Theory (J. Roughgarden, R.M. May and S.A. Levin, eds), pp. 157–177. Princeton University Press, Princeton, NJ.Google Scholar
  52. Price, H.J. (1989) Swimming behaviour of krill in response to algal patches: A mesocosm study. Limnol. Oceanogr. 34, 649–659.Google Scholar
  53. Priddle, J., Watkins, J., Morris, D., Rickets, C. and Buchholz, F. (1990) Variation of feeding by krill in swarms. J. Plank. Res. 12, 1189–1205.Google Scholar
  54. Prins, H.H. (1989) Buffalo herd structure and its repercussions for condition of individual African buffalo cows. Ethology 81, 47–71.Google Scholar
  55. Ranta, E. and Kaitala, V. (1991) School size affects individual feeding success in three-spined sticklebacks (Gasterosteus aculeatus L.). J. Fish. Biol. 39, 733–737.Google Scholar
  56. Rose, G.A. and Legget, W.C. (1990) The importance of scale to predator-prey spatial correlations: An example of Atlantic fishes. Ecology 71, 33–43.Google Scholar
  57. Royce, W.F. (1972) Introduction to the Fishery Sciences. Academic Press, London.Google Scholar
  58. Ryer, C.H. and Olla, B.H. (1991) Information transfer and the facilitation and inhibition of feeding in a schooling fish. Env. Biol. Fish. 30, 317–323.Google Scholar
  59. Schultz, S., Breuel, G., Lass, U., Matthäus, W., Nehring, D. and Postel, L. (1989). The patchy distribution of oceanological parameters during the spring bloom in the Baltic proper. In Reproduction, Genetics, and Distributions of Marine Organisms (J.S. Ryland and P.A. Tyler, eds), pp. 423–429. Olsen and Olsen, Fredensborg, Denmark.Google Scholar
  60. Sinclair, A.R.E. (1977) The African Buffalo. University of Chicago Press, Chicago, IL.Google Scholar
  61. Smith, M.F.L. and Warburton, K. (1992) Predator shoaling moderates the confusion effect in blue-green chromis, Chromis viridis. Behav. Ecol. Sociobiol. 30, 103–107.Google Scholar
  62. Steele, C.W., Scarfe, A.D. and Owens, D.W. (1991) Effects of group size on the responsiveness of zebrafish, Brachydanio rerio (Hamilton Buchanan), to alanine, a chemical attractant. J. Fish. Biol. 38, 553–564.Google Scholar
  63. Strand, S.W. and Hamner, W.M. (1990) Schooling behaviour of Antarctic krill (Euphausia superba) in lab-oratory aquaria: Reactions to chemical and visual stimuli. Mar. Biol. 106, 355–359.Google Scholar
  64. Tranquillo, R.T. (1990) Models of chemical gradient sensing by cells. In Biological Motion (W. Alt and G. Hoffmann, eds), pp. 415–442. Lecture Notes in Biomathematics Vol. 89. Springer-Verlag, Berlin.Google Scholar
  65. Veit, R.R., Silverman, E.D. and Everson, I. (1993) Aggregation patterns of pelagic predators and their principal prey, Antarctic krill near South Georgia. J. Anim. Ecol. 62, 551–564.Google Scholar
  66. Veit, R.R., Silverman, E.D., Hewitt, R.P. and Demer, D.A. (1995) Spatial and behavioural responses by foraging seabirds to Antarctic krill swarms. Ant. J. U.S. 29, 164–166.Google Scholar
  67. Waloff, Z. (1972) Observations on the airspeeds of freely flying locusts. Anim. Behav. 20, 367–372.Google Scholar
  68. Warburton, K. and Lazarus, J. (1991) Tendency-distance models of social cohesion in animal groups. J. Theor. Biol. 150, 473–488.Google Scholar
  69. Weber, L.H., El-Sayed, S.Z. and Hampton, I. (1986) The variance spectra of phytoplankton, krill and water temperature in the Antarctic Ocean south of Africa. Deep-Sea Res. 33, 1327–1343.Google Scholar
  70. Witek, Z., Kalinowski, J., Grelowski, A. and Wolnomiejski, N. (1981) Studies of aggregations of krill (Eu-phausia superba). Meeresforsch. 28, 228–243.Google Scholar
  71. Wolf, N.G. (1987) Schooling tendency and foraging benefit in the ocean surgeonfish. Behav. Ecol. Sociobiol. 21, 59–63.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Daniel GrÜnbaum
    • 1
  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada

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