Abstract
We construct and analyze a nonlocal continuum model for group formation with application to self-organizing collectives of animals in homogeneous environments. The model consists of a hyperbolic system of conservation laws, describing individual movement as a correlated random walk. The turning rates depend on three types of social forces: attraction toward other organisms, repulsion from them, and a tendency to align with neighbors. Linear analysis is used to study the role of the social interaction forces and their ranges in group formation. We demonstrate that the model can generate a wide range of patterns, including stationary pulses, traveling pulses, traveling trains, and a new type of solution that we call zigzag pulses. Moreover, numerical simulations suggest that all three social forces are required to account for the complex patterns observed in biological systems. We then use the model to study the transitions between daily animal activities that can be described by these different patterns.
Similar content being viewed by others
References
Beecham, J.A., Farnsworth, K.D., 1999. Animal group forces resulting from predator avoidance and competition minimization. J. Theor. Biol. 198, 533–548.
Breder, C.M., 1954. Equations descriptive of fish schools and other animal aggregations. Ecology 35 361–370.
Bressloff, P.C., 2004. Euclidean shift-twist symmetry in population models of self-aligning objects. SIAM J. Appl. Math. 64, 1668–1690.
Buchanan, J.B., Schick, C.T., Brennan, L.A., Herman, S.G., 1988. Merlin predation on wintering dunlins: Hunting success and dunlin escape tactics. Wilson Bull. 100, 108–118.
Bumann, D., Krause, J., 1993. Front individuals lead in shoals of three-spined sticklebacks (Gasterosteus aculeatus) and juvenile roach (Rutilus rutilus). Behaviour 125, 189–198.
Couzin, I.D., Krause, J., James, R., Ruxton, G.D., Franks, N.R., 2002. Collective memory and spatial sorting in animal groups. J. Theor. Biol. 218, 1–11.
Davis, M., 1980. The coordinated aerobatics of dunlin flocks. Anim. Behav. 28, 668–673.
Edelstein-Keshet, L., Watmough, J., Grünbaum, D., 1998. Do travelling band solutions describe cohesive swarms? An investigation for migratory locusts. J. Math. Biol. 36(6), 515–549.
Flierl, G., Grünbaum, D., Levin, S., Olson, D., 1999. From individuals to aggregations: The interplay between behavior and physics. J. Theor. Biol. 196, 397–454.
Gazi, V., Passino, K.M., 2002. Stability analysis of swarms. In: Proc. Am. Control Conf. Anchorage, AK, pp. 8–10.
Grünbaum, D., 1998. Schooling as a strategy for taxis in a noisy environment. Evol. Ecol. 12, 503–522.
Gueron, S., Levin, S.A., Rubenstein, D.I., 1996. The dynamics of herds: From individuals to aggregations. J. Theor. Biol. 182, 85–98.
Helfman, G., 1993. Fish behaviour by day, night and twilight. In: Pitcher, T. (Ed.), Behaviour of Teleost Fishes. Chapman & Hall, London, pp. 479–512.
Humphries, D.A., Driver, P.M., 1970. Protean defence by prey animals. Oecologia (Berl.) 5, 285–302.
Huth, A., Wissel, C., 1994. The simulation of fish schools in comparison with experimental data. Ecol. Model. 75/76, 135–145.
Kac, M., 1974. A stochastic model related to the telegrapher's equation. Rocky Mt. J. Math. 4, 497–509.
Kerner, B.S., Konhäuser, P., 1994. Structure and parameters of clusters in traffic flow. Phys. Rev. E 50, 54–83.
Kube, C.R., Zhang, H., 1993. Collective robotics: From social insects to robots. Adapt. Behav. 2, 189–218.
LeVeque, R., 1992. Numerical Methods for Conservation Laws. Birkhäuser, Basel, Switzerland.
Lutscher, F., 2002. Modeling alignment and movement of animals and cells. J. Math. Biol. 45, 234–260.
Lutscher, F., Stevens, A., 2002. Emerging patterns in a hyperbolic model for locally interacting cell systems. J. Nonlinear Sci. 12, 619–640.
Marler, P., 1967. Animal communication signals. Science 157, 769–774.
Mogilner, A., Edelstein-Keshet, L., 1996. Spatio-angular order in populations of self-aligning objects: Formation of oriented patches. Physica D 89, 346–367.
Mogilner, A., Edelstein-Keshet, L., 1999. A non-local model for a swarm. J. Math. Biol. 38, 534–570.
Mogilner, A., Edelstein-Keshet, L., Bent, L., Spiros, A., 2003. Mutual interactions, potentials, and individual distance in a social aggregation. J. Math. Biol. 47, 353–389.
Okubo, A., 1986. Dynamical aspects of animal grouping: Swarms, school, flocks and herds In: Kotani, M. (Ed.). Adv. Biophys. 22, 1–94.
Okubo, A., Grünbaum, D., Edelstein-Keshet, L., 2001. The dynamics of animal grouping. In: Okubo, A., Levin, S. (Eds.), Diffusion and Ecological Problems: Modern Perspectives. Springer, New York, pp. 197–237.
Othmer, H.G., Dunbar, S.R., Alt, W., 1988. Models of dispersal in biological systems. J. Math. Biol. 26, 263–298.
Parrish, J.K., 1999. Using behavior and ecology to exploit schooling fishes. Environ. Biol. Fish. 55, 157–181.
Partan, S.R., Marler, P., 2005. Issues in the classification of multimodal communication signals. Am. Nat. 166, 231–245.
Partridge, B.L., Pitcher, T., Cullen, J.M., Wilson, J., 1980. The three-dimensional structure of fish schools. Behav. Ecol. Sociobiol. 6, 277–288.
Pfistner, B., 1990. A one dimensional model for the swarming behavior of Myxobakteria. In: Alt, W., Hoffmann, G. (Eds.), Biological Motion, Lecture Notes on Biomathematics, vol. 89. Springer, New York, pp. 556–563.
Pfistner, B., 1995. Simulation of the dynamics of myxobacteria swarms based on a one-dimensional interaction model. J. Biol. Syst. 3, 579–588.
Pomeroy, H., Heppner, F., 1992. Structure of turning in airborne rock dove (Columba Livia) flocks. The Auk 109, 256–267.
Potts, W.K., 1984. The chorus-line hypothesis of manoeuvre coordination in avian flocks. Nature 309, 344–345.
Radakov, D.V., 1973. Schooling in the Ecology of Fish. Wiley, New York.
Reuter, H., Breckling, B., 1994. Self organization of fish schools: An object-oriented model. Ecol. Model. 75/76, 147–159.
Reynolds, C.W., 1987. Flocks, herds and schools: A distributed behavioral model. Comput. Graph. 21, 25–34.
Robbins, T., 2003. Seed dispersal and biological invasion: A mathematical analysis. PhD thesis, University of Utah.
Segel, L.A., 1977. A theoretical study of receptor mechanisms in bacterial chemotaxis. SIAM J. Appl. Math. 32, 653–665.
Simpson, S.J., McCaffery, A.R., Hägele, B.F., 1999. A behavioural analysis of phase change in the desert locust. Biol. Rev. 74, 461–480.
Topaz, C.M., Bertozzi, A.L., 2004. Swarming patterns in a two-dimensional kinematic model for biological groups. SIAM J. Appl. Math. 65, 152–174.
Topaz, C.M., Bertozzi, A.L., Lewis, M.A., 2005. A nonlocal continuum model for biological aggregation. Bull. Math. Biol. 68, 1601–1623.
Uvarov, B., 1966. Grasshoppers and Locusts. Centre for Overseas Pest Research, London.
Vabø, R., Nøttestad, L., 1997. An individual based model of fish school reactions: Predicting antipredator behaviour as observed in nature. Fish. Oceanogr. 6, 155–171.
Vicsek, T., Czirok, A., Farkas, I.J., Helbing, D., 1999. Application of statistical mechanics to collective motion in biology. Physica A 274, 182–189.
Warburton, K., Lazarus, J., 1991. Tendency-distance models of social cohesion in animal groups. J. Theor. Biol. 150, 473–488.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Eftimie, R., de Vries, G., Lewis, M.A. et al. Modeling Group Formation and Activity Patterns in Self-Organizing Collectives of Individuals. Bull. Math. Biol. 69, 1537–1565 (2007). https://doi.org/10.1007/s11538-006-9175-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-006-9175-8