Abstract
It is hard to find animals in nature that do not aggregate for one reason or another. The details of such aggregations are important because they influence numerous fundamental processes like mate-finding, prey-detection, predator avoidance, and disease transmission. Yet, despite the near universality of aggregation and its profound consequences, biologists have only recently begun to probe its underlying mechanisms. In this chapter we review theoretical approaches to animal aggregation, concentrating on aggregations which are caused by social interactions. We emphasize methods and limitations, and suggest what we think are the most promising avenues for future research. For an earlier review article on dynamical aspects of animal grouping consult Okubo (1986).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Alt, W. (1980) Biased random walk models for Chemotaxis and related diffusion approximations. J. Math. Biol. 9:147–177.
Alt, W. (1985) Degenerate diffusion equations with drift functionals modelling aggregation. Nonlinear Analysis, Theory, Methods & Applications. 9:811–836.
Alt, W. (1990) Correlation analysis of two-dimensional locomotion paths. In: Biological Motion (eds. W. Alt and G. Hoffmann). Lecture Notes in Biomathematics. Vol. 89, Springer-Verlag, pp.584–565.
Anderson, J. (1980) A stochastic model for the size offish schools. Fish. Bull. 79(2):315–323.
Aoki, I. (1982) A simulation study on the schooling mechanism in fish. Bulletin of the Japanese Society of Scientific Fisheries. 48:1081–1088.
Aoki, I. (1984) Internal dynamics of fish schools in relation to inter-fish distance. Bulletin of the Japanese Society of Scientific Fisheries. 50:751–758.
Bovet, P. and S. Benhamou. (1988) Spatial analysis of animal’s movements using a correlated random walk model. J. Theor. Biol. 133:419–433.
Britton, N.F. (1989) Aggregation and the competitive exclusion principle. J. Theor. Biol. 136:57–66.
Cantrell, R.S. and C. Cosner (1991) The effects of spatial heterogeneity in population dynamics. J. Math. Biol. 29:315–338.
Childress, S. and J.K. Percus (1981) Nonlinear aspects of Chemotaxis. Math. Biosci. 56:217–237.
Cohen, D.S. and J.D. Murray (1981) A generalized diffusion model for growth and dispersal in a population. J. Math. Biol. 12:237–49.
Dal Passo, R. and P. de Mottoni (1984) Aggregative effects for a reaction-advection equation. J. Math. Biol. 20:103–112.
Dill, L.M., C.S. Holling and L.H. Palmer (1994) Predicting the 3-dimensional structure of animal aggregations from functional considerations: the role of information. In: Animal Aggregation: analysis, theory, and modelling (tentative title) (eds. J. Parrish and W. Hamner). Cambridge University Press, (in press).
Duffy, D.C. and C. Wissel (1988) Models of fish school size in relation to environmental productivity. Ecol. Mod. 40:201–211.
Edelstein-Keshet, L. (1989) Mathematical Models in Biology. Random House, 586pp.
Fife, P.C. (1979) Mathematical aspects of reacting and diffusing systems. Lecture Notes in Biomathematics. Vol. 28. Springer-Verlag, 186pp.
Gardiner, C.W. (1983) Handbook of Stochastic Methods. Springer-Verlag, 442pp.
Goldsmith, A., H.C. Chiang, and A. Okubo (1980) Turning motion of individual midges. Annals of Entomol. Soc. America. 74:48–50.
Grindrod, P. (1991) Patterns and Waves. Oxford University Press, 239pp.
Grünbaum, D. (1992) Aggregation models of individuals seeking a target density. Ph.D. dissertation, Cornell University.
Grünbaum, D. (1994a) Translating stochastic density-dependent individual behavior to a continuum model of animal swarming. J. Math. Biol., (in press).
Grünbaum, D. (1994b) Gradient-following in schools and swarms. In: Animal Aggregation: analysis, theory, and modelling (tentative title) (eds. J. Parrish and W. Hamner). Cambridge University Press, (in press).
Gueron, S. and S.A. Levin (1993) Self organization of front patterns in large wildebeest herds. (manuscript).
Gueron, S. and N. Liron (1989) A model of herd grazing as a travelling wave, Chemotaxis and stability. J. Math. Biol. 27:595–608.
Hara, I. (1984) Shape and size of Japanese sardine school in the waters off southeastern Hokkaido on the basis of acoustic and aerial surveys. Bulletin of the Japanese Society of Scientific Fisheries. 51:41–46.
Heppner, F. and U. Grenander (1990) A stochastic nonlinear model for coordinated bird flocks. In: The Ubiquity of Chaos (ed. S. Krusna). AAAS Publications, Washington, D.C., pp. 233–238.
Hilborn, R. (1991) Modeling the stability of fish schools: exchange of individual fish between schools of skipjack tuna (Katsuwonu pelamis). Can. J. Fish. Aquat. Sci. 48:1081–1091.
Holmes, E. E. (1993) Are diffusion models too simple? A comparison with telegraph models of invasion. Amer. Nat. (in press).
Holmes, E. E., M. A. Lewis, J. E. Banks, R. R. Veit (1993) Partial differential equations in ecology: spatial interactions and population dynamics. Ecology, (in press).
Huth, A. and C. Wissel (1990) The movement of fish schools: a simulation model. In: Biological Motion (eds. W. Alt and G. Hoffmann). Lecture Notes in Biomathematics. Vol. 89, Springer-Verlag, pp. 577–590.
Huth, A. and C. Wissel (1992) The simulation of the movement offish schools. J. Theor. Biol. 156:365–385.
Inagaki, T., W. Sakamoto and T. Kuroki (1976) Studies on the schooling behavior of fish — II: Mathematical modelling of schooling form depending on the intensity of mutual force between individuals. Bulletin of the Japanese Society of Scientific Fisheries. 42(3):265–270.
Ikawa, T. and H. Okabe (1994) Reconstructing three-dimensional positions: a study of swarming mosquitoes. In: Animal Aggregation: analysis, theory, and modelling (tentative title) (eds. J. Parrish and W. Hamner). Cambridge University Press, (in press).
Kareiva, P. and G. Odell (1987) Swarms of predators exhibit “preytaxis” if individual predators use area-restricted search. Amer. Nat. 130:207–228.
Kawasaki, K. (1978) Diffusion and the formation of spatial distribution. Mathematical Sciences. 16(183):47–52.
Keller, E.F. and L.A. Segel (1971) Model for Chemotaxis. J. Theor. Biol. 30: 225–234.
Khait, A. and L.A. Segel (1984) A model for the establishment of pattern by positional differentiation with memory. J. Theor. Biol. 110:1135–53.
Lara Ochoa, F. (1984) A generalized reaction diffusion model for spatial structure formed by mobile cells. Biosystems. 17:35–50.
Levin, S.A. (1974) Dispersion and population interactions. Amer. Nat. 108:207–228.
Levin, S.A. (1977) A more functional response to predator-prey stability. Amer. Nat. 111:381–383.
Levin, S.A. (1981) Models of population dispersal. In: Differential Equations and Applications in Ecology, Epidemics and Population Problems (eds. S. N. Busenberg and K. Cooke). Academic Press, pp. 1–18.
Levin, S.A. (1986). Random walk models and their implications. In: Mathematical Ecology (eds. T. G. Hallam and S. A. Levin) Biomathematics. Vol. 17, Springer-Verlag, pp. 149–154.
Levin, S.A. and L.A. Segel (1976) Hypothesis for origin of planktonic patchiness. Nature. 259:659.
Levin, S.A. and L. A. Segel (1982) Models of the influence of predation on aspect diversity in prey population. J. Math. Biol. 14:253–284.
Levin, S.A., A. Morin and T. M. Powell (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).
Levine, H. and W. Reynolds (1991) Streaming instability of aggregating slime mold amoebae. Phys. Rev. Lett. 66:2400–2403.
Maini, P.K., D.L. Benson and J.A. Sherratt (1992) Pattern formation in reaction-diffusion models with spatially inhomogeneous diffusion coefficients. IMA J. Mathematics Applied in Medicine and Biology. 9:197–213.
Malchow, H. (1988) Spatial patterning of interacting and dispersing populations. Mem. Fac. Sci. Kyoto University (Ser. Biology). 13:83–100.
Mangel, M. (1987) Simulation of southern ocean krill fisheries. SC-CAMLR-VII/BG. 22, Report for CCAMLR.
Marsh, L.M. and R.E. Jones (1988) The form and consequences of random walk movement models. J. Theor. Biol. 133:113–131.
Matuda, K. and N. Sannomiya (1980) Computer simulation of fish behavior in relation to fishing gear —I: Mathematical model of fish behavior. Bulletin of the Japanese Society of Scientific Fisheries. 46(6):689–697.
Matuda, K. and N. Sannomiya (1985) Computer simulation of fish behavior in relation to a trap model. Bulletin of the Japanese Society of Scientific Fisheries. 51(1):33–39.
McFarland, W.N. and S.A. Moss (1967) Internal behavior in fish schools. Science. 156(3772):260–262.
Mimura, M. and M. Yamaguti (1982) Pattern formation in interacting and diffusing systems in population biology. Advances in Biophysics. 15:19–65.
Morin, A., A. Okubo and K. Kawasaki (1989) Acoustic data analysis and models of krill spatial distribution. Science Committee for the Conservation of Antarctic Marine Living Resources, Selected Scientific Papers, pp.311–329, Hobart, Australia.
Mullen, A.J. (1989) Aggregation of fish through variable diffusivity. Fishery Bulletin, USA. 87:353–362.
Murray, J.D. (1989) Mathematical Biology. Springer-Verlag, 767pp.
Nagai, T. and M. Mimura (1983) Asymptotic behavior for a nonlinear degenerate diffusion equation in population dynamics. SIAM J. Appl. Math. 43:449–464.
Niwa, H. (1991) Features offish grouping and migration determined by random movement of the composite individuals. In: Noise in Physical Systems and 1/f Fluctuations (eds. T. Musha, S. Sato and M. Yamamoto). Ohm-Sha, Tokyo, pp.415–418.
Niwa, H. (1993) Self-organizing dynamic model offish schooling. J. Theor. Biol (submitted).
O’Brien, D.P. (1989) Analysis of the internal arrangement of individuals within crustacean aggregations (Euphauiascea, Mysidacea). J. Exp. Mar. Biol. Ecol. 128:1–30.
Odendaal, F.J., P. Turchin and F.R. Stermitz (1988) An incidental-effect hypothesis explaining aggregation of males in a population of Euphydryas anicia. Amer. Nat. 132:735–749.
Okubo, A. (1980) Diffusion and Ecological Problems: Mathematical Models. Biomathematics Vol. 10. Springer-Verlag. 254pp.
Okubo, A. (1986) Dynamical aspects of animal grouping: swarms, schools, flocks and herds. Advances in Biophysics 22:1–94.
Okubo, A., D.J. Bray, and H.C. Chiang (1980) Use of shadows for studying the three dimensional structure of insect swarms. Annals of Entomol. Soc. America. 74:48–50.
Okubo, A. and J.J. Anderson (1984) Mathematical models for Zooplankton swarms: their formation and maintenance. The Oceanography Report, EOS, Amer. Geophy. Union, Washington, pp.731–733.
Othmer, H.G., S.R. Dunbar and W. Alt (1988) Models of dispersal in biological systems. J. Math. Biol. 26:263–298.
Parrish, J. (1989) Re-examining the selfish herd: are central fish safer? Anim. Behav. 38:1048–1053.
Papoulis, A. (1984) Probability, Random Variables, and Stochastic Processes. McGraw-Hill. 576pp.
Pfistner, B. (1990) A one dimensional model for the swarming behavior Myxobacteria. In: Biological Motion (eds. W. Alt and G. Hoffmann). Lecture Notes in Biomathematics. Vol.89, Springer-Verlag, pp.584–565.
Pfistner, B. and W. Alt (1990) A two dimensional random walk model for swarming behavior. In: Biological Motion (eds. W. Alt and G. Hoffmann). Lecture Notes in Biomathematics. Vol. 89, Springer-Verlag, pp.584–565.
Pitcher, T. (1983) Heuristic definitions of shoaling behavior. Anim. Behav. 31:611–613.
Pitcher, T. (1986) Functions of shoaling behavior in teleosts. In: The Behavior of Teleost Fishes (ed. Pitcher, T.J.), Johns Hopkins University Press, Baltimore, Md. pp.415–418.
Prins, H.H. (1989) Buffalo herd structure and its repercussions for condition of individual African buffalo cows. Ethology. 81:47–71.
Risken, H. (1984) The Fokker-Planck Equation. Springer-Verlag. 454pp.
Rivera, M.A., R.T. Tranquillo, H.M. Buettner, and D.A. Lauffenburger (1989) Transport models for chemotactic cell populations based on individual cell behavior. Chem. Engr. Sci. 44:2881–2897.
Rubinow, S.I., L.A. Segel, and W. Ebel (1981) A mathematical framework for the study of morphogenetic development in the slime mold. J. Theor. Biol. 91:99–113.
Sakai, S. (1973) A model for group structure and its behavior. Biophysics. 13:82–90.
Satsuma, J. (1983) Exact solutions of nonlinear equations with singular integral terms. In: Proceedings of RIMS Symposium on Non-Linear Integrable Systems: classical theory and quantum theory (eds. M. Jimbo and T. Miwa). World Scientific Publ. Co., Singapore, pp. 183–193.
Segel, L.A. (1982) Taxes in cellular ecology. In: Mathematical Ecology (eds. S. A. Levin and T. G. Hallam). Lecture Notes in BioMathematics. Vol. 54, Springer-Verlag, pp.407–424.
Segel, L.A. (1978) Mathematical models for cellular behavior. In: Studies in Mathematical Biology. Part I: Cellular Behavior and the Development of Pattern (ed. S.A. Levin). Math. Assoc. Amer. pp. 156–191.
Segel, L.A. and J.L. Jackson (1972) Dissipative structure: an explanation and an ecological example. J. Theor. Biol. 37:545–559.
Seno, H. (1990) A density-dependent diffusion model of shoaling of nesting fish. Ecol. Mod. 51:217–226.
Shigesada, N. and E. Teramoto (1978) A consideration on the theory of environmental density. Japanese J. Ecol. 28:1–8.
Shigesada, N. K. Kawasaki and E. Teramoto (1979) Spatial segregation of interacting species. J. Theor. Biol. 79:83–99.
Shinn, E.A. and G.E. Long (1986) Technique for 3-D analysis of Cheumatopsyche pettiti (Trichoptera:Hydropsychidae) swarms. Environmental Entomology. 15:355–359.
Sinclair, A.R.E. (1977) The African Buffalo. University of Chicago Press, 355pp.
Skellam, J.G. (1951) Random dispersal in theoretical populations. Biometrika. 38:196–218.
Smith, P.E. (1970) The horizontal dimension and abundance of fish schools in the upper mixed layer as measured by sonar. In: Proceedings of an International Conference on Biological Sound Scattering in the Ocean (ed. G.B. Farquhar). U.S. Gov’t Printing Office, Washington D.C.
Steele, J.H. and E.W. Henderson (1992) A simple model for plankton patchiness. J. Plankton Research. 14:1397–1403.
Strand, S.W. and W.M. Hamner. (1990) Schooling behavior of Antarctic krill (Euphausia superba) in laboratory aquaria: reactions to chemical and visual stimuli. Mar. Biol. 106:355–359.
Suzuki, R. and S. Sakai (1973) Movement of a group of animals. Biophysics. 13:281–282.
Timm, U. and A. Okubo (1992) Diffusion-driven instability in a predator-prey system with time-varying diffusivities. J. Math. Biol. 30:307–320.
Tranquilo, R.T. (1990) Models of chemical gradient sensing by cells. In: Biological Motion (eds. W. Alt and G. Hoffmann). Lecture Notes in Biomathematics. Vol. 89, Springer-Verlag, pp.415–441.
Turchin, P. (1989) Population consequences of aggregative movement. J. Animal Ecol. 58:75–100.
Turchin, P. and J. Parrish (1994) Analyzing movements of animals in congregations. In: Animal Aggregation: analysis, theory, and modelling (tentative title), (eds. J. Parrish and W. Hamner) Cambridge University Press, (in press).
Turing, A.M. (1952) The chemical basis of morphogenisis. Phil. Trans. Roy. Soc. Lond. B 237:37–72.
Tyson, J.J., K.A. Alexander, V.S. Manoranjan, and J.D. Murray (1989) Spiral waves of cyclic AMP in a model of slime mold aggregation. Physica D. 34:193–207.
van Kampen, N.G. (1992) Stochastic Processes in Physics and Chemistry (revised and enlarged edition). North-Holland Publ. Co., Amsterdam, 464pp.
Warburton, K. and J. Lazarus (1991) Tendency-distance models of social cohesion in animal groups. J. Theor. Biol. 150:473–488.
Watkins, J.L. (1986) Variations in the size of Antarctic krill, Euphausia superba Dana, in small swarms. Mar. Ecol. Prog. Ser. 31:67–73.
Watkins, J.L., D.J. Morris, C. Ricketts and J. Priddle (1986) Differences between swarms of Antarctic krill and some implications for sampling krill populations. Mar. Biol. 93:137–146.
Weber, L.H., S.Z. El-Sayed and I. Hampton (1986) The variance spectra of phytoplankton, krill, and water temperature in the Antarctic Ocean south of Africa. Deep-Sea Research. 33:1327–1343.
Yen, J. and E.A. Bryant (1994) Aggregative behavior in zooplankton:phototactic swarming in 4 developmental stages of Scottolana canadensis (Copepoda,harpacticoida). In: Animal Aggregation: analysis, theory, and modelling (tentative title) (eds. J. Parrish and W. Hamner). Cambridge University Press, (in press).
Zaferman, M.L. and L.I. Serebrov (1987) A geometric model of the distribution of fish in a school. J. Icthyol. 27(6): 145–48.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Grünbaum, D., Okubo, A. (1994). Modelling Social Animal Aggregations. In: Levin, S.A. (eds) Frontiers in Mathematical Biology. Lecture Notes in Biomathematics, vol 100. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-50124-1_18
Download citation
DOI: https://doi.org/10.1007/978-3-642-50124-1_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-50126-5
Online ISBN: 978-3-642-50124-1
eBook Packages: Springer Book Archive