Bulletin of Mathematical Biology

, Volume 42, Issue 5, pp 647–679 | Cite as

Applied catastrophe theory in the social and biological sciences

  • M. A. B. Deakin


Catastrophe theory is a mathematical theory which, allied with a new and controversial methodology, has claimed wide application, particularly in the biological and the social sciences. These claims have recently been heatedly opposed. This article describes the debate and assesses the merits of the different arguments advanced.


Nerve Impulse Catastrophe Theory Catastrophe Model Cusp Catastrophe Cusp Surface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Amson, J. C. 1975. “Catastrophe Theory: A Contribution to the Study of Urban Systems?”Envir. Planning,B2, 177–221.Google Scholar
  2. Arnol’d, V. I. 1972. “Integrals of Rapidly Oscillating Functions and Singularities of Projections of Lagrangian Submanifolds.” (English translation)Funct. Analyt. Applic.,6, 222–224.CrossRefzbMATHGoogle Scholar
  3. Benham, C. J. and J. J. Kozak. 1976. “Denaturation: An Example of a Catastrophe. II. Two-state transitions.”J. theor. Biol.,63, 125–149.CrossRefGoogle Scholar
  4. — and J. J. Kozak. 1977. “Denaturation: An Example of a Catastrophe. III. Phase Diagrams for Multistate Transitions.”J. theor. Biol.,66, 679–693.CrossRefGoogle Scholar
  5. — and J. J. Kozak. 1978. “Catastrophes, in Statistical Biophysics.”Behavl Sci.,23, 355–359.Google Scholar
  6. Berlinski, D. 1975. “Mathematical Models of the World.”Synthèse,31, 211–227.MathSciNetCrossRefGoogle Scholar
  7. Berry, M. V. 1976. “Waves and Thom’s Theorem.”Adv. Phys.,25, 1–25.MathSciNetCrossRefGoogle Scholar
  8. Bröcker, Th. 1975. “Differentiable Germs and Catastrophes” (Trans. L. Lander).London Mathl Soc. Lecture Notes, Vol. 17. Cambridge University Press.Google Scholar
  9. Chillingworth, D. R. J. 1976. “Structural Stability of Mathematical Models: The Role of the Catastrophe Method.” InMathematical Modelling, Ed. J. G. Andrews and R. R. McLone. London: Butterworths.Google Scholar
  10. Cobb, L. 1978. “Stochastic Catastrophe Models and Multimodal Distributions.”Behavl Sci.,23, 360–374.MathSciNetGoogle Scholar
  11. Cobb, L. in press a. “Parameter Estimation for Ensembles of Nonlinear Stochastics Systems.”Google Scholar
  12. Cobb, L. in press b. “Parameter Estimation for Cusp Catastrophe Models.”Google Scholar
  13. Cooke, J. and E. C. Zeeman. 1976. “A Clock and Wavefront Model for Control of the Number of Repeated Structures During Animal Morphogenesis.”J. theor. Biol.,58, 455–476.Google Scholar
  14. Croll, J. 1976. “Is Catastrophe Theory Dangerous?”New Scient.,70, 630–632.Google Scholar
  15. Deakin, M. A. B. 1977. “Catastrophe Theory and Its Applications.”Mathl Scient.,2, 73–94.zbMATHGoogle Scholar
  16. Deakin, M. A. B. to appear. “The Impact of Catastrophe Theory on the Philosophy of Science.” Preprint. Monash University Mathematics Department.Google Scholar
  17. Dodgson, M. M. 1975. “Quantum Evolution and the Fold Catastrophe.”Evolut. Theory.,1, 107–118.Google Scholar
  18. — 1976. “Darwin’s Law of Natural Selection and Thom’s Theory of Catastrophes.”Mathl Biosci. 28, 243–274.CrossRefGoogle Scholar
  19. —. 1977. “Catastrophe Theory.”Nature,270, 658.CrossRefGoogle Scholar
  20. — and A. Hallam. 1977. “Allopatric Speciation and the Fold Catastrophe.”Am. Nat.,111, 415–433.CrossRefGoogle Scholar
  21. Duistermaat, J. J. 1974. “Oscillatory Integrals, Lagrange Immersions, and Unfoldings of Singularities.”Communs pure appl. Math.,27, 207–281.zbMATHMathSciNetGoogle Scholar
  22. Elsdale, T. R., M. J. Pearson and M. Whitehead. 1976. “Abnormalities in Somite Segregation Induced by Heat Shocks toXenopus embryo.”J. Embryol. exp. Morph.,35, 625–635.Google Scholar
  23. Fowler, D. H. 1972. “The Riemann-Hugoniot Catastrophe and Van der Waals’ Equation.” InTowards a Theoretical Biology, Vol. 4: Essays, Ed. C. H. Waddington. Edinburgh University Press.Google Scholar
  24. Fox, V. 1971. “Why prisoners riot.”Fed. Prob.,35, 9–14.Google Scholar
  25. Golubitsky, M. 1978. “An Introduction to Catastrophe Theory and its Applications”SIAM Rev.,20, 352–387.zbMATHMathSciNetCrossRefGoogle Scholar
  26. Guckenheimer, J. 1978. “The Catastrophe Controversy.”Math. Int.,1, 15–20.zbMATHMathSciNetCrossRefGoogle Scholar
  27. Hardin, G. (1960). “The Competitive Exclusion Principle.”Science,131, 1292–1297.Google Scholar
  28. Hodgkin, A. L. and A. F. Huxley. 1952. “A Quantitative Description of Membrane Current and its Applications to Conduction and Excitation in Nerve.”J. Physiol.,117, 500–544.Google Scholar
  29. Isnard, C. A. and E. C. Zeeman. 1976. “Some Models from Catastrophe Theory in the Social Sciences.” InThe Use of Models in the Social Sciences, Ed. L. Collins. London: Tavistock.Google Scholar
  30. Klahr, D. and J. G. Wallace. 1976.Cognitive Development—An Information-Processing View. New York: Wiley. (Note, in particular, pp. 201–208).Google Scholar
  31. Kolata, G. B. 1977. “Catastrophe Theory: The Emperor has no Clothes.”Science,196, 287, 350–351.MathSciNetGoogle Scholar
  32. Kozak, J. J. and C. J. Benham. 1974. “Denaturation: An Example of a Catastrophe.”Proc. natn. Acad. Sci. U.S.A.,71, 1977–1981.zbMATHCrossRefGoogle Scholar
  33. Lavis, D. A. and G. M. Bell. 1977. “Thermodynamic Phase Changes and Catastrophe Theory.”Bull. Inst. math. Applic.,13, 34–42.MathSciNetGoogle Scholar
  34. Lewis, M. 1977. “Catastrophe Theory.”Science,196, 1271.Google Scholar
  35. Lorenz, K. 1966.On Agression. London: Methuen.Google Scholar
  36. Lu, Y.-C. 1976.Singularity Theory and an Introduction to Catastrophe Theory. Berlin: Springer.zbMATHGoogle Scholar
  37. Mees, A. I. 1975. “The Revival of Cities in Medieval Europe: An Application of Catastrophe Theory.”Reg. Sci. Urb. Econ.,5, 403–425.CrossRefGoogle Scholar
  38. Onsager, L. 1944. “Crystal Statistics. I. A Two-Dimensional Model with Order-Disorder Transition.”Phys. Rev.,65, 117–149.zbMATHMathSciNetCrossRefGoogle Scholar
  39. Panati, C. 1976. “Catastrophe Theory.”Newsweek, 19/1/76, 46-47.Google Scholar
  40. Popper, K. 1959.The Logic of Scientific Discovery (English Translation). New York: Basic Books.zbMATHGoogle Scholar
  41. Poston, T. 1978. “The Elements of Catastrophe Theory or the Honing of Occam’s Razor.” InTransformations: Mathematical Approaches to Culture Change, Ed. K. Cooke and C. Renfrew. New York: Academic Press.Google Scholar
  42. Poston, T. in press. “On Deducing the Presence of Catastrophes.”Math. Sci. Hum. Google Scholar
  43. Poston, T. and I. Stewart. 1976.Taylor Expansions and Catastrophes. Research Notes in Mathematics Vol. 7. London: Pitman.Google Scholar
  44. — 1978.Catastrophe Theory and Its Applications. Surveys and Reference Works in Mathematics Vol. 2. London: Pitman.Google Scholar
  45. — and A. G. Wilson 1977. “Facility Size vs Distance Travelled: Urban Services and the Fold Catastrophe.”Envir. Planning,A9, 681–686.Google Scholar
  46. Renfrew, A. C. and T. Poston. 1978. “Discontinuities in the Endogenous Change of Settlement Pattern.” InTransformations: Mathematical Approaches to Culture Change, Ed. K. Cooke and C. Renfrew. New York: Academic Press.Google Scholar
  47. Rosenhead, J. 1976. “Prison ‘Catastrophe’.”New Scient.,71, 140.Google Scholar
  48. Rybak, B. and J. J. Béchet. 1961. “Recherches sur l’Électromécanique Cardiaque.”Path. Biol.,9, 1861-1871, 2035–2054.Google Scholar
  49. Senechal, M., M. Lewis, R. Rosen and M. A. B. Deakin (separately). 1977. “Catastrophe Theory.” (Letters to the editor.)Science,196, 1271–1272.Google Scholar
  50. Sewell, M. J. 1975. “Kitchen Catastrophe.”Mathl. Gaz.,59, 246–249.Google Scholar
  51. Stewart, I. N. (1975). “The Seven Elementary Catastrophes.”New Scient. 68, 447–454.Google Scholar
  52. Stewart, I. N. and A. E. R. Woodcock. In press. “On Zeeman’s Equation for the Propagation of the Nerve Impulse.”Google Scholar
  53. Sussman, H. J. 1975. “Catastrophe Theory.”Synthèse,31, 229–270.CrossRefGoogle Scholar
  54. — 1976. “Catastrophe Theory—A Preliminary Critical Study.” InPSA 1976: Proceedings of the 1976 Biennial Meeting of the Philosophy of Science Association, Ed. F. Suppe and P. Asquith. East Lansing: Phil. Sci. Assn.Google Scholar
  55. — 1977. “Catastrophe Theory: A Skeptic.” (Letter to the Editor.)Science,197, 820–821.Google Scholar
  56. — 1979. Review ofCatastrophe Theory: Selected Papers 1972-1977 (by E. C. Zeeman).SIAM Review,21, 268–276.CrossRefGoogle Scholar
  57. — and Zahler, R. S. 1977. “Catastrophe Theory: Mathematics Misused.”The Sciences,17 (6), 20–23.Google Scholar
  58. — and-— 1978. “Catastrophe Theory as Applied to the Social and Biological Sciences: A Critique.”Synthèse 37, 117–216. Reprinted in modified form as “A Critique of Applied Catastrophe Theory in the Social Sciences.”Behav. Sci.,23, 383–389, 1978.CrossRefGoogle Scholar
  59. Thom, R. 1972a.Stabilité Structurelle et Morphogénèse. Reading, MA: Benjamin.Google Scholar
  60. Thom, R. 1972b. “Structuralism and Biology.” InTowards a Theoretical Biology, Vol. 4: Essays, Ed. C. H. Waddington. Edinburgh University Press.Google Scholar
  61. — 1975. “Answer to Christopher Zeeman’s Reply.” InDynamical Systems—Warwick 1974 Ed. A. Manning.Lecture Notes in Mathematics Vol. 468. Berlin: Springer.Google Scholar
  62. — 1976a. “The Two-fold way of Catastrophe Theory.” InStructural Stability, the Theory of Catastrophes, and Applications in the Sciences Ed. P. Hilton.Lecture Notes in Mathematics Vol. 525. Berlin: Springer.Google Scholar
  63. Thom, R. 1976a. “Applications of Catastrophe Theory.” Compiled from notes taken by A. W.-C. Lun, Ed. M. A. B. Deakin, Preprint, Monash University Mathematics Department.Google Scholar
  64. Thom, R. 1977. “Structural Stability, Catastrophe Theory, and Applied Mathematics.” (The John von Neumann Lecture, 1976.)SIAM Rev.,19, 189–201.zbMATHMathSciNetCrossRefGoogle Scholar
  65. — and M. Dodgson (separately). 1977. “Catastrophe Theory.” (Letters to the Editor.)Nature,270, 658.CrossRefGoogle Scholar
  66. Thompson, M. 1976. “Class, Caste, the Curriculum Cycle and the Cusp Catastrophe.”Stud. Higher Ed.,1, 31–46.CrossRefGoogle Scholar
  67. Thompson, M. to appear. “The Geometry of Confidence.”Google Scholar
  68. Woodcock, A. E. R. and M. Davis. 1978.Catastrophe Theory. New York: Dutton.zbMATHGoogle Scholar
  69. Zahler, R. S. 1978. “Catastrophe Theory Reply.” (Letter to the Editor.)Nature,271, 401.CrossRefGoogle Scholar
  70. — and H. J. Sussmann. 1977. “Claims and Accomplishments of Applied Catastrophe Theory.”Nature,269, 759–763.CrossRefGoogle Scholar
  71. Zeeman, E. C. 1971. “The Geometry of Catastrophe.”Times Literary Supplement 10/12/71, pp. 1556-1557.Google Scholar
  72. Zeeman, E. C. 1972a. “Different Equations for the Heartbeat and Nerve Impulse.” InTowards a Theoretical Biology, Vol. 4: Essays, Ed. C. H. Waddington. Edinburgh University Press.Google Scholar
  73. Zeeman, E. C. 1972b. “A Catastrophe Machine.” InTowards a Theoretical Biology, Vol. 4: Essays Ed. C. H. Waddington. Edinburgh University Press.Google Scholar
  74. — 1974a. “Primary and Secondary Waves in Developmental Biology.” InSome Mathematical Questions in Biology VI, Ed. S. A. Levin.Lectures on Mathematics in the Life Sciences, Vol. 7. Providence, R.I.: Am. Math. Soc.Google Scholar
  75. — 1974b. “On the Unstable Behaviour of Stock Exchanges.”J. Mathl Econ.,1, 39–49.zbMATHMathSciNetCrossRefGoogle Scholar
  76. — 1976a. “Catastrophe Theory.”Scient. Am.,234 (4), 65–83.CrossRefGoogle Scholar
  77. — 1976b. “Euler Buckling.” InStructural Stability, the Theory of Catastrophes, and Applications in the Sciences Ed. P. Hilton.Lecture Notes in Mathematics Vol. 525. Berlin: Springer.Google Scholar
  78. — 1976c. “Gastrulation and Formation of Somites and Amphibia and Birds.” InStructural Stability, the Theory of Catastrophes, and Applications in the Sciences, Ed. P. Hilton).Lecture Notes in Mathematics Vol. 525. Berlin: Springer.Google Scholar
  79. — 1976d. “A Mathematical Model for Conflicting Judgements Caused by Stress, Applied to Possible Misestimations of Speed Caused by Alcohol.”Br. J. math. statist Psychol.,29, 19–31.zbMATHGoogle Scholar
  80. — 1976e. “Brain Modelling.”In Structural Stability, the Theory of Catastrophes, and Applications in the Sciences, Ed. P. Hilton.Lecture Notes in Mathematics Vol. 525. Berlin: Springer.Google Scholar
  81. — 1976f. “Duffing’s Equation in Brain Modelling.”Bull. Inst. math. Applic.,12, 207–214MathSciNetGoogle Scholar
  82. — 1977a. “Catastrophe Theory: Draft for a Scientific American Article.” InCatastrophe Theory: Selected Papers 1972–1977 (by E. C. Zeeman). London: Addison-Wesley.Google Scholar
  83. — 1977b.Catastrophe Theory: Selected Papers 1972–1977. London: Addison-Wesley. [This reprints the eleven preceding papers and the next one in this reference list.]zbMATHGoogle Scholar
  84. —, C. Hall, P. J. Harrison, H. Marriage and P. Shapland. 1976. “A Model for Institutional Disturbances.”Br. J. math. statist. Psychol.,29, 66–80.zbMATHGoogle Scholar
  85. —, R. Bellairset al., I. Stewart, M. Berry, J. Guckenheimer, and A. E. P. Woodcock (separately). 1977. “In Support of Catastrophe Theory.” (Letters to the Editor.)Nature,270, 381–384.CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 1980

Authors and Affiliations

  • M. A. B. Deakin
    • 1
  1. 1.Mathematics DepartmentMonash UniversityClaytonAustralia

Personalised recommendations