Journal of Mathematical Biology

, Volume 28, Issue 5, pp 529–565 | Cite as

The velocity of spatial population expansion

  • F. van den Bosch
  • J. A. J. Metz
  • O. Diekmann


We consider the velocity with which an invading population spreads over space. For a general linear model, originally due to Diekmann and Thieme, it is shown that the asymptotic velocity of population expansion can be calculated if information is available on: (i) the net-reproduction, Ro; i.e. the expected number of offspring produced by one individual throughout its life, and (ii) the (normalized) reproduction-and-dispersal kernel, β(a, χ − ξ); i.e. the density of newborns produced per unit of time at position χ by an individual of age a born at ξ By means of numerical examples we study the effect of the net-reproduction and the shape of the reproduction-and-dispersal kernel on the velocity of population expansion. The reproduction-and-dispersal kernel is difficult to measure in full. This leads us to derive approximation formulas in terms of easily measurable parameters. The relation between the velocity of population expansion calculated from the general model and that from the Fisher/Skellam diffusion model is discussed. As a final step we use the model to analyse some real-life examples, thus showing how it can be put to work.

Key words

Space time Integral equation Dispersal Asymptotic velocity of propagation Approximation formulae Cumulant generating function Fisher/Skellam diffusion model 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ammerman, A. J., Cavalli-Sforza, L. L.: The neolithic transition and the genetics of populations in Europe. Princeton University Press 1984.Google Scholar
  2. Anderson, R. M. (ed.): Population dynamics of infectious diseases. Theory and applications. London: Chapman and Hall 1982.Google Scholar
  3. Andow, D. A., Kareiva, P. M., Levin, S. A., Okubo, A.: Spread of invading organisms: patterns of spread. In: Kim, K. C. (ed.) Evolution of insect pests: the pattern of variations. New York: Wiley.Google Scholar
  4. Andow, D. A., Kareiva, P. M., Levin, S. A., Okubo, A.: Spread of invading organisms, submitted.Google Scholar
  5. Andral, L., Artois, M., Aubert, M. F. A., Blancou, J.: Radio-pistage de renards enrages. Comp. Immunol. Microbiol. Infect. Diseases 5, 284–291 (1982).Google Scholar
  6. Andral, L., Toma, B.: La rage en France en 1976. Rec. Med. vet. 153, 503–508 (1977).Google Scholar
  7. Anonymous: Ecology of biological invasions. SCOPE Newsletter 23, 1–5 (1985).Google Scholar
  8. Aronson, D. G., Weinberger, H. F.: Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Goldstein, J. A. (ed.) Partial differential equations and related topics. (Lect. Notes Math., vol. 446, pp. 5–49) Berlin Heidelberg New York: Springer 1975.Google Scholar
  9. Aronson, D. G., Weinberger, H. F.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30, 33–76 (1978).Google Scholar
  10. Artimo, A.: The dispersal and the acclimatisation of the muskrat Ondatra zibetica (L.), in Finland. Papers on game research 21, 1–101 (1960).Google Scholar
  11. Bacon, P. J. (ed.): Population dynamics of rabies in wildlife. New York: Academic Press 1985.Google Scholar
  12. Ball, F. G.: Some statistical problems in the epidemiology of fox rabies. Thesis 1981, University of Nottingham.Google Scholar
  13. Becker, K.: Populationsstudien an Bismratten (Ondatra zibethica L.) I Zoologische Beitrage 13, 369–396 (1967).Google Scholar
  14. Berger, J.: Model of rabies control. In: Berger, J., Buhler, W., Repges, R., Tautu, P. (eds.) Mathematical models in medicine. (Lest. Notes Biomath., vol. 11, pp. 75–88) Berlin Heidelberg New York: Springer 1976.Google Scholar
  15. Bögel, K., Moegle, H.: Characteristics of the spread of a wildlife rabies epidemic in Europe. Biogeographica 8, 251–258 (1980).Google Scholar
  16. Bramson, M.: Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Am. Math. Soc. 44, 190 (1983).Google Scholar
  17. Broadbent, S. R., Kendall, D. G.: The random walk of Trichostrongylus retortaeformis. Biometrika 9, 460–465 (1953).Google Scholar
  18. Browning, J. A., Frey, K. J.: Multiline cultivars as a means of disease control. Annu. Rev. Phytopathol. 7, 355–382 (1969).Google Scholar
  19. Caughley, G.: Liberation, dispersal and distribution of Himalayas Thar (Hemitragus jemlahicus) in New Zealand. New Zealand J. Sci. 13, 220–239 (1970).Google Scholar
  20. Creegan, P., Lui, R.: Some ramarks about the wave speed and travelling wave solutions of a nonlinear integral generator. J. Math. Biol. 20, 59–68 (1984).Google Scholar
  21. Diekmann, O.: Thresholds and travelling waves for the geographical spread of infection. J. Math. Biol. 6, 109–130 (1978).Google Scholar
  22. Diekmann, O.: Run for your life. A note on the asymptotic speed of propagation of an epidemic. J. Differ. Equations 33, 58–73 (1979).Google Scholar
  23. Diekmann, O.: Dynamics in biomathematical perspective. In: Hazewinkel, M., Lenstra, J. K., Meertens, L. G. L. (eds.) Mathematics and computer sicence II. (CWI Monographs vol. 4, pp. 23–50) 1986.Google Scholar
  24. Diekmann, O., Temme, N. M.: Nonlinear diffusion problems. Amsterdam: Mathematical Centre 1976.Google Scholar
  25. Doude van Troostwijk, W. J.: The muskrat (Ondatra zibethicus L.) in the Netherlands, its ecological aspects and their consequences for man. Thesis, State University of Leiden.Google Scholar
  26. Errington, P. L.: Muskrat populations. Iowa: Iowa State University.Google Scholar
  27. Fisher, R. A.: The wave of advance of advantageous genes. Ann. Eugen. 7, 355–369 (1937).Google Scholar
  28. Hadeler, K. P., Rothe, F.: Travelling fronts in nonlinear diffusion equations. J. Math. Biol. 2, 251–263 (1975).Google Scholar
  29. Hengeveld, R.: Dynamics of biological invasions. London: Chapman and Hall 1989.Google Scholar
  30. Hoffman, M.: Die Bisamratte. Leipzig: Academische Verlagsgesellschaft 1958.Google Scholar
  31. Källen, A., Arcuri, P., Murray, J. D.: A simple model for the spatial spread and control of rabies. J. Theor. Biol. 116, 377–393 (1985).Google Scholar
  32. Kendall, M. G., Stuart, A.: The advanced theory of statistics, vol. I. London: Griffin 1958.Google Scholar
  33. Kendall, D. G.: Mathematical models of the spread of infection. In: Mathematics and computer sicence in biology and medicine (Medical Research Council, London, pp. 213–224) 1965.Google Scholar
  34. Keyfitz, N.: Introduction to the mathematics of population. Reading, Mass.: Addison Wesley 1968.Google Scholar
  35. Kolmogorov, A., Petrovsky, I., Piscounov, N.: Etude de l'équation de la diffusion avec croissance de la quantité de matiere et son application a un probleme biologique. Mosc, Univ. Math. Bull. 1, 1–25 (1937).Google Scholar
  36. Kornberg, H., Williamson, M. H.: Quantitative aspects of the ecology of biological invasions. London: Royal Society, 1987.Google Scholar
  37. Lambinet, D., Boisvieux, J. F., Mallet, A., Artois, M., Andral, L.: Modele mathématique de la propagation d'une épizootie de rage vulpine. Rev. Epidém. et Santé Publ. 26, 9–28 (1978).Google Scholar
  38. Levin, S. A.: Analysis of risk for invasions and control programs. In: Drake, J., Castri, F. di, Groves, R., Kruger, F., Mooney, H., Rejamenk, M., Williamson, M. (eds.) Biological invasions: a global perspective. Chichester: Wiley, in press.Google Scholar
  39. Lloyd, H. G.: Wildlife rabies in Europe and the British situation. Trans. R. Soc. Trop. Med. Hyg. 70, 179–187 (1976).Google Scholar
  40. Lubina, J. A., Levin, S. A.: The spread of a reinvading species: Range expansion in the California Sea Otter. Am. Nat. 131, 526–543 (1988).Google Scholar
  41. MacDonald, D. W.: Rabies and wildlife. Oxford: Oxford University Press 1980.Google Scholar
  42. MacDonald, D. W., Bacon, P. J.: Fox society, contact rate and rabies epizootiology. Comp. Immunol. Microbiol. Infect. Dis. 5, 247–256 (1982).Google Scholar
  43. Mallach, N.: Markierungsversuche zur Analyse des Aktionsrau und der Ortsbewegungen des Bisams (Ondatra zibethica L.) Anzeiger für Schädlingskunde and Pflanzenschutz XLIV9, 129–136 (1971).Google Scholar
  44. Metz, J. A. J., Diekmann, O.: The dynamics of physiologically structured populations. (Lect. Notes Biomath., vol. 68) Berlin Heidelberg New York: Springer 1986.Google Scholar
  45. Minogue, K. P., Frey, W. E.: Models for the spread of disease: model description. Phytopathology 73, 1168–1173 (1983a).Google Scholar
  46. Minogue, K. P., Frey, W. E.: Models for the spread of plant disease: some experimental results. Phytophathology 73, 1173–1176 (1983b).Google Scholar
  47. Moens, R.: Etude bio-écologique du rat musqué en Belgique. Parasitica 34, 57–121 (1978).Google Scholar
  48. Mollison, D.: The rate of spatial propagation of simple epidemics. In: Le Cam, L. M., Neyman, J., Scott, E. L. (eds.) Proc. Sixth Berkeley Symposium, III, Univ. of California Press, pp. 579–614 (1972).Google Scholar
  49. Mollison, D.: Spatial contact models for ecological and epidemic spread. J. Roy. Statist. Soc. B39, 283–326 (1977).Google Scholar
  50. Mollison, D., Kuulasmaa, K.: Spatial epidemic models: theory and simulations. In: Bacon, P. J. (ed.) Population dynamics of rabies in wildlife, pp. 291–309. New York: Academic Press 1985.Google Scholar
  51. Mollison, D.: Modelling biological invasions: chance, explanation, prediction. Philos. Trans. R. Soc. Lond. B, 314, 675–693 (1986).Google Scholar
  52. Mooney, H. A., Drake, J. A. (eds.): Ecology of biological invasions of North America and Hawaii. (Ecological Studies, vol. 58) Berlin Heidelberg New York: Springer 1986.Google Scholar
  53. Nobel, J. V.: Geographic and temporal development of plagues. Nature 250, 726–729 (1974).Google Scholar
  54. Okubo, A.: Diffusion-type models for avian range expansion. In: Quellet, H. (ed.) Acta XIX Congress Internationa lis Ornithologici, vol. 1, pp. 1038–1049. National Museum of Natural Sciences, University of Ottawa Press, Ontario, Canada 1988.Google Scholar
  55. Othmer, H. G., Dunbar, S. R., Alt, W.: Models of dispersal in biological systems. J. Math. Biol. 26, 263–298 (1988).Google Scholar
  56. Roughgarden, J.: Theory of population genetics and evolutionary ecology: an introduction. New York: MacMillan 1979.Google Scholar
  57. Sikes, R. K.: Pathogenesis of rabies in wildlife. I. Comparative effect of varying doses of rabies virus inoculated into foxes and skunks. Am. J. Vet. Res. 23, 1041–1047 (1962).Google Scholar
  58. Skellam, J. G.: Random dispersal in theoretical populations. Biometrica 38, 196–218 (1951).Google Scholar
  59. Smith, A. D. M.: A continuous time dterministic model of temporal rabies. In: Bacon, P. J. (ed.) Population dynamics of rabies in wildlife. New York: Academic Press 1985.Google Scholar
  60. Steck, F., Wandeler, A.: The epidemiology of fox rabies in Europe. Epidemiol. Rev. 2, 71–96 (1980).Google Scholar
  61. Thieme, H. R.: A model for the spatial spread of an epidemic. J. Math. Biol 4, 337–351 (1977a).Google Scholar
  62. Thieme, H. R.: The asymptotic behaviour of solutions of nonlinear integral equations. Math. Z. 157, 141–154 (1977b).Google Scholar
  63. Thieme, H. R.: Asymptotic estimates of the solutions of non-linear integral equations and asymptotic speeds for the spread of populations. Jal Reine Angew. Math. 306, 94–121 (1979a).Google Scholar
  64. Thieme, H. R.: Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread. J. Math. Biol. 8, 173–187 (1979b).Google Scholar
  65. Van den Bosch, F., Zadoks, J. C., Metz, J. A. J.: Focus expansion in plant disease. I: The constant rate of focus expansion. Phytopathology 78, 54–58 (1988).Google Scholar
  66. Van den Bosch, F., Zadoks, J. C., Metz, J. A. J.: Focus expansion in plant disease. II: Realistic parameter-sparse models. Phytopathology 78, 59–64 (1988).Google Scholar
  67. Van den Bosch, F., Frinking, H. D., Metz, J. A. J., Zadoks, J. C.: Focus expansion in plant disease. III: Two experimental examples. Phytopathology 78, 919–925 (1988).Google Scholar
  68. Van den Bosch, F., Verhaar, M. A., Buiel, A. A. M., Hoogkamer, W., Zadoks, J. C.: Focus expansion in plant disease. IV: Expansion rates in mixtures of resistant and susceptible hosts. Phytopathology, in press.Google Scholar
  69. Van den Bosch, F., Hengeveld, R., Metz, J. A. J.: Analysing animal range expansion. Preprint (1988).Google Scholar
  70. Verkaik, A. J.: The muskrat in the Netherlands. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen 90, 67–72 (1987).Google Scholar
  71. Vincent, J. P., Quéré, J. P.: Quelques donées sur la reproduction et sur la dynamique des populations du rat musqué dans le nord de la France. Ann. Zool. Ecol. anim. 4, 395–415 (1972).Google Scholar
  72. Watt, K. E. F.: Ecology and resource management. New York: McGraw-Hill 1968.Google Scholar
  73. Weinberger, H. F.: Asymptotic behaviour of a model in population genetics. In: Chadam, J. M. (ed.) Nonlinear partial differential equations and applications. (Lect. Notes in Maths., vol. 648, pp. 47–98) Berlin Heidelber New York: Springer 1978.Google Scholar
  74. Weinberger, H. F.: Long-time behaviour of a class of biological models. SIAM J. Math. Anal. 13, 353–396 (1982).Google Scholar
  75. Williamson, E. J.: The distribution of larvae of randomly moving insects. Aust. J. Biol. Sci. 14, 598–604 (1961).Google Scholar
  76. Williamson, M. H., Brown, K. C.: The analysis and modelling of British invasions. Phil. Trans. R. Soc. London B314, 505–522 (1986).Google Scholar
  77. Wolfe, M. S.: The current status and prospects of multiline cultivars and variety mixtures for disease resistance. Annu. Rev. Phytopathol. 23, 251–273 (1985).Google Scholar
  78. Zadoks, J. C., Kampmeijer, P.: The role of crop populations and their development, illustrated by means of a simulator Epimul 76. Ann. N.Y. Acad. Sci. 287, 164–190 (1977).Google Scholar
  79. Zawolek, M. W.: A physical theory of focus development in plant disease. Agric. Univ. Wageningen Papers. Pudoc, in press.Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • F. van den Bosch
    • 1
  • J. A. J. Metz
    • 1
  • O. Diekmann
    • 1
    • 2
  1. 1.Institute of Theoretical BiologyState University of LeidenLeidenThe Netherlands
  2. 2.Centre for Mathematics and Computer ScienceAmsterdamThe Netherlands

Personalised recommendations