Journal of Mathematical Biology

, Volume 8, Issue 3, pp 217–258 | Cite as

Spatial patterning of the spruce budworm

  • D. Ludwig
  • D. G. Aronson
  • H. F. Weinberger


The spatial and temporal distribution of the spruce budworm is modelled by a nonlinear diffusion equation. Two questions are considered:
  1. 1.

    What is the critical size of a patch of forest which can support an outbreak?

  2. 2.

    What is the width of an effective barrier to spread of an outbreak?


Answers to these questions are obtained with the aid of comparison methods for nonlinear diffusion equations.

Key words

Nonlinear diffusion Spatial patterning Spruce Budworm Comparison methods 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aronson, D. G., Weinberger, H. F.: Nonlinear diffusion in population genetics, combustion and nerve propagation. Partial differential equations and related topics, Lecture Notes in Mathematics, vol. 446, Berlin: Springer-Verlag, 1975Google Scholar
  2. Aronson, D. G., Weinberger, H. F.: Multidimensional nonlinear diffusions arising in population genetics. Adv. in Math. 30, 33–76 (1978)Google Scholar
  3. Kierstead, H., Slobodkin, L. B.: The size of water masses containing plankton blooms. J. Mar. Res. 12, 141–147 (1953)Google Scholar
  4. Levin, S. A.: Population dynamic models in heterogeneous environments. Ann. Rev. Ecol. Syst. 7, 287–310 (1976)Google Scholar
  5. Levin, S. A.: Population models and community structure in heterogeneous environments. Studies in Mathematical Biology, M.A.A. Studies in Math. Vol. 16, (S. A. Levin, ed.), 439–476, 1978Google Scholar
  6. Ludwig, D., Jones, D. D., Holling, C. S.: Qualitative analysis of insect outbreak systems: the spruce budwork and the forest. J. Anim. Ecol. 47, 315–332 (1978)Google Scholar
  7. McMurtrie, R.: Persistence and stability of single-species and prey-predator systems in spatially heterogeneous environments. Math. Biosci. 39, 11–51 (1978)Google Scholar
  8. Okubo, A.: Ecology and Diffusion, (in Japanese), Tsukiji Shokan, Tokyo (1975). An English edition will be published by Springer-VerlagGoogle Scholar
  9. Protter, M. H., Weinberger, H. F.: Maximum Principles in Differential Equations. Englewood Cliffs, N.J.: Prentice-Hall, 1967Google Scholar
  10. Skellam, J. G.: Random dispersal in theoretical populations. Biometrika 38, 196–218 (1951)Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • D. Ludwig
    • 1
  • D. G. Aronson
    • 2
  • H. F. Weinberger
    • 2
  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  2. 2.School of MathematicsUniversity of MinnesotaMinneapolisUSA

Personalised recommendations