Bulletin of Mathematical Biology

, Volume 55, Issue 2, pp 365–384 | Cite as

Diffusion driven instability in an inhomogeneous domain

  • Debbie L. Benson
  • Jonathan A. Sherratt
  • Philip K. Maini


Diffusion driven instability in reaction-diffusion systems has been proposed as a mechanism for pattern formation in numerous embryological and ecological contexts. However, the possible effects of environmental inhomogeneities has received relatively little attention. We consider a general two species reaction-diffusion model in one space dimension, with one diffusion coefficient a step function of the spatial coordinate. We derive the dispersion relation and the solution of the linearized system. We apply our results to Turing-type models for both embryogenesis and predator-prey interactions. In the former case we derive conditions for pattern to be isolated in one part of the domain, and in the latter we introduce the concept of “environmental instability”. Our results suggest that environmental inhomogeneity could be an important regulator of biological pattern formation.


Dispersion Relation Marginal Stability Linear Solution Flux Boundary Condition Homogeneous Steady State 
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  1. Auchmuty, J. F. G. and G. Nicolis. 1975. Bifurcation analysis of nonlinear reaction-diffusion equations—I. Evolution equations and the steady state solutions.Bull. math. Biol. 37, 323–365.zbMATHMathSciNetGoogle Scholar
  2. Arcuri, P. and J. D. Murray. 1986. Pattern sensitivity to boundary and initial conditions in reactions-diffusion models.J. math. Biol. 24, 141–165.zbMATHMathSciNetCrossRefGoogle Scholar
  3. Britton, N. F. 1986.Reaction-Diffusion Equations and their Applications to Biology. London: Academic Press.Google Scholar
  4. Cantrell, R. S. and C. Cosner. 1991. The effects of spatial heterogeneity in population dynamics.J. math. Biol. 29, 315–338.zbMATHMathSciNetCrossRefGoogle Scholar
  5. Dillon, R., P. K. Maini and H. G. Othmer. 1992. Pattern formation in generalized Turing systems. I, Steady-state patterns in systems with mixed boundary conditions. In preparation.Google Scholar
  6. Gear, C. W. 1971.Numerical Initial Value Problems in Ordinary Differential Equations. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
  7. Hunding, A. and P. G. Sørenson. 1988. Size adaptation of Turing prepatterns.J. math. Biol. 26, 27–39.zbMATHMathSciNetGoogle Scholar
  8. Levin, S. A. 1976. Population dynamic models in heterogeneous environments.A. Rev. ecol. Syst. 7, 287–310.CrossRefGoogle Scholar
  9. Levin, S. A. 1986. Population models and community structure in heterogeneous environments. InLecture Notes in Biomathematics 17, T. G. Hallam and S. A. Levin (Eds), pp. 259–263. Berlin, Heidelberg, New York: Springer-Verlag.Google Scholar
  10. Meinhardt, H. 1982.Models of Biological Pattern Formation. London: Academic Press.Google Scholar
  11. Murray, J. D. 1982.Mathematical Biology. Heidelberg: Springer-Verlag.Google Scholar
  12. Okubo, A. 1980.Diffusion and Ecological Problems: Mathematical Models. Heidelberg: Springer-Verlag.Google Scholar
  13. Othmer, H. G. and E. Pate. 1980. Scale-invariance in reaction-diffusion models of spatial pattern formation.Proc. natn. Acad. Sci. U.S.A. 77, 4180–4184.CrossRefGoogle Scholar
  14. Pacala, S. W. and J. Roughgarden. 1982. Spatial heterogeneity and interspecific competition.Theor. Pop. Biol. 21, 92–113.zbMATHMathSciNetCrossRefGoogle Scholar
  15. Pate, E. and H. G. Othmer. 1984. Applications of a model for scale-invariant pattern formation in developing systems.Differentiation 28, 1–8.CrossRefGoogle Scholar
  16. Schnackenberg, J. 1979. Simple chemical reaction systems with limit cycle behaviour.J. theor. Biol. 81, 389–400.CrossRefGoogle Scholar
  17. Segel, L. A. and J. L. Jackson, 1972. Dissipative structure: an explanation and an ecological example.J. theor. Biol. 37, 545–559.CrossRefGoogle Scholar
  18. Shigesada, N. 1984. Spatial distribution of rapidly dispersing animals in heterogeneous environments. InLecture Notes in Biomathematics 54, S. A. Levin and T. G. Hallam (Eds), pp. 478–491. Heidelberg: Springer-Verlag.Google Scholar
  19. Turing, A. M. 1952. The chemical basis of morphogenesis.Phil. Trans. R. Soc. Lond. B 237, 37–72.Google Scholar
  20. Wolpert, L. 1969. Positional information and the spatial pattern of cellular differentiation.J. theor. Biol. 25, 1–47.CrossRefGoogle Scholar
  21. Wolpert, L. 1981. Positional information and pattern formation.Phil. Trans. R. Soc. Lond. B295, 441–450.Google Scholar
  22. Wolpert, L. and A. Hornbruch. 1990. Double anterior chick limb buds and models for cartilage rudiment specification.Development 109, 961–966.Google Scholar

Copyright information

© Society for Mathematical Biology 1992

Authors and Affiliations

  • Debbie L. Benson
    • 1
  • Jonathan A. Sherratt
    • 1
  • Philip K. Maini
    • 1
  1. 1.Centre for Mathematical BiologyMathematical InstituteOxfordUK

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