Skip to main content
SpringerLink
Log in

Diffusion driven instability in an inhomogeneous domain

  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literature

  • 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.

    MATH  MathSciNet  Google Scholar 

  • Arcuri, P. and J. D. Murray. 1986. Pattern sensitivity to boundary and initial conditions in reactions-diffusion models.J. math. Biol. 24, 141–165.

    Article  MATH  MathSciNet  Google Scholar 

  • Britton, N. F. 1986.Reaction-Diffusion Equations and their Applications to Biology. London: Academic Press.

    Google Scholar 

  • Cantrell, R. S. and C. Cosner. 1991. The effects of spatial heterogeneity in population dynamics.J. math. Biol. 29, 315–338.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

  • Gear, C. W. 1971.Numerical Initial Value Problems in Ordinary Differential Equations. Englewood Cliffs, NJ: Prentice-Hall.

    Google Scholar 

  • Hunding, A. and P. G. Sørenson. 1988. Size adaptation of Turing prepatterns.J. math. Biol. 26, 27–39.

    MATH  MathSciNet  Google Scholar 

  • Levin, S. A. 1976. Population dynamic models in heterogeneous environments.A. Rev. ecol. Syst. 7, 287–310.

    Article  Google Scholar 

  • 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 

  • Meinhardt, H. 1982.Models of Biological Pattern Formation. London: Academic Press.

    Google Scholar 

  • Murray, J. D. 1982.Mathematical Biology. Heidelberg: Springer-Verlag.

    Google Scholar 

  • Okubo, A. 1980.Diffusion and Ecological Problems: Mathematical Models. Heidelberg: Springer-Verlag.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Pacala, S. W. and J. Roughgarden. 1982. Spatial heterogeneity and interspecific competition.Theor. Pop. Biol. 21, 92–113.

    Article  MATH  MathSciNet  Google Scholar 

  • Pate, E. and H. G. Othmer. 1984. Applications of a model for scale-invariant pattern formation in developing systems.Differentiation 28, 1–8.

    Article  Google Scholar 

  • Schnackenberg, J. 1979. Simple chemical reaction systems with limit cycle behaviour.J. theor. Biol. 81, 389–400.

    Article  Google Scholar 

  • Segel, L. A. and J. L. Jackson, 1972. Dissipative structure: an explanation and an ecological example.J. theor. Biol. 37, 545–559.

    Article  Google Scholar 

  • 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 

  • Turing, A. M. 1952. The chemical basis of morphogenesis.Phil. Trans. R. Soc. Lond. B 237, 37–72.

    Google Scholar 

  • Wolpert, L. 1969. Positional information and the spatial pattern of cellular differentiation.J. theor. Biol. 25, 1–47.

    Article  Google Scholar 

  • Wolpert, L. 1981. Positional information and pattern formation.Phil. Trans. R. Soc. Lond. B295, 441–450.

    Google Scholar 

  • Wolpert, L. and A. Hornbruch. 1990. Double anterior chick limb buds and models for cartilage rudiment specification.Development 109, 961–966.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre for Mathematical Biology, Mathematical Institute, 24-29 St Giles', OX1 3LB, Oxford, UK

    Debbie L. Benson, Jonathan A. Sherratt & Philip K. Maini

Authors
  1. Debbie L. Benson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jonathan A. Sherratt
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Philip K. Maini
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Benson, D.L., Sherratt, J.A. & Maini, P.K. Diffusion driven instability in an inhomogeneous domain. Bltn Mathcal Biology 55, 365–384 (1993). https://doi.org/10.1007/BF02460888

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02460888

Keywords

Search

Navigation