Abstract
Recent examples of biological pattern formation where a pattern changes qualitatively as the underlying domain grows have given rise to renewed interest in the reaction-diffusion (Turing) model for pattern formation. Several authors have now reported studies showing that with the addition of domain growth the Turing model can generate sequences of patterns consistent with experimental observations. These studies demonstrate the tendency for the symmetrical splitting or insertion of concentration peaks in response to domain growth. This process has also been suggested as a mechanism for reliable pattern selection. However, thus far authors have only considered the restricted case where growth is uniform throughout the domain.
In this paper we generalize our recent results for reaction-diffusion pattern formation on growing domains to consider the effects of spatially nonuniform growth. The purpose is twofold: firstly to demonstrate that the addition of weak spatial heterogeneity does not significantly alter pattern selection from the uniform case, but secondly that sufficiently strong nonuniformity, for example where only a restricted part of the domain is growing, can give rise to sequences of patterns not seen for the uniform case, giving a further mechanism for controlling pattern selection. A framework for modelling is presented in which domain expansion and boundary (apical) growth are unified in a consistent manner. The results have implications for all reaction-diffusion type models subject to underlying domain growth.
Similar content being viewed by others
References
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.
Benson, D. L., P. K. Maini and J. A. Sherratt (1993). Analysis of pattern formation in reaction diffusion models with spatially inhomogeneous diffusion coefficients. Math. Comput. Modell. 17, 29–34.
Borckmans, P., A. De Wit and G. Dewel (1992). Competition in ramped Turing structures. Physica A 188, 137–157.
Chaplain, M. A. J., M. Ganesh and I. G. Graham (2001). Spatio-temporal pattern formation on spherical surfaces: Numerical simulation and application to solid tumour growth. J. Math. Biol. 42, 387–423.
Crampin, E. J., E. A. Gaffney and P. K. Maini (1999). Pattern formation through reaction and diffusion on growing domains: Scenarios for robust pattern formation. Bull. Math. Biol. 61, 1093–1120.
Crampin, E. J., E. A. Gaffney and P. K. Maini (2002). Mode doubling and tripling in reaction-diffusion patterns on growing domains: A piecewise linear model. J. Math. Biol. 44, 107–128.
Dewel, G. and P. Borckmans (1989). Effects of slow spatial gradients on dissipative structures. Phys. Lett. A 138, 189–192.
Dillon, R. and H. G. Othmer (1999). A mathematical model for outgrowth and spatial patterning of the vertebrate limb bud. J. Theor. Biol. 197, 295–330.
Dulos, E., P. Davies, B. Rudovics and P. De Kepper (1996). From quasi-2D to 3D Turing structures in ramped systems. Physica D 98, 53–66.
Gierer, A. and H. Meinhardt (1972). A theory of biological pattern formation. Kybernetik 12, 30–39.
Harrison, L. G. and M. Kolář (1988). Coupling between reaction-diffusion prepattern and expressed morphogenesis, applied to desmids and dasyclads. J. Theor. Biol. 130, 493–515.
Harrison, L. G., S. Wehner and D. M. Holloway (2002). Complex morphogenesis of surfaces: theory and experiment on coupling of reaction-diffusion patterning to growth. Faraday Discuss. 120, 277–293.
Herschkowitz-Kaufman, M. (1975). Bifurcation anaylsis of nonlinear reaction-diffusion equations—II. Steady state solutions and comparison with numerical simulations. Bull. Math. Biol. 37, 589–636.
Holloway, D. M. and L. G. Harrison (1999). Algal morphogenesis: modelling interspecific variation in Micrasteras with reaction-diffusion patterned catalysis of cell surface growth. Phil. Trans. R. Soc. B 354, 417–433.
Kondo, S. and R. Asai (1995). A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus. Nature 376, 765–768.
Kulesa, P. M., G. C. Cruywagen, S. R. Lubkin, P. K. Maini, J. Sneyd, M. W. J. Ferguson and J. D. Murray (1996). On a model mechanism for the spatial pattering of teeth primordia in the alligator. J. Theor. Biol. 180, 287–296.
Lacalli, T. C. (1981). Dissipative structures and morphogenetic pattern in unicellular algae. Phil. Trans. R. Soc. B 294, 547–588.
Liaw, S. S., C. C. Yang, R. T. Liu and J. T. Hong (2001). Turing model for the patterns of lady beetles. Phys. Rev. E 64, article no. 041909.
Madzvamuse, A., R. D. K. Thomas, P. K. Maini and A. J. Wathen (2002). A numerical approach to the study of spatial pattern formation in the ligaments of arcoid bivalves, submitted.
Maini, P. K., D. L. Benson and J. A. Sherratt (1992). Pattern formation in reaction-diffusion models with spatially inhomogeneous diffusion coefficients. IMA J. Math. Appl. Med. Biol. 9, 197–213.
May, A., P. A. Firby and A. P. Bassom (1999). Diffusion driven instability in an inhomogeneous circular domain. Math. Comput. Modell. 29, 53–66.
Meinhardt, H. (1982). Models of Biological Pattern Formation, London: Academic Press.
Meinhardt, H. (1995). The Algorithmic Beauty of Sea Shells, Heidelberg: Springer.
Meinhardt, H., A.-J. Koch and G. Bernasconi (1998). Models of pattern formation applied to plant development, in Symmetry in Plants, R. V. Jean and D. Barabé (Eds), Singapore: World Scientific, pp. 723–758.
Monod, J. (1942). Recherches sur la Croissance des Cultures Bacteriennes, Paris: Herman.
Morton, K. W. and D. F. Mayers (1994). Numerical Solution of Partial Differential Equations, Cambridge: Cambridge University Press.
Murray, J. D. (1993). Mathematical Biology, 2nd edn, Berlin: Springer.
Oster, G. F. and J. D. Murray (1989). Pattern formation models and developmental constraints. J. Exp. Zool. 251, 186–202.
Painter, K. J., P. K. Maini and H. G. Othmer (1999). Stripe formation in juvenile Pomacanthus explained by a generalised Turing mechanism with chemotaxis. Proc. Natl. Acad. Sci. USA 96, 5549–5554.
Plaza, R., F. Sánchez-Garduño, P. Padilla, R. A. Barrio and P. K. Maini (2002). The effect of growth and curvature on pattern formation, in preparation.
Schnakenberg, J. (1979). Simple chemical reaction systems with limit cycle behaviour. J. Theor. Biol. 81, 389–400.
Smith, H. L. and P. Waltman (1995). The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge: Cambridge University Press.
Turing, A. M. (1952). The chemical basis of morphogenesis. Phil. Trans. R. Soc. B 237, 37–72.
Varea, C., J. L. Aragón and R. A. Barrio (1999). Turing patterns on a sphere. Phys. Rev. E 60, 4588–4592.
Wolpert, L. (1969). Positional information and the spatial pattern of cellular differentiation. J. Theor. Biol. 25, 1–47.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Crampin, E.J., Hackborn, W.W. & Maini, P.K. Pattern formation in reaction-diffusion models with nonuniform domain growth. Bull. Math. Biol. 64, 747–769 (2002). https://doi.org/10.1006/bulm.2002.0295
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1006/bulm.2002.0295