Skip to main content
SpringerLink
Log in

Asymptotic Methods for Reaction-Diffusion Systems: Past and Present

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

Abstract

A brief historical survey of the development of asymptotic and analytical methodologies for the analysis of spatio-temporal patterns in reaction-diffusion (RD) and related systems is given. Although far from complete, the bibliography is hopefully representative of some of the advances in this area over the past 40 years. Within the scope of this survey, some of the key research contributions of Lee Segel are highlighted.

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

References

  • Alikakos, N., Bates, P., Chen, X., 1994. Convergence of the Cahn-Hilliard equation to the Hele–Shaw model. Arch. Ration. Mech. Anal. 128(2), 165–205.

    Article  MATH  MathSciNet  Google Scholar 

  • Alikakos, N., Bates, P., Fusco, G., 1991. Slow motion for the Cahn–Hilliard equation in one space dimension. J. Differential Equations 90(1), 81–135.

    Article  MATH  MathSciNet  Google Scholar 

  • Bank, R., 1998. PLTMG: A Software Package for Solving Elliptic Partial Differential Equations: Users Guide 8.0, Software, Environments and Tools 5. SIAM, Philadelphia.

  • Barkley, D., 1992. Linear stability analysis of spiral waves in excitable media. Phys. Rev. Lett. 68, 2090–2093.

    Article  Google Scholar 

  • Beyn, W., Champneys, A., Sandstede, B., Scheel, A., 2002. Numerical continuation, and computation of normal forms. In: Handbook of Dynamical Systems, vol. 2. North-Holland, Amsterdam, pp. 149–219.

  • Bonami, A., Hilhorst, D., Logak, E., Mimura, M., 2001. Singular limit of a chemotaxis growth model. Adv. Differential Equations 6(10), 1173–1218.

    MATH  MathSciNet  Google Scholar 

  • Borckmans, P., Dewel, G., Wit, A.D., Walgraef, D., 1995. Turing patterns and pattern selection. In: Kapral, R., Showalter, K. (Eds.), Chemical Waves and Patterns. Kluwer, pp. 323–363.

  • Bressloff, P., 2005. Weakly interacting pulses in synaptically coupled neural media. SIAM J. Appl. Math. 66(1), 57–81.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  • Budd, C., Carretero-Gonzalez, R., Russell, R., 2005. Precise computations of chemotactic collapse using moving mesh methods. J. Comput. Phys. 202(2), 463–487.

    Article  MATH  MathSciNet  Google Scholar 

  • Cahn, J.W., Hilliard, J.E., 1958. Free energy of a non-uniform system i. Interfacial free energy. J. Chem. Phys. 28, 258–267.

    Google Scholar 

  • Carr, J., 1981. Applications of Center Manifold Theory. Springer-Verlag, New York, Heidelberg, Berlin.

    Google Scholar 

  • Carr, J., Pego, R., 1989. Metastable patterns in solutions of u t = ε2 u xx − f(u). Commun. Pure Appl. Math. 42(5), 523–576.

    Article  MATH  MathSciNet  Google Scholar 

  • Champneys, A., Kuznetsov, Y., Sandstede, B., 1996. A numerical toolbox for homoclinic bifurcation analysis. Int. J. Bifur. Chaos Appl. Sci. Eng. 6(5), 867–887.

    Article  MATH  MathSciNet  Google Scholar 

  • Childress, S., Percus, J., 1981. Nonlinear aspects of chemotaxis. Math. Biosci. 56, 217–237.

    Article  MATH  MathSciNet  Google Scholar 

  • Chow, S. N., Hale, J., 1982. Methods of Bifurcation Theory. Springer-Verlag, New York.

    MATH  Google Scholar 

  • Crampin, E.J., Gaffney, E.A., Maini, P.K., 1999. Reaction and diffusion on growing domains: Scenarios for robust pattern formation. Bull. Math. Biol. 61, 1093–1120.

    Article  Google Scholar 

  • Crampin, E.J., Gaffney, E.A., Maini, P.K., 2002. Mode doubling and tripling in reaction-diffusion patterns on growing domains: a piece-wise linear model. J. Math. Biol. 44, 107–128.

    Article  MATH  MathSciNet  Google Scholar 

  • Crandall, M., Rabinowitz, P., 1973. Bifucation, perturbation of simple eigemvalues, and linearized stability. Arch. Rational Mech. Anal. 52, 161–180.

    Article  MATH  MathSciNet  Google Scholar 

  • Cross, M., Hohenburg, P., 1993. Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112.

    Article  Google Scholar 

  • Dancer, N., Hilhorst, D., Mimura, M., Peletier, L.A., 1999. Spatial segregation limit of a competition-diffusion system. Eur. J. Appl. Math. 10(2), 97–115.

    Article  MATH  MathSciNet  Google Scholar 

  • de Wit, A., 1999. Spatial patterns and spatiotemporal dynamics in chemical physics. Adv. Chem. Phys. 109, 435–513.

    Article  Google Scholar 

  • Doedel, E., Tuckerman, L. S., 2000. Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems, IMA Volumes in Mathematics and its Applications, vol. 119. Springer, New York.

  • Doedel, E., Wang, X.J., 1994. Auto94: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations. Applied Mathematics Report, California Institute of Technology.

  • Doelman, A., Gardner, R.A., Kaper, T.J., 1998. Stability analysis of singular patterns in the 1d Gray–Scott model: a matched asymptotics approach. Physica D 122, 1–36.

    Google Scholar 

  • Doelman, A., Gardner, R.A., Kaper, T.J., 2001. Large stable pulse solutions in reaction-diffusion equations. Indiana Univ. Math. J. 50(1), 443–507.

    Article  MATH  MathSciNet  Google Scholar 

  • Doelman, A., van der Ploeg, H., 2002. Homoclinic stripe patterns. SIAM J. Appl. Dyn. Syst. 1(1), 65–104.

    Article  MATH  MathSciNet  Google Scholar 

  • Dolak, Y., Schmeiser, C., 2006. The Keller-Segel model with logistic sensitivity function and small diffusivity. SIAM J. Appl. Math. 66(1), 286–308.

    Article  MathSciNet  Google Scholar 

  • Ei, S., 2002. The motion of weakly interacting pulses in reaction-diffusion systems. J. Dynam. Differential Equations 14(1), 85–137.

    Article  MATH  MathSciNet  Google Scholar 

  • Ei, S., Nishiura, Y., Ueda, K., 2001. 2n splitting or edge splitting: a manner of splitting in dissipative systems. Jpn. J. Ind. Appl. Math. 18(2), 181–205.

    Article  MATH  MathSciNet  Google Scholar 

  • Ermentrout, B., 1991. Stripes or spots? Nonlinear effects in bifurcation of reaction-diffusion equations on the square. Proc. R. Soc. Lond, Ser. A 434(1891), 413–417.

    Google Scholar 

  • Ermentrout, B., 2002. Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, Software, Environments, and Tools 14. SIAM, Philadelphia.

  • Fiedler, B., Scheel, A., 2003. Spatio-temporal dynamics of reaction-diffusion systems. In: Trends in Nonlinear Analysis. Springer-Verlag, Berlin, pp. 23–152.

  • Fife, P., 1988. Dynamics of Internal Layers and Diffusive Interfaces, CBMS-NSF Regional Conference Series in Applied Mathematics vol. 53. SIAM, Philadelphia.

  • Fife, P., 2002. Pattern formation in gradient systems. In: Handbook of Dynamical Systems, vol. 2. North-Holland, Amsterdam, pp. 677–722.

  • Filippas, S., Kohn, R.V., 1992. Refined asymptotics for the blowup of u t −δ u = u p. Commun. Pure Appl. Math. 45(7), 821–869.

  • Fusco, G., Hale, J., 1989. Slow-motion manifolds, dormant instability, and singular perturbations. J. Dynam. Differential Equations 1(1), 75–94.

    Article  MATH  MathSciNet  Google Scholar 

  • Gierer, A., Meinhardt, H., 1972. A theory of biological pattern formation. Kybernetik 12, 30–39.

    Article  Google Scholar 

  • Goldstein, R.E., Muraki, D.J., Petrich, D.M., 1996. Interface proliferation and the growth of labyrinths in a reaction-diffusion system. Phys. Rev. E. 53, 3933–3957.

    Article  MathSciNet  Google Scholar 

  • Golubitsky, M.I., Stewart, I., Schaeffer, D.G., 1988. Singularities and Groups in Bifurcation Theory, vol. II, Applied Mathematical Sciences 69. Springer, New York.

    Google Scholar 

  • Grindrod, P., 1996. The Theory and Application of Reaction-Diffusion Equations. Oxford University Press, Oxford.

    Google Scholar 

  • Habetler, G., Matkowsky, B.J., 1975. On the validity of a nonlinear dynamic stability theory. Arch. Rational Mech. Anal. 57, 166–188.

    MathSciNet  Google Scholar 

  • Herrero, M.A., Velázquez, J.J.L., 1996. Chemotactic collapse for the Keller-Segel model. J. Math. Biol. 35, 583–623.

    Article  Google Scholar 

  • Horstmann, D., 2003. From 1970 until present: the Keller–Segel model in chemotaxis and its consequences. I. Jahresber. Deutsch. Math.-Verein. 105(3), 103–165.

    MATH  MathSciNet  Google Scholar 

  • Iron, D., Ward, M.J., Wei, J., 2001. The stability of spike solutions to the one-dimensional Gierer–Meinhardt model. Physica D 150, 25–62.

    Article  MATH  MathSciNet  Google Scholar 

  • Jones, C.K.R.T., 1994. Geometric singular perturbation theory. In: Springer Lecture Notes in Mathematics, 1609. Springer, New York, pp. 44–118.

  • Keener, J.P., 1976. Secondary bifurcation in nonlinear diffusion reaction equations. Studies Appl. Math. 55, 187–211.

    MathSciNet  MATH  Google Scholar 

  • Keener, J.P., 1978. Activators and inhibitors in pattern formation. Studies Appl. Math. 59, 1–23.

    MathSciNet  Google Scholar 

  • Keller, E.F., Segel, L.A., 1970. The initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415.

    Article  Google Scholar 

  • Keller, E.F., Segel, L.A., 1971. Model for chemotaxis. J. Theor. Biol. 30, 225–234.

    Article  Google Scholar 

  • Keller, H.B., 1987. Lectures on Numerical Methods in Bifurcation Problems, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, Bombay, 79. Springer-Verlag, Berlin.

  • Kevorkian, J., Cole, J., 1981. Perturbation Methods in Applied Mathematics, Applied Mathematical Sciences, 34. Springer-Verlag, New York, Berlin.

    Google Scholar 

  • Knobloch, E., 2003. Outstanding problems in the theory of pattern formation. In: Hogan, S.J., et al. (Eds.), Nonlinear dynamics and chaos. Where do we go from here? Institute of Physics Publishing, Bristol, UK, pp. 117–166.

  • Koch, A.J., Meinhardt, H., 1994. Biological pattern formation from basic mechanisms to complex structures. Rev. Mod. Phys. 66(4), 1481–1507.

    Article  Google Scholar 

  • Kolokolnikov, T., Ward, M., Wei, J., 2005a. The existence and stability of spike equilibria in the one-dimensional Gray–Scott model: the low feed-rate regime. Studies Appl. Math. 115(1), 21–71.

    Article  MathSciNet  MATH  Google Scholar 

  • Kolokolnikov, T., Ward, M., Wei, J., 2005b. The existence and stability of spike equilibria in the one-dimensional Gray–Scott model: the pulse-splitting regime. Physica D 202, 258–293.

    Article  MATH  MathSciNet  Google Scholar 

  • Kolokolnikov, T., Ward, M.J., 2003. Reduced wave green’s functions and their effect on the dynamics of a spike for the Gierer–Meinhardt model. Eur. J. Appl. Math. 14(5), 513–545.

    Article  MATH  MathSciNet  Google Scholar 

  • Kondo, S., Asai, R., 1995. A viable reaction-diffusion wave on the skin of Pomacanthus, a marine Angelfish. Nature 376, 765–768.

    Article  Google Scholar 

  • Lee, K.J., McCormick, W.D., Pearson, J.E., Swinney, H.L., 1994. Experimental observation of self-replicating spots in a reaction-diffusion system. Nature 369, 215–218.

    Article  Google Scholar 

  • Lee, K.J., Swinney, H., 1995. Lamellar structures and self-replicating spots in a reaction-diffusion system. Phys. Rev. E. 51(3), 1899–1915.

    Article  Google Scholar 

  • Liang, C., Troy, W., 2003. PDE methods for nonlocal models. SIAM J. Appl. Dyn. Sys. 2(3), 487–516.

    Article  Google Scholar 

  • Lin, C.C., Segel, L.A., 1974. Mathematics Applied to Deterministic Problems in the Natural Sciences. Macmillan, New York.

    MATH  Google Scholar 

  • Madzvamuse, A., Maini, P.K., Wathen, A.J., 2005. A moving grid finite element method for the simulation of pattern generation by Turing models on growing domains. J. Sci. Comput. 24(2), 247–262.

    Article  MATH  MathSciNet  Google Scholar 

  • Madzvamuse, A., Wathen, A.J., Maini, P.K., 2003. A moving grid finite element method applied to a model biological pattern generator. J. Comput. Phys. 190(2), 478–500.

    Article  MATH  MathSciNet  Google Scholar 

  • Maini, P., Painter, K.J., Chau, H., 1997. Spatial pattern formation in chemical and biological systems. J. Chem. Soc., Faraday Trans. 93(20), 3601–3610.

    Article  Google Scholar 

  • Margerit, D., Barkley, D., 2002. Cookbook asymptotics for spiral and scroll waves in excitable media. Chaos 12(3), 636–649.

    Google Scholar 

  • Matkowsky, B.J., 1970. Nonlinear dynamic stability: a formal theory. SIAM J. Appl. Math. 18, 872–883.

    Article  MATH  Google Scholar 

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

    Google Scholar 

  • Meinhardt, H., 1995. The Algorithmic Beauty of Sea Shells. Springer-Verlag, Berlin.

    Google Scholar 

  • Mimura, M., 2003. Reaction-diffusion systems arising in biological and chemical systems: applications of singular limit procedures. In: Mathematical Aspects of Evolving Interfaces (Funchal, 2000), Lecture Notes in Mathematics, 1812. Springer, Berlin, pp. 89–112.

  • Mogilner, A., Edelstein-Keshet, L., 1996. Spatio-angular order in populations of self-aligning objects: formation of oriented patches. Physica D 89, 346–367.

    Article  MATH  MathSciNet  Google Scholar 

  • Mogilner, A., Edelstein-Keshet, L., Ermentrout, B., 1996. Selecting a common direction ii: peak-like solutions representing total alignment of cell clusters. J. Math. Biol. 34, 811– 842.

    MATH  MathSciNet  Google Scholar 

  • Muratov, C., Osipov, V.V., 2000. Static spike autosolitons in the Gray–Scott model. J. Phys. A: Math Gen. 33, 8893–8916.

    Article  MathSciNet  Google Scholar 

  • Muratov, C., Osipov, V.V., 2002. Stability of the static spike autosolitons in the Gray–Scott model. SIAM J. Appl. Math. 62(5), 1463–1487.

    Article  MATH  MathSciNet  Google Scholar 

  • Murray, J.D., 2003. Mathematical Biology II: Spatial Models and Biomedical Applications, Interdisciplinary Applied Mathematics, vol. 18. Springer, New York.

    Google Scholar 

  • Nanjundiah, V., 1973. Chemotaxis, signal relaying, and aggregation morphology. J. Theor. Biol. 42, 63–105.

    Article  Google Scholar 

  • Newell, A.C., Whitehead, J.A., 1969. Finite bandwidth, finite amplitude convection. J. Fluid Mech. 38, 279–303.

    Article  MATH  Google Scholar 

  • Ni, W.M., 1998. Diffusion, cross-diffusion, and their spike-layer steady-states. Notices Am. Math. Soc. 45(1), 9–18.

    MATH  Google Scholar 

  • Nicolis, G., Prigogine, I., 1977. Self-Organization in Non-Equilibrium Systems: From Dissipative Structures to Order Through Fluctuations. Wiley, New York.

  • Nishiura, Y., 2002. Far-From-Equilibrium Dynamics, Translations of Mathematical Monographs, vol. 209. AMS publications, Providence, Rhode Island.

  • Nishiura, Y., Fujii, H., 1987. Stability of singularly perturbed solutions to systems of reaction-diffusion equations. SIAM J. Math. Anal. 18, 1726–1770.

    Article  MATH  MathSciNet  Google Scholar 

  • Nishiura, Y., Teramoto, T., Ueda, K., 2003. Scattering and separators in dissipative systems. Phys. Rev. E. 67(5), 56210.

    Google Scholar 

  • Nishiura, Y., Ueyama, D., 1999. A skeleton structure of self-replicating dynamics. Physica D 130, 73–104.

    Article  MATH  Google Scholar 

  • Nishiura, Y., Ueyama, D., 2001. Spatio-temporal chaos for the Gray–Scott model. Physica D 150, 137–162.

    Article  MATH  Google Scholar 

  • Novick-Cohen, A., 1998. The Cahn–Hilliard equation: mathematical and modeling perspectives. Adv. Math. Sci. Appl. 8(2), 965–985.

    MATH  MathSciNet  Google Scholar 

  • Novick-Cohen, A., Segel, L., 1984. Nonlinear aspects of the Cahn–Hilliard equation. Physica D 10(3), 277–298.

    Article  MathSciNet  Google Scholar 

  • Painter, K., Hillen, T., 2002. Volume-filling and quorom-sensing in models for chemosensitive movement. Can. Appl. Math. Q. 10(4), 501–543.

    MATH  MathSciNet  Google Scholar 

  • Painter, K.J., Maini, P.K., Othmer, H.G., 1999. Stripe formation in juvenile pomacanthus explained by a generalized Turing mechanism with chemotaxis. Proc. Natl. Acad. Sci. USA, Dev. Biol. 96, 5549–5554.

    Google Scholar 

  • Pearson, J.E., 1993. Complex patterns in a simple system. Science 216, 189–192.

    Article  Google Scholar 

  • Pego, R., 1989. Front migration in the nonlinear Cahn–Hilliard equation. Proc. R. Soc. Lond. Ser. A 422(1863), 261–278.

    Google Scholar 

  • Peletier, L.A., Troy, W.C., 2001. Higher Order Models in Physics and Mechanics, Progress in Nonlinear Differential Equations and Their Applications, 45. Birhauser Boston, Boston, MA.

  • Potapov, A., Hillen, T., 2005. Metastability in chemotaxis models. J. Dynam. Differential Equations 17(2), 293–330.

    Article  MATH  MathSciNet  Google Scholar 

  • Rademacher, J., Sandstede, B., Scheel, S., 2005. Computing absolute and essential spectra using continuation. submitted, SIAM J. Sci. Comput.

  • Ren, W., Wang, X.P., 2000. An iterative grid redistribution method for singular problems in multiple dimensions. J. Comput. Phys. 159(2), 246–273.

    Article  MATH  MathSciNet  Google Scholar 

  • Rubinstein, J., Sternberg, P., Keller, J.B., 1989. Fast reaction, slow diffusion, and curve shortening. SIAM J. Appl. Math. 49(1), 116–133.

    Article  MATH  MathSciNet  Google Scholar 

  • Saarloos, W.V., 1994. The complex Ginzburg–Landau equation for beginners. In: Cladis, P.E., Palffy-Muhoray, P. (Eds.), Proceedings of the Santa Fe Workshop on Spatio-Temporal Patterns in Nonequilibrium Complex Systems. Addison-Wesley, Chicago, pp. 19–31.

  • Saarloos, W.V., 2003. Front propagation into unstable states. Phys. Rep. 386, 29–222.

    Article  MATH  Google Scholar 

  • Sakamoto, K., 1998. Internal layers in high-dimensional domains. Proc. R. Soc. Edinb. Sect. A 128(2), 359–401.

    Google Scholar 

  • Sandstede, B., 2002. Stability of traveling waves. In: Handbook of Dynamical Systems, vol. 2. North-Holland, Amsterdam, pp. 983–1055.

  • Sandstede, B., Scheel, A., 2000a. Absolute and convective instabilities of waves on unbounded and large bounded domains. Physica D 145, 233–277.

    Article  MATH  MathSciNet  Google Scholar 

  • Sandstede, B., Scheel, A., 2000b. Absolute versus convective instability of spiral waves. Phys. Rev. E. 62, 7708–7714.

    Article  MathSciNet  Google Scholar 

  • Sandstede, B., Scheel, A., Wulff, C., 1999. Bifurcations and dynamics of spiral waves. J. Nonlinear Sci. 9, 439–478.

    Article  MATH  MathSciNet  Google Scholar 

  • Segel, L.A., 1966. The importance of asymptotic analysis in applied mathematics. Am. Math. Monthly 73, 7–14.

    Google Scholar 

  • Segel, L.A., 1969. Distant sidewalls cause slow amplitude modulation of cellular convection. J. Fluid Mech. 38, 203–224.

    Article  MATH  Google Scholar 

  • Segel, L.A., 1972. Simplification and scaling. SIAM Rev. 14(4), 547–571.

    Article  MATH  MathSciNet  Google Scholar 

  • Segel, L.A., Jackson, J.L., 1972. Dissipative structure: an explanation and an ecological example. J. Theor. Biol. 37(3), 545–559.

    Article  Google Scholar 

  • Segel, L.A., Slemrod, M., 1989. The quasi-steady state assumption: a case study in perturbation. SIAM Rev. 31(3), 446–477.

    Article  MATH  MathSciNet  Google Scholar 

  • Sun, W., Ward, M.J., Russell, R., 2005. The slow dynamics of two-spike solutions for the Gray-Scott and Gierer-Meinhardt systems: competition and oscillatory instabilities. SIAM J. Appl. Dyn. Sys. 4(4), 904–953.

    Article  MATH  MathSciNet  Google Scholar 

  • Taniguchi, M., Nishiura, Y., 1994. Instability of planar interfaces in reaction-diffusion systems. SIAM J. Math. Anal. 25(1), 99–134.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  • Tyson, J.J., Keener, J.P., 1989. Singular perturbation theory of spiral waves in excitable media. Physica D 32, 327–361.

    Article  MathSciNet  Google Scholar 

  • van Hecke, H., Hohenburg, P.C., van Saarloos, W., 1994. Amplitude equations for pattern forming systems. In: van Beijeren, H., Ernst, M. H. (Eds.), Fundamental Problems in Statistical Mechanics VIII. North-Holland, Amsterdam, pp. 245–278.

  • Velazquez, J.J.L., 2004. Point dyamics in a singular limit of the Keller–Segel model: I and ii. SIAM J. Appl. Math. 64(4), 1198–1248.

    Article  MATH  MathSciNet  Google Scholar 

  • Walgraef, D., 1997. Spatio-Temporal Pattern Formation, With Examples from Physics, Chemistry, and Materials Science. Springer, New York.

    Google Scholar 

  • Ward, M.J., 1998. Exponential asymptotics and convection–diffusion–reaction models. In: Cronin, J., O’Malley, R. (Eds.), Analyzing Multiscale Phenomena Using Singular Perturbation Methods. Proceedings of Symposia in Applied Mathematics, vol. 56, AMS Short Course, AMS publications, Providence, Rhode Island, pp. 151–184.

  • Ward, M.J., 2005. Spikes for singularly perturbed reaction-diffusion systems and carrier’s problem. In: Hua, C., Wong, R. (Eds.), Differential Equations and Asymptotic Theory in Mathematical Physics. Series in Analysis, vol. 2. World Scientific, Singapore, pp. 100–188.

  • Ward, M.J., Wei, J., 2003. Hopf bifurcations and oscillatory instabilities of spike solutions for the one-dimensional Gierer-Meinhardt model. J. Nonlinear Sci. 13(2), 209–264.

    Article  MATH  MathSciNet  Google Scholar 

  • Wei, J., Winter, M., 2001. Spikes for the two-dimensional Gierer-Meinhardt system: the weak coupling case. J. Nonlinear Sci. 11(6), 415–458.

    Article  MATH  MathSciNet  Google Scholar 

  • Wheeler, P., Barkley, D., 2006. Computation of spiral spectra. to appear, SIAM J. Appl. Dyn. Sys.

  • Winfree, A., 1991. Varieties of spiral wave behavior: An experimentalist’s approach to the theory of excitable media. Chaos 1(3), 303–334.

    Article  MathSciNet  Google Scholar 

  • Wittenberg, R., Holmes, P., 1997. The limited effectiveness of normal forms: a critical review and extension of local bifurcation studies of the Brusselator pde. Physica D 100, 1–40.

    Article  MATH  MathSciNet  Google Scholar 

  • Woodward, D.E., Tyson, R.C., Murray, J.D., Budrene, E.O., Berg, H. 1995. Spatio-temporal patterns generated by Salmonella Typhimurium. Biophysical J. 68, 2181–2189.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, V6T 1Z2, Canada

    Michael J. Ward

Authors
  1. Michael J. Ward
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael J. Ward.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ward, M.J. Asymptotic Methods for Reaction-Diffusion Systems: Past and Present. Bull. Math. Biol. 68, 1151–1167 (2006). https://doi.org/10.1007/s11538-006-9091-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-006-9091-y

Keywords

Search

Navigation