\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations

Abstract Related Papers Cited by
  • In this paper we study pattern formation arising in a system of a single reaction-diffusion equation coupled with subsystem of ordinary differential equations, describing spatially-distributed growth of clonal populations of precancerous cells, whose proliferation is controlled by growth factors diffusing in the extracellular medium and binding to the cell surface. We extend the results on the existence of nonhomogenous stationary solutions obtained in [9] to a general Hill-type production function and full parameter set. Using spectral analysis and perturbation theory we derive conditions for the linearized stability of such spatial patterns.
    Mathematics Subject Classification: Primary: 35K57, 35J57; Secondary: 92B99.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    V. I. Arnold, "Ordinary Differential Equations," MIT Press, Cambridge, 1978.

    [2]

    K. I. Chueh, C. Conley and J. Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Ind. Univ. Math. J., 26 (1977), 373-392.doi: 10.1512/iumj.1977.26.26029.

    [3]

    A. Doelman, R. A. Gardner and T. J. Kaper, Stability analysis of singular patterns in the 1-D Gray-Scott model: A matched asymptotics approach, Phys. D, 122 (1998), 1-36.doi: 10.1016/S0167-2789(98)00180-8.

    [4]

    A. Doelman, R. A. Gardner and T. J. Kaper, Large stable pulse solutions in reaction-diffusion equations, Indiana Univ. Math. J., 50 (2001), 443-507.doi: 10.1512/iumj.2001.50.1873.

    [5]

    D. Henry, "Geomertic Theory of Semilinear Parabolic Equations," Springer-Verlag, 1981.

    [6]

    T. Kato, "Perturbation Theory for Linear Operators," Springer-Verlag, New York Inc, 1966.

    [7]

    A. Marciniak-Czochra and M. Kimmel, Modelling of early lung cancer progression: Influence of growth factor production and cooperation between partially transformed cells, Math. Mod. Meth. Appl. Sci., 17 (2007), 1693-1719.doi: 10.1142/S0218202507002443.

    [8]

    A. Marciniak-Czochra and M. Kimmel, Dynamics of growth and signalling along linear and surface structures in very early tumours, Comp. Math. Meth. Med., 7 (2006), 189-213.doi: 10.1080/10273660600969091.

    [9]

    A. Marciniak-Czochra and M. Kimmel, Reaction-diffusion model of early carcinogenesis: The effects of influx of mutated cells, Math. Model. Nat.Phenom., 3 (2008), 90-114.doi: 10.1051/mmnp:2008043.

    [10]

    J. D. Murray, "Mathematical Biology," Springer-Verlag, 2003.

    [11]

    P. K. Maini, In "On Growth and Form. Spatio-Temporal Pattern Formation in Biology," John Wiley & Sons, 1999.

    [12]

    F. Rothe, "Global Solutions of Reaction-Diffusion Systems," Springer-Verlag, Berlin, 1994.

    [13]

    J. Smoller, "Shock-Waves and Reaction-Diffusion Equations," Springer-Verlag, New York Heidelberg Berlin, 1994.

    [14]

    A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. B, 237 (1952), 37-72.doi: 10.1098/rstb.1952.0012.

    [15]

    J. Wei, On the interior spike layer solutions for some singular perturbation problems, Proc. Royal Soc. Edinb., 128A (1998), 849-874.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(89) Cited by(0)

Access History

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return