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Spot Self-Replication and Dynamics for the Schnakenburg Model in a Two-Dimensional Domain

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Abstract

The dynamical behavior of multi-spot solutions in a two-dimensional domain Ω is analyzed for the two-component Schnakenburg reaction–diffusion model in the singularly perturbed limit of small diffusivity ε for one of the two components. In the limit ε→0, a quasi-equilibrium spot pattern in the region away from the spots is constructed by representing each localized spot as a logarithmic singularity of unknown strength S j for j=1,…,K at unknown spot locations x j ∈Ω for j=1,…,K. A formal asymptotic analysis, which has the effect of summing infinite logarithmic series in powers of −1/log ε, is then used to derive an ODE differential algebraic system (DAE) for the collective coordinates S j and x j for j=1,…,K, which characterizes the slow dynamics of a spot pattern. This DAE system involves the Neumann Green’s function for the Laplacian. By numerically examining the stability thresholds for a single spot solution, a specific criterion in terms of the source strengths S j , for j=1,…,K, is then formulated to theoretically predict the initiation of a spot-splitting event. The analytical theory is illustrated for spot patterns in the unit disk and the unit square, and is compared with full numerical results computed directly from the Schnakenburg model.

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References

  • Anderson, E., et al.: LAPACK User’s Guide, 3rd edn. SIAM, Philadelphia (1999)

    MATH  Google Scholar 

  • Ascher, U., Christiansen, R., Russell, R.: Collocation software for boundary value ODE’s. Math. Comput. 33, 659–679 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  • Astrov, Y.A., Purwins, H.G.: Plasma spots in a gas discharge system: birth, scattering and formation of molecules. Phys. Lett. A 283(3–4), 349–354 (2001)

    Article  Google Scholar 

  • Astrov, Y.A., Purwins, H.G.: Spontaneous division of dissipative solitions in a planar gas-discharge system with high ohmic electrode. Phys. Lett. A 358(5–6), 404–408 (2006)

    Article  MATH  Google Scholar 

  • Barrass, I., Crampin, E.J., Maini, P.K.: Mode transitions in a model reaction–diffusion system driven by domain growth and noise. Bull. Math. Biol. 68, 981–995 (2006)

    Article  MathSciNet  Google Scholar 

  • Blom, J.G., Trompert, R.A., Verwer, J.G.: Algorithm 758: VLUGR2: a vectorizable adaptive-grid solver for PDEs in 2D. ACM Trans. Math. Softw. 22(3), 302–328 (1996)

    Article  MATH  Google Scholar 

  • Bode, M., Liehr, A.W., Schenk, C.P., Purwins, H.G.: Interactions of dissipative solitons: particle-like behavior of localized structures in a three component reaction–diffusion system. Physica D 161(1–2), 45–66 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • Bressloff, P.C., Earnshaw, B.E., Ward, M.J.: Diffusion of protein receptors on a cylindrical dendritic membrane with partially absorbing traps. SIAM J. Appl. Math. 68(5), 1223–1241 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  • Chen, X., Kowalczyk, M.: Dynamics of an interior spike in the Gierer–Meinhardt system. SIAM J. Math. Anal. 33(1), 172–193 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  • Davis, P.W., Blanchedeau, P., Dullos, E., De Kepper, P.: Dividing blobs, chemical flowers, and patterned islands in a reaction–diffusion system. J. Phys. Chem. A 102(43), 8236–8244 (1998)

    Article  Google Scholar 

  • Doelman, A., Gardner, R.A., Kaper, T.J.: Stability analysis of singular patterns in the 1D Gray–Scott model: a matched asymptotics approach. Physica D 122(1–4), 1–36 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  • Doelman, A., Kaper, T.J., Peletier, L.A.: Homoclinic bifurcations at the onset of pulse self-replication. J. Differ. Equ. 231(1), 359–423 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • E, W.: Dynamics of vortices in Ginzburg–Landau theories with applications to superconductivity. Physica D 77(4), 383–404 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  • Ei, S., Wei, J.: Dynamics of metastable localized patterns and its application to the interaction of spike solutions for the Gierer–Meinhardt system in two space dimensions. Jpn. J. Ind. Appl. Math. 19(2), 181–226 (2002)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Ei, S., Mimura, M., Nagayama, M.: Pulse–pulse interaction in reaction–diffusion systems. Physica D 165(3–4), 176–198 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • Ei, S., Mimura, M., Nagayama, M.: Interacting spots in reaction–diffusion systems. Discrete Contin. Dyn. Syst. 14(1), 31–62 (2006)

    MATH  MathSciNet  Google Scholar 

  • Gaffney, E.A., Monk, N.A.M.: Gene expression time delays and turing pattern formation systems. Bull. Math. Biol. 68(1), 99–130 (2006)

    Article  MathSciNet  Google Scholar 

  • Gradshteyn, I.M., Ryzhik, I.M.: Table of Integrals, Series, and Products, Corrected and Enlarged edn. Academic Press, New York (1980)

    Google Scholar 

  • Gueron, S., Shafrir, I.: On a discrete variational problem involving interacting particles. SIAM J. Appl. Math. 60(1), 1–17 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  • Gustafson, S., Sigal, I.M.: Effective dynamics of magnetic vortices. Adv. Math. 109(2), 448–498 (2006)

    Article  MathSciNet  Google Scholar 

  • Iron, D., Wei, J., Winter, M.: Stability analysis of turing patterns generated by the Schnakenburg model. J. Math. Biol. 49(4), 358–390 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  • Kolokolnikov, T., Ward, M.J.: 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 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • Kolokolnikov, T., Ward, M., Wei, J.: The stability of spike equilibria in the one-dimensional Gray–Scott model: the pulse-splitting regime. Physica D 202(3–4), 258–293 (2005a)

    Article  MATH  MathSciNet  Google Scholar 

  • Kolokolnikov, T., Titcombe, M.S., Ward, M.J.: Optimizing the fundamental Neumann eigenvalue for the Laplacian in a domain with small traps. Eur. J. Appl. Math. 16(2), 161–200 (2005b)

    Article  MATH  MathSciNet  Google Scholar 

  • Kolokolnikov, T., Sun, W., Ward, M.J., Wei, J.: The stability of a stripe for the Gierer–Meinhardt model and the effect of saturation. SIAM J. Appl. Dyn. Sys. 5(2), 313–363 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  • Lin, C.S., Ni, W.M., Takagi, I.: Large amplitude stationary solutions to a chemotaxis system. J. Differ. Equ. 72, 1–27 (1988)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Madzvamuse, A., Maini, P.K., Wathen, A.J.: 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 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • Marshall, S.L.: A rapidly convergent modified Green’s function for Laplace’s equation in a rectangular domain. Proc. R. Soc. Lond. A 455, 1739–1766 (1999)

    MATH  Google Scholar 

  • McCann, R.C., Hazlett, R.D., Babu, D.K.: Highly accurate approximations of Green’s and Neumann functions on rectangular domains. Proc. R. Soc. Lond. A 457, 767–772 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Muratov, C., Osipov, V.V.: Spike autosolitons and pattern formation scenarios in the two-dimensional Gray–Scott model. Eur. Phys. J. B 22, 213–221 (2001)

    Article  Google Scholar 

  • Nasuno, S.: Dancing “atoms” and “molecules” of luminous gas-discharge spots. Chaos 13(3), 1010–1013 (2003)

    Article  Google Scholar 

  • Nishiura, Y., Ueyama, D.: A skeleton structure of self-replicating dynamics. Physica D 130(1–2), 73–104 (1999)

    Article  MATH  Google Scholar 

  • Nishiura, Y., Ueyama, D.: Spatio-temporal chaos for the Gray–Scott model. Physica D 150(3–4), 137–162 (2001)

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  • Straube, R., Ward, M.J., Falcke, M.: Reaction rate of small diffusing molecules on a cylindrical membrane. J. Stat. Phys. 129(2), 377–406 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Sun, W., Ward, M.J., Russell, R.: 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 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • Titcombe, M.S., Ward, M.J.: An asymptotic study of oxygen transport from multiple capillaries to skeletal muscle tissue. SIAM J. Appl. Math. 60(5), 1767–1788 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Ueyama, D.: Dynamics of self-replicating patterns in the one-dimensional Gray–Scott model. Hokkaido Math. J. 28(1), 175–210 (1999)

    MATH  MathSciNet  Google Scholar 

  • Vanag, V.K., Epstein, I.R.: Localized patterns in reaction–diffusion systems. Chaos 17(3), 037110 (2007)

    Article  MathSciNet  Google Scholar 

  • Ward, M.J., Wei, J.: The existence and stability of asymmetric spike patterns for the Schnakenburg model. Stud. Appl. Math. 109(3), 229–264 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • Ward, M.J., Henshaw, W.D., Keller, J.: Summing logarithmic expansions for singularly perturbed eigenvalue problems. SIAM J. Appl. Math. 53(3), 799–828 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  • Ward, M.J., McInerney, D., Houston, P., Gavaghan, D., Maini, P.: The dynamics and pinning of a spike for a reaction–diffusion system. SIAM J. Appl. Math. 62(4), 1297–1328 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • Wei, J.: Existence, stability, and metastability of point condensation patterns generated by the Gray–Scott system. Nonlinearity 12, 593–616 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  • Wei, J.: Pattern formations in two-dimensional Gray–Scott model: existence of single-spot solutions and their stability. Physica D 148(1–2), 20–48 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Wei, J., Winter, M.: Spikes for the two-dimensional Gierer–Meinhardt system: the strong coupling case. J. Differ. Equ. 178(2), 478–518 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • Wei, J., Winter, M.: Existence and stability of multiple spot solutions for the Gray–Scott model in ℝ2. Physica D 176(3–4), 147–180 (2003a)

    Article  MATH  MathSciNet  Google Scholar 

  • Wei, J., Winter, M.: Asymmetric spotty patterns for the Gray–Scott model in R 2. Stud. Appl. Math. 110(1), 63–102 (2003b)

    Article  MATH  MathSciNet  Google Scholar 

  • Wei, J., Winter, M.: Stationary multiple spots for reaction–diffusion systems. J. Math. Biol. 57(1), 53–89 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Dalhousie University, Halifax, Nova Scotia, B3H 3J5, Canada

    T. Kolokolnikov

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

    M. J. Ward

  3. Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong

    J. Wei

Authors
  1. T. Kolokolnikov
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  2. M. J. Ward
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  3. J. Wei
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Correspondence to M. J. Ward.

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Communicated by F. Verhulst.

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Kolokolnikov, T., Ward, M.J. & Wei, J. Spot Self-Replication and Dynamics for the Schnakenburg Model in a Two-Dimensional Domain. J Nonlinear Sci 19, 1–56 (2009). https://doi.org/10.1007/s00332-008-9024-z

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  • DOI: https://doi.org/10.1007/s00332-008-9024-z

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