The slow dynamics of two-spike solutions for the gray-scott and gierer-meinhardt systems: Competition and oscillatory instabilities

SIAM Journal on Applied Dynamical Systems

Volumn 4, Issue 4, 2005, Pages 904-953

  Sun, Wentao ^a   Ward, Michael J ^b   Russell, Robert ^c  

^a SIMON FRASER UNIVERSITY   (Canada)

^b SHANDONG UNIVERSITY   (China)

^c UNIVERSITY OF BRITISH COLUMBIA   (Canada)

Author keywords

Competition and oscillatory instabilities; Hopf bifurcation; Nonlocal eigenvalue problem; Spikes

Indexed keywords

BIFURCATION (MATHEMATICS); DIFFERENTIAL EQUATIONS; DIFFUSION; EIGENVALUES AND EIGENFUNCTIONS; MATHEMATICAL MODELS; STABILITY;

COMPETITION AND OSCILLATORY INSTABILITIES; HOPF BIFURCATION; NONLOCAL EIGENVALUE PROBLEM; SPIKES;

OSCILLATIONS;

EID: 20044376798     PISSN: 15360040     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/040620990     Document Type: Article

References (45)

1. M. CROSS AND P. HOHENBURG, Pattern formation outside of equilibrium, Rev. Modern Phys., 65 (1993), pp. 851-1112.
2. A. DOELMAN, W. ECKHAUS, AND T. J. KAPER, Slowly modulated two-pulse solutions in the Gray-Scott model I: Asymptotic construction and stability, SIAM J. Appl. Math., 61 (2000), pp. 1080-1102.
3. A. DOELMAN, W. ECKHAUS, AND T. J. KAPER, Slowly modulated two-pulse solutions in the Gray-Scott model II: Geometric theory, bifurcations, and splitting dynamics, SIAM J. Appl. Math., 61 (2001), pp. 2036-2062.
4. A. DOELMAN, R. A. GARDNER, AND T. J. KAPER, Stability analysis of singular patterns in the 1D Gray-Scott model: A matched asymptotics approach, Phys. D, 122 (1998), pp. 1-36.
5. A. DOELMAN, R. A. GARDNER, AND T. J. KAPER, A stability index analysis of 1-D patterns of the Gray-Scott model, Mem. Amer. Math. Soc., 155 (737) (2002).
6. A. DOELMAN, R. A. GARDNER, AND T. KAPER., Large stable pulse solutions in reaction-diffusion equations, Indiana, Univ. Math. J., 50 (2001), pp. 443-507.
7. A. DOELMAN, T. J. KAPER, AND P. ZEGELING, Pattern formation in the one-dimensional Gray-Scott model, Nonlinearity, 10 (1997), pp. 523-563.
8. A. DOELMAN AND H. VAN DER PLOEG, Homoclinic stripe patterns, SIAM J. Appl. Dyn. Syst., 1 (2002), pp. 65-104.
9. A. DOELMAN AND T. J. KAPER, Semistrong pulse interactions in a class of coupled reaction-diffusion equations, SIAM J. Appl. Dyn. Syst., 2 (2003), pp. 53-96.
10. T. ERNEUX, E. L. REISS, L. J. HOLDEN, AND M. GEORGIOU, Slow passage through bifurcation and limit points. Asymptotic theory and applications, in Dynamic Bifurcations (Luminy 1990), Lecture Notes in Math. 1493, Springer-Verlag, Berlin, 1991, pp. 14-28.
11. P. FIFE, Pattern formation in gradient systems, in Handbook of Dynamical Systems, Vol. 2, North-Holland, Amsterdam, 2002, pp. 677-722.
12. P. FREITAS AND C. ROCHA, Lyapunov functional and stability for Fitzhugh-Nagumo systems, J. Differential Equations, 169 (2001), pp. 208-227.
13. A. GIERER AND H. MEINHARDT, A theory of biological pattern formation, Kybernetik, 12 (1972), pp. 30-39.
14. P. GRAY AND S. K. SCOTT, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilities in the system A + 2B → 3B, B → C, Ohem. Eng. Sci., 39 (1984), pp. 1087-1097.
15. W. HUANG AND R. RUSSELL, A moving collocation method for the solution of time dependent partial differential equations, Appl. Numer. Math., 20 (1996), pp. 101-116.
16. D. IRON AND M. J. WARD, The dynamics of multispike solutions to the one-dimensional Gierer-Meinhardt model, SIAM J. Appl. Math., 62 (2002), pp. 1924-1951.
17. D. IRON, M. J. WARD, AND J. WEI, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Phys. D, 150 (2001), pp. 25-62.
18. T. KOLOKOLNIKOV, M. WARD, AND J. WEI, The stability of spike equilibria in the one-dimensional Gray-Scott model: The low feed-rate regime, Stud. Appl. Math., 115 (2005), pp. 21-71.
19. T. KOLOKOLNIKOV, M. WARD, AND J. WEI, The stability of spike equilibria in the one-dimensional Gray-Scott model: The pulse-splitting regime, Phys. D, 202 (2005), pp. 258-293.
20. T. KOLOKOLNIKOV, M. WARD, AND J. WEI, Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model, Interfaces Free Bound., submitted.
21. T. KOLOKOLNIKOV, M. WARD, AND J. WEI, Pulse-splitting for some reaction-diffusion systems in one-space dimension, Stud. Appl. Math., 114 (2005), pp. 115-165.
22. E. KNOBLOCH, Outstanding problems in the theory of pattern formation, in Nonlinear Dynamics and Chaos. Where Do We Go from Here?, S. J. Hogan et al., eds., Institute of Physics Publishing, Bristol, UK, 2003, pp. 117-166.
23. K. J. LEE, W. D. MCCORMICK, J. E. PEARSON, AND H. L. SWINNEY, Experimental observation of self-replicating spots in a reaction-diffusion system, Nature, 369 (1994), pp. 215-218.
24. K. J. LEE AND H. L. SWINNEY, Lamellar structures and self-replicating spots in a reaction-diffusion system, Phys. Rev. E (3), 51 (1995), pp. 1899-1915.
25. C. S. LIN, W. M. NI, AND I. TAKAGI, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), pp. 1-27.
26. H. MEINHARDT, The Algorithmic Beauty of Sea Shells, Springer-Verlag, Berlin, 1995.
27. C. MURATOV AND V. V. OSIPOV, Traveling spike auto-solitons in the Gray-Scott model, Phys. D, 155 (2001), pp. 112-131.
28. C. B. MURATOV AND V. V. OSIPOV, Stability of the static spike autosolitons in the Gray-Scott model, SIAM J. Appl. Math., 62 (2002), pp. 1463-1487.
29. C. MURATOV AND V. V. OSIPOV, Static spike autosolitons in the Gray-Scott model, J. Phys. A, 33 (2000), pp. 8893-8916.
30. NAG FORTRAN LIBRARY, Mark 17, routine D03PCF, Numerical Algorithms Group Ltd., Oxford, UK, 1995.
31. W. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), pp. 9-18.
32. Y. NISHIURA AND D. UEYAMA, A skeleton structure of self-replicating dynamics, Phys. D, 130 (1999), pp. 73-104.
33. Y. NISHIURA AND D. UEYAMA, Spatio-temporal chaos for the Gray-Scott model, Phys. D, 150 (2001), pp. 137-162.
34. Y. NISHIURA, Far-from-Equilibrium Dynamics, Transl. Math. Monogr. 209, AMS, Providence, RI, 2002.
35. Y. NISHIURA, Global bifurcational approach to the onset of spatio-temporal chaos in reaction-diffusion systems, Methods Appl. Anal., 8 (2001), pp. 321-332.
36. J. E. PEARSON, Complex patterns in a simple system, Science, 216 (1993), pp. 189-192.
37. V. PETROV, S. K. SCOTT, AND K. SHOWALTER, Excitability, wave reflection, and wave splitting in a cubic autocatalysis reaction-diffusion system, Philos. Trans. Roy. Soc. London Ser. A, 347 (1994), pp. 631-642.
38. W. N. REYNOLDS, S. PONCE-DAWSON, AND J. E. PEARSON, Dynamics of self-replicating patterns in reaction-diffusion systems, Phys. Rev. Lett., 72 (1994), pp. 2797-2800.
39. W. N. REYNOLDS, S. PONCE-DAWSON, AND J. E. PEARSON, Dynamics of self-replicating spots in reaction-diffusion systems, Phys. Rev. E (3), 56 (1997), pp. 185-198.
40. W. SUN, T. TANG, M. J. WARD, AND J. WEI, Numerical challenges for resolving spike dynamics for two reaction-diffusion systems, Stud. Appl. Math., 111 (2003), pp. 41-84.
41. A. TURING, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. B, 327 (1952), pp. 37-72.
42. D. UEYAMA, Dynamics of self-replicating patterns in the one-dimensional Gray-Scott model, Hokkaido Math J., 28 (1999), pp. 175-210.
43. M. J. WARD AND J. WEI, Hopf bifurcations and oscillatory instabilities of spike solutions for the one-dimensional Gierer-Meinhardt model, J. Nonlinear Sci., 13 (2003), pp. 209-264.
44. M. J. WARD AND J. WEI, Hopf bifurcations of spike solutions for the shadow Gierer-Meinhardt model, European J. Appl. Math., 14 (2003), pp. 677-711.
45. J. WEI, On single interior spike solutions for the Gierer-Meinhardt system: Uniqueness and stability estimates, European J. Appl. Math., 10 (1999), pp. 353-378.