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Self-organization analysis for a nonlocal convective Fisher equation. (English) Zbl 1227.35058

Summary: Using both an analytical method and a numerical approach we have investigated pattern formation for a nonlocal convective Fisher equation with constant and spatial velocity fields. We analyze the limits of the influence function due to nonlocal interaction and we obtain the phase diagram of critical velocities \(v_c\) as function of the width \(\mu \) of the influence function, which characterize the self-organization of a finite system.

MSC:

35B36 Pattern formations in context of PDEs
35K57 Reaction-diffusion equations
35K55 Nonlinear parabolic equations
92D25 Population dynamics (general)
76R50 Diffusion
76S05 Flows in porous media; filtration; seepage
76V05 Reaction effects in flows
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References:

[1] Cross, M. C.; Hohenberg, P. C., Rev. Mod. Phys., 65, 851 (1993) · Zbl 1371.37001
[2] Fife, P. C., J. Chem. Phys., 64, 554 (1976)
[3] Zhabotinsky, A. M.; Dolnik, M.; Epstein, I. R., J. Chem. Phys., 103, 10306 (1995)
[4] Legawiec, B.; Kawczynski, A. L., J. Phys. Chem. A, 101, 8063 (1997)
[5] Vanag, V. K.; Zhabotinsky, A. M.; Epstein, I. R., Phys. Rev. Lett., 86, 552 (2000)
[6] De Kepper, P.; Dulos, E.; Boissonade, J.; De Wit, A.; Dewel, G.; Borckmans, P., J. Stat. Phys., 101, 495 (2000) · Zbl 0973.80010
[7] Kawczynski, A. L.; Legawiec, B., Phys. Rev. Lett., 63, 021405 (2001)
[8] Delprato, A. M.; Samadani, A.; Kudrolli, A.; Tsimring, L. S., Phys. Rev. Lett., 87, 158102 (2001)
[9] Fuentes, M. A.; Kuperman, M. N.; Kenkre, V. M., Phys. Rev. Lett., 91, 158104 (2003)
[10] Mikhailov, A. S.; Showalter, K., Phys. Rep., 425, 79 (2006)
[11] Debnath, L., Nonlinear Differential Equations for Scientists and Engineers (1997), Birkhäuser: Birkhäuser Boston · Zbl 0892.35001
[12] Aoki, K., J. Math. Biol., 25, 453 (1987) · Zbl 0622.92014
[13] Petty, H. R.; Worth, R. G.; Kindzelskii, A. L., Phys. Rev. Lett., 84, 2754 (2000)
[14] Canosa, J., J. Res. Develop., 17, 307 (1973) · Zbl 0266.65080
[15] Berge, P.; Dubois, M., Phys. Rev. Lett., 32, 1041 (1974)
[16] Reichhardt, C.; Olson, C. J.; Gronbech-Jensen, N.; Nori, F., Phys. Rev. Lett., 86, 4354 (2001)
[17] Neicu, T.; Pradhan, A.; Larochelle, D. A.; Kudrolli, A., Phys. Rev. E, 62, 1059 (2000)
[18] Lin, A. L.; Mann, B. A.; Torres-Olviedo, G.; Lincoln, B.; Kas, J.; Swinney, H. L., Biophys. J., 87, 75 (2004)
[19] Kenkre, V. M., Physica A, 342, 242 (2004)
[20] Murray, J. D., Mathematical Biology (1993), Springer: Springer New York · Zbl 0779.92001
[21] Giuggioli, L.; Kenkre, V. M., Physica D, 183, 245 (2003) · Zbl 1059.92043
[22] Morgado, R.; Oliveira, F. A.; Batrouni, G. G.; Hansen, A., Phys. Rev. Lett., 89, 100601 (2001)
[23] Lapas, L. C.; Costa, I. V.L.; Vainstein, M. H.; Oliveira, F. A., Europhys. Lett., 77, 37004 (2007)
[24] Lapas, L. C.; Morgado, R.; Vainstein, M. H.; Rubi, J. M.; Oliveira, F. A., Phys. Rev. Lett., 101, 230602 (2008)
[25] Costa, I. V.L.; Morgado, R.; Lima, M. V.T.; Oliveira, F. A., Europhys. Lett., 63, 173 (2003)
[26] William, H. P.; Teukolsky, A. S.; Vetterling, T. V.; Flannery, P. B., Numerical Recipes in C (1992), Cambridge Univ. Press: Cambridge Univ. Press New York · Zbl 0778.65003
[27] Fuentes, M. A.; Kuperman, M. N.; Kenkre, V. M., J. Phys. Chem. B, 108, 10505 (2004)
[28] Infeld, E.; Rowlands, G., Nonlinear Waves, Solitons and Chaos (1990), Cambrigde Univ. Press: Cambrigde Univ. Press New York · Zbl 0726.76018
[29] Majora, A. M.; Ciesla, M.; Longa, L.; Oliveira, F. A., Phys. Rev. E, 63, 061801 (2001)
[30] Odell, J. A.; Taylor, M. A., Biopolymers, 34, 1483 (1994)
[31] Morgado, R.; Ciesla, M.; Longa, L.; Oliveira, F. A., Europhys. Lett., 79, 10002 (2007)
[32] Longa L, L.; Curado, E. M.F.; Oliveira, F. A., Phys. Rev. E, 54, R2201 (1996)
[33] Ciesla, M.; Dias, S. P.; Longa, L.; Oliveira, F. A., Phys. Rev. E, 63, 065202 (2001)
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