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Spatial dynamics of a nonlocal and delayed population model in a periodic habitat
The existence and structure of large spiky steady states for S-K-T competition systems with cross-diffusion
1. | Department of Mathematics, Capital Normal University, Beijing 100048, China, China |
References:
[1] |
Y. S. Choi, R. Lui and Y. Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 719-730.
doi: doi:10.3934/dcds.2004.10.719. |
[2] |
H. Kuiper and L. Dung, Global attractors for cross diffusion systems on domains of arbitrary dimension, Rocky Mountain J. Math., 37 (2007), 1645-1668.
doi: doi:10.1216/rmjm/1194275939. |
[3] |
Y. Kan-on, Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics, Hiroshima Math. J., 23 (1993), 509-536. |
[4] |
K. Kishimoto and H. F. Weinberger, The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains, J. Differential Equations, 58 (1985), 15-21.
doi: doi:10.1016/0022-0396(85)90020-8. |
[5] |
T. Kolokolnikov, M. Ward and J. Wei, The existence and stability of spike equilibria in the one-dimensional Gray-Scott model, the pulse-splitting regime, Phys. D, 202 (2005), 258-293.
doi: doi:10.1016/j.physd.2005.02.009. |
[6] |
C.-S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.
doi: doi:10.1016/0022-0396(88)90147-7. |
[7] |
Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.
doi: doi:10.1006/jdeq.1996.0157. |
[8] |
Y. Lou and W. M. Ni, Diffusion vs cross-diffusion: An elliptic approach, J. Differential Equations, 154 (1999), 157-190.
doi: doi:10.1006/jdeq.1998.3559. |
[9] |
Y. Lou, W. M. Ni and Y. Wu, On the global existence of a cross-diffusion system, Discrete Contin. Dyn. Syst., 4 (1998), 193-203.
doi: doi:10.3934/dcds.1998.4.193. |
[10] |
Y. Lou, W. M. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion, Discrete Contin. Dyn. Syst, 10 (2004), 435-458.
doi: doi:10.3934/dcds.2004.10.435. |
[11] |
M. Mimura, Y. Nishiura, A. Tesei and T. Tsujikawa, Coexistence problem for two competing species models with density-dependent diffusion, Hiroshima Math. J., 14 (1984), 425-449. |
[12] |
W. M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. |
[13] |
W. M. Ni, I. Takagi and E. Yanagida, Stability analysis of point condensation solutions to a reaction-diffusion system proposed by Gierer and Meinhardt, Tohoku Math. J., to appear. |
[14] |
N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83-99.
doi: doi:10.1016/0022-5193(79)90258-3. |
[15] |
B. D. Sleeman, M. J. Ward and J. Wei, The existence and stability of spike patterns in a chemotaxis model, SIAM J. Appl. Math., 65 (2005), 790-817.
doi: doi:10.1137/S0036139902415117. |
[16] |
J.Wei, Existence and stability of spikes for the Gierer-Meinhardt systems, "Handbook of Differential Equations: Stationary Partial Differential Equations," Elsevier/ North-Holland, Amsterdam, V (2008), 487-585. |
[17] |
Y. Wu, Existence of stationary solutions with transition layers for a class of cross-diffusion systems, Proc. of Royal Soc. Edinburg, Sect. A, 132 (2002), 1493-1511. |
[18] |
Y. Wu and X. Zhao, The existence and stability of traveling waves with transition layers for some singular cross-diffuion systems, Phys. D, 200 (2005), 325-358.
doi: doi:10.1016/j.physd.2004.11.010. |
[19] |
Y. Wu and Y. Zhao, The existence and stability of traveling waves with transition layers for S-K-T competition model with cross-diffusion, Sci. China Math., 53 (2010), 1161-1184.
doi: doi:10.1007/s11425-010-0141-4. |
show all references
References:
[1] |
Y. S. Choi, R. Lui and Y. Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 719-730.
doi: doi:10.3934/dcds.2004.10.719. |
[2] |
H. Kuiper and L. Dung, Global attractors for cross diffusion systems on domains of arbitrary dimension, Rocky Mountain J. Math., 37 (2007), 1645-1668.
doi: doi:10.1216/rmjm/1194275939. |
[3] |
Y. Kan-on, Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics, Hiroshima Math. J., 23 (1993), 509-536. |
[4] |
K. Kishimoto and H. F. Weinberger, The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains, J. Differential Equations, 58 (1985), 15-21.
doi: doi:10.1016/0022-0396(85)90020-8. |
[5] |
T. Kolokolnikov, M. Ward and J. Wei, The existence and stability of spike equilibria in the one-dimensional Gray-Scott model, the pulse-splitting regime, Phys. D, 202 (2005), 258-293.
doi: doi:10.1016/j.physd.2005.02.009. |
[6] |
C.-S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.
doi: doi:10.1016/0022-0396(88)90147-7. |
[7] |
Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.
doi: doi:10.1006/jdeq.1996.0157. |
[8] |
Y. Lou and W. M. Ni, Diffusion vs cross-diffusion: An elliptic approach, J. Differential Equations, 154 (1999), 157-190.
doi: doi:10.1006/jdeq.1998.3559. |
[9] |
Y. Lou, W. M. Ni and Y. Wu, On the global existence of a cross-diffusion system, Discrete Contin. Dyn. Syst., 4 (1998), 193-203.
doi: doi:10.3934/dcds.1998.4.193. |
[10] |
Y. Lou, W. M. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion, Discrete Contin. Dyn. Syst, 10 (2004), 435-458.
doi: doi:10.3934/dcds.2004.10.435. |
[11] |
M. Mimura, Y. Nishiura, A. Tesei and T. Tsujikawa, Coexistence problem for two competing species models with density-dependent diffusion, Hiroshima Math. J., 14 (1984), 425-449. |
[12] |
W. M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. |
[13] |
W. M. Ni, I. Takagi and E. Yanagida, Stability analysis of point condensation solutions to a reaction-diffusion system proposed by Gierer and Meinhardt, Tohoku Math. J., to appear. |
[14] |
N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83-99.
doi: doi:10.1016/0022-5193(79)90258-3. |
[15] |
B. D. Sleeman, M. J. Ward and J. Wei, The existence and stability of spike patterns in a chemotaxis model, SIAM J. Appl. Math., 65 (2005), 790-817.
doi: doi:10.1137/S0036139902415117. |
[16] |
J.Wei, Existence and stability of spikes for the Gierer-Meinhardt systems, "Handbook of Differential Equations: Stationary Partial Differential Equations," Elsevier/ North-Holland, Amsterdam, V (2008), 487-585. |
[17] |
Y. Wu, Existence of stationary solutions with transition layers for a class of cross-diffusion systems, Proc. of Royal Soc. Edinburg, Sect. A, 132 (2002), 1493-1511. |
[18] |
Y. Wu and X. Zhao, The existence and stability of traveling waves with transition layers for some singular cross-diffuion systems, Phys. D, 200 (2005), 325-358.
doi: doi:10.1016/j.physd.2004.11.010. |
[19] |
Y. Wu and Y. Zhao, The existence and stability of traveling waves with transition layers for S-K-T competition model with cross-diffusion, Sci. China Math., 53 (2010), 1161-1184.
doi: doi:10.1007/s11425-010-0141-4. |
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