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The existence and stability of nontrivial steady states for S-K-T competition model with cross diffusion
1. | Center for PDE, East China Normal University, Shanghai, 200241, China |
2. | College of Mathematical Sciences, Capital Normal University, Beijing 100048, China |
3. | Department of Basic Courses, Beijing Union University, Beijing 100101, 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: 10.3934/dcds.2004.10.719. |
[2] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. |
[3] |
M. Iida, M. Mimura and H. Ninomiya, Diffusion, cross diffusion and competitive interaction, J. Math. Biol., 53 (2006), 617-641.
doi: 10.1007/s00285-006-0013-2. |
[4] |
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: 10.1216/rmjm/1194275939. |
[5] |
Y. Kan-on, Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics, Hiroshima Math. J., 23 (1993), 509-536. |
[6] |
H. Kielhofer, Bifurcation Theory: An Introduction with Applications to PDEs, Applied Math. Sci. Vol. 156, Springer Verlag, New York Inc. 2004.
doi: 10.1007/b97365. |
[7] |
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: 10.1016/0022-0396(85)90020-8. |
[8] |
Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross diffusion, J. Differential Equations, 131 (1996), 79-131.
doi: 10.1006/jdeq.1996.0157. |
[9] |
Y. Lou and W. M. Ni, Diffusion vs cross diffusion: An elliptic approach, J. Differential Equations, 154 (1999), 157-190.
doi: 10.1006/jdeq.1998.3559. |
[10] |
Y. Lou, W. M. Ni and Y. Wu, On the global existence of a cross diffusion system, Discrete and Continuous Dynamical Systems, 4 (1998), 193-203. |
[11] |
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. |
[12] |
H. Matano and M. Mimura, Pattern formation in competition-diffusion systems in nonconvex domains, Publ. RIMS. Kyoto Univ., 19 (1983), 1049-1079. |
[13] |
M. Mimura, Stationary pattern of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. J., 11 (1981), 621-635. |
[14] |
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. |
[15] |
W. M. Ni, Qualitative properties of solutions to elliptic problems, Stationary Partial Differential Equations, Hand. Differ. Equ., North-Holland, Amesterdam, 1 (2004), 157-233.
doi: 10.1016/S1874-5733(04)80005-6. |
[16] |
N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83-99.
doi: 10.1016/0022-5193(79)90258-3. |
[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, The instability of spiky steady states for a competing species model with cross diffusion, J. Differential Equations, 213 (2005), 289-340.
doi: 10.1016/j.jde.2004.08.015. |
[19] |
Y. Wu and Q. Xu, The Existence and structure of large spiky steady states for S-K-T competition system with cross diffusion, Discrete Contin. Dyn. Syst., 29 (2011), 367-385.
doi: 10.3934/dcds.2011.29.367. |
[20] |
Y. Wu and Y. Zhao, The existence and stability of traveling waves with transition layers for the S-K-T competition model with cross diffusion, Science China, 53 (2010), 1161-1184.
doi: 10.1007/s11425-010-0141-4. |
[21] |
Y. Yamada, Positive solutions for Lotka-Volterra systems with cross diffusion, Stationary Partial Differential Equations, Hand. Differ. Equ., Elsevier, Amsterdam, VI (2008), 411-501.
doi: 10.1016/S1874-5733(08)80023-X. |
[22] |
Y. Yamada, Global solutions for the Shigesada-Kawasaki-Teramoto model with cross diffusion, Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, 282-299, World Sci. Publ. Hackensack, NJ, (2009).
doi: 10.1142/9789812834744_0013. |
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: 10.3934/dcds.2004.10.719. |
[2] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. |
[3] |
M. Iida, M. Mimura and H. Ninomiya, Diffusion, cross diffusion and competitive interaction, J. Math. Biol., 53 (2006), 617-641.
doi: 10.1007/s00285-006-0013-2. |
[4] |
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: 10.1216/rmjm/1194275939. |
[5] |
Y. Kan-on, Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics, Hiroshima Math. J., 23 (1993), 509-536. |
[6] |
H. Kielhofer, Bifurcation Theory: An Introduction with Applications to PDEs, Applied Math. Sci. Vol. 156, Springer Verlag, New York Inc. 2004.
doi: 10.1007/b97365. |
[7] |
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: 10.1016/0022-0396(85)90020-8. |
[8] |
Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross diffusion, J. Differential Equations, 131 (1996), 79-131.
doi: 10.1006/jdeq.1996.0157. |
[9] |
Y. Lou and W. M. Ni, Diffusion vs cross diffusion: An elliptic approach, J. Differential Equations, 154 (1999), 157-190.
doi: 10.1006/jdeq.1998.3559. |
[10] |
Y. Lou, W. M. Ni and Y. Wu, On the global existence of a cross diffusion system, Discrete and Continuous Dynamical Systems, 4 (1998), 193-203. |
[11] |
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. |
[12] |
H. Matano and M. Mimura, Pattern formation in competition-diffusion systems in nonconvex domains, Publ. RIMS. Kyoto Univ., 19 (1983), 1049-1079. |
[13] |
M. Mimura, Stationary pattern of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. J., 11 (1981), 621-635. |
[14] |
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. |
[15] |
W. M. Ni, Qualitative properties of solutions to elliptic problems, Stationary Partial Differential Equations, Hand. Differ. Equ., North-Holland, Amesterdam, 1 (2004), 157-233.
doi: 10.1016/S1874-5733(04)80005-6. |
[16] |
N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83-99.
doi: 10.1016/0022-5193(79)90258-3. |
[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, The instability of spiky steady states for a competing species model with cross diffusion, J. Differential Equations, 213 (2005), 289-340.
doi: 10.1016/j.jde.2004.08.015. |
[19] |
Y. Wu and Q. Xu, The Existence and structure of large spiky steady states for S-K-T competition system with cross diffusion, Discrete Contin. Dyn. Syst., 29 (2011), 367-385.
doi: 10.3934/dcds.2011.29.367. |
[20] |
Y. Wu and Y. Zhao, The existence and stability of traveling waves with transition layers for the S-K-T competition model with cross diffusion, Science China, 53 (2010), 1161-1184.
doi: 10.1007/s11425-010-0141-4. |
[21] |
Y. Yamada, Positive solutions for Lotka-Volterra systems with cross diffusion, Stationary Partial Differential Equations, Hand. Differ. Equ., Elsevier, Amsterdam, VI (2008), 411-501.
doi: 10.1016/S1874-5733(08)80023-X. |
[22] |
Y. Yamada, Global solutions for the Shigesada-Kawasaki-Teramoto model with cross diffusion, Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, 282-299, World Sci. Publ. Hackensack, NJ, (2009).
doi: 10.1142/9789812834744_0013. |
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