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On limit systems for some population models with cross-diffusion
1. | Department of Communication Engineering and Informatics, The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu-shi, Tokyo 182-8585, Japan |
2. | Department of Applied Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan |
References:
[1] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340. |
[2] |
E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729-743.
doi: 10.1090/S0002-9947-1984-0743741-4. |
[3] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin-Heidelberg, 1998. |
[4] |
C. Gui and Y. Lou, Uniqueness and nonuniqueness of coexistence states in the Lotka-Volterra competition model, Comm. Pure. Appl. Math., 47 (1994), 1571-1594. |
[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] |
Y. Kan-on and E. Yanagida, Existence of non-constant stable equilibria in competition-diffusion equations, Hiroshima Math. J., 23 (1993), 193-221. |
[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] |
K. Kuto and Y. Yamada, Positive solutions for Lotka-Volterra competition systems withcross-diffusion, Applicable Anal., 89 (2010), 1037-1066. |
[9] |
C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27. |
[10] |
Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131. |
[11] |
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. |
[12] |
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: 10.3934/dcds.2004.10.435. |
[13] |
H. Matano and M. Mimura, Pattern formation in competition-diffusion systems in nonconvex domains, Publ. RIMS. Kyoto Univ., 19 (1983), 1049-1079.
doi: 10.2977/prims/1195182020. |
[14] |
M. Mimura, Stationary pattern of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. J., 11 (1981), 621-635. |
[15] |
M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64. |
[16] |
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. |
[17] |
W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. |
[18] |
W.-M. Ni, Qualitative properties of solutions to elliptic systems, in "Stationary Partial Differential Equations. Vol. 1" (eds. M. Chipot and P. Quittner), Handb. Differ. Equ., North-Holland, Amsterdam, (2004), 157-233. |
[19] |
A. Okubo and S. A. Levin, "Diffusion and Ecological Problems: Modern Perspectives," Second edition, Interdisciplinary Applied Mathematics, Vol. 14, Springer-Verlag, New York, 2001. |
[20] |
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Springer-Verlag, New York, 1984. |
[21] |
W. H. Ruan, Positive steady-state solutions of a competing reaction-diffusion system with large cross-diffusion coefficients, J. Math. Anal. Appl., 197 (1996), 558-578.
doi: 10.1006/jmaa.1996.0039. |
[22] |
K. Ryu and I. Ahn, Positive steady-states for two interacting species models with linear self-cross diffusions, Discrete Contin. Dynam. Systems, 9 (2003), 1049-1061. |
[23] |
K. Ryu and I. Ahn, Coexistence theorem of steady states for nonlinear self-cross diffusion systems with competitive dynamics, J. Math. Anal. Appl., 283 (2003), 46-65.
doi: 10.1016/S0022-247X(03)00162-8. |
[24] |
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. |
[25] |
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. |
[26] |
Y. Yamada, Positive solutions for Lotka-Volterra systems with cross-diffusion, in "Handbook of Differential Equations: Stationary Partial Differential Equations. Vol. VI" (ed. M. Chipot), Elsevier/North-Holland, Amsterdam, (2008), 411-501. |
show all references
References:
[1] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340. |
[2] |
E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729-743.
doi: 10.1090/S0002-9947-1984-0743741-4. |
[3] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin-Heidelberg, 1998. |
[4] |
C. Gui and Y. Lou, Uniqueness and nonuniqueness of coexistence states in the Lotka-Volterra competition model, Comm. Pure. Appl. Math., 47 (1994), 1571-1594. |
[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] |
Y. Kan-on and E. Yanagida, Existence of non-constant stable equilibria in competition-diffusion equations, Hiroshima Math. J., 23 (1993), 193-221. |
[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] |
K. Kuto and Y. Yamada, Positive solutions for Lotka-Volterra competition systems withcross-diffusion, Applicable Anal., 89 (2010), 1037-1066. |
[9] |
C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27. |
[10] |
Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131. |
[11] |
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. |
[12] |
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: 10.3934/dcds.2004.10.435. |
[13] |
H. Matano and M. Mimura, Pattern formation in competition-diffusion systems in nonconvex domains, Publ. RIMS. Kyoto Univ., 19 (1983), 1049-1079.
doi: 10.2977/prims/1195182020. |
[14] |
M. Mimura, Stationary pattern of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. J., 11 (1981), 621-635. |
[15] |
M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64. |
[16] |
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. |
[17] |
W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. |
[18] |
W.-M. Ni, Qualitative properties of solutions to elliptic systems, in "Stationary Partial Differential Equations. Vol. 1" (eds. M. Chipot and P. Quittner), Handb. Differ. Equ., North-Holland, Amsterdam, (2004), 157-233. |
[19] |
A. Okubo and S. A. Levin, "Diffusion and Ecological Problems: Modern Perspectives," Second edition, Interdisciplinary Applied Mathematics, Vol. 14, Springer-Verlag, New York, 2001. |
[20] |
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Springer-Verlag, New York, 1984. |
[21] |
W. H. Ruan, Positive steady-state solutions of a competing reaction-diffusion system with large cross-diffusion coefficients, J. Math. Anal. Appl., 197 (1996), 558-578.
doi: 10.1006/jmaa.1996.0039. |
[22] |
K. Ryu and I. Ahn, Positive steady-states for two interacting species models with linear self-cross diffusions, Discrete Contin. Dynam. Systems, 9 (2003), 1049-1061. |
[23] |
K. Ryu and I. Ahn, Coexistence theorem of steady states for nonlinear self-cross diffusion systems with competitive dynamics, J. Math. Anal. Appl., 283 (2003), 46-65.
doi: 10.1016/S0022-247X(03)00162-8. |
[24] |
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. |
[25] |
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. |
[26] |
Y. Yamada, Positive solutions for Lotka-Volterra systems with cross-diffusion, in "Handbook of Differential Equations: Stationary Partial Differential Equations. Vol. VI" (ed. M. Chipot), Elsevier/North-Holland, Amsterdam, (2008), 411-501. |
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