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A relation between cross-diffusion and reaction-diffusion
1. | Graduate School of Science and Engineering for Research, University of Toyama, 3190 Gofuku, Toyama 930-8555, Japan |
References:
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L. Chen and A. Jüngel, Analysis of a multidimensional parabolic population model with strong cross-diffusion, SIAM J. Math. Anal., 36 (2006), 301-322.
doi: 10.1137/S0036141003427798. |
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L. Chen and A. Jüngel, Analysis of a parabolic cross-diffusion population model without self-diffusion, J. Differential Equations, 224 (2006), 39-59.
doi: 10.1016/j.jde.2005.08.002. |
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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. |
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M. Iida and H. Ninomiya, A reaction-diffusion approximation to a cross-diffusion system, in "Recent Advances on Elliptic and Parabolic Issues" (eds. M. Chipot and H. Ninomiya), World Scientific, (2006), 145-164.
doi: 10.1142/9789812774170_0007. |
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T. Kadota and K. Kuto, Positive steady states for a prey-predator model with some nonlinear diffusion terms, J. Math. Anal. Appl., 323 (2006), 1387-1401.
doi: 10.1016/j.jmaa.2005.11.065. |
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E. H. Kerner, Further considerations on the statistical mechanics of biological associations, Bull. Math. Biophys., 21 (1959), 217-255.
doi: 10.1007/BF02476361. |
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H. Murakawa, Reaction-diffusion system approximation to degenerate parabolic systems, Nonlinearity, 20 (2007), 2319-2332.
doi: 10.1088/0951-7715/20/10/003. |
[9] |
H. Murakawa, A solution of nonlinear diffusion problems by semilinear reaction-diffusion systems, Kybernetika, 45 (2009), 580-590. |
[10] |
H. Murakawa, Discrete-time approximation to nonlinear degenerate parabolic problems using a semilinear reaction-diffusion system,, preprint., ().
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[11] |
A. Okubo and S. A. Levin, "Diffusion and Ecological Problems: Modern Perspectives. Second Edition," Interdisciplinary Applied Mathematics, 14, Springer-Verlag, New York, 2001. |
[12] |
P. Y. H. Pang and M. X. Wang, Strategy and stationary pattern in a three-species predator-prey model, J. Differential Equations, 200 (2004), 245-273.
doi: 10.1016/j.jde.2004.01.004. |
[13] |
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. |
[14] |
R. Temam, "Navier-Stokes Equation Theory and Numerical Analysis," AMS Chelsea Publishing, Providence, RI., 2001. |
show all references
References:
[1] |
L. Chen and A. Jüngel, Analysis of a multidimensional parabolic population model with strong cross-diffusion, SIAM J. Math. Anal., 36 (2006), 301-322.
doi: 10.1137/S0036141003427798. |
[2] |
L. Chen and A. Jüngel, Analysis of a parabolic cross-diffusion population model without self-diffusion, J. Differential Equations, 224 (2006), 39-59.
doi: 10.1016/j.jde.2005.08.002. |
[3] |
M. E. Gurtin, Some mathematical models for population dynamics that lead to segregation, Quart. Appl. Math, 32 (1974), 1-9. |
[4] |
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. |
[5] |
M. Iida and H. Ninomiya, A reaction-diffusion approximation to a cross-diffusion system, in "Recent Advances on Elliptic and Parabolic Issues" (eds. M. Chipot and H. Ninomiya), World Scientific, (2006), 145-164.
doi: 10.1142/9789812774170_0007. |
[6] |
T. Kadota and K. Kuto, Positive steady states for a prey-predator model with some nonlinear diffusion terms, J. Math. Anal. Appl., 323 (2006), 1387-1401.
doi: 10.1016/j.jmaa.2005.11.065. |
[7] |
E. H. Kerner, Further considerations on the statistical mechanics of biological associations, Bull. Math. Biophys., 21 (1959), 217-255.
doi: 10.1007/BF02476361. |
[8] |
H. Murakawa, Reaction-diffusion system approximation to degenerate parabolic systems, Nonlinearity, 20 (2007), 2319-2332.
doi: 10.1088/0951-7715/20/10/003. |
[9] |
H. Murakawa, A solution of nonlinear diffusion problems by semilinear reaction-diffusion systems, Kybernetika, 45 (2009), 580-590. |
[10] |
H. Murakawa, Discrete-time approximation to nonlinear degenerate parabolic problems using a semilinear reaction-diffusion system,, preprint., ().
|
[11] |
A. Okubo and S. A. Levin, "Diffusion and Ecological Problems: Modern Perspectives. Second Edition," Interdisciplinary Applied Mathematics, 14, Springer-Verlag, New York, 2001. |
[12] |
P. Y. H. Pang and M. X. Wang, Strategy and stationary pattern in a three-species predator-prey model, J. Differential Equations, 200 (2004), 245-273.
doi: 10.1016/j.jde.2004.01.004. |
[13] |
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. |
[14] |
R. Temam, "Navier-Stokes Equation Theory and Numerical Analysis," AMS Chelsea Publishing, Providence, RI., 2001. |
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