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Finite element approximation of a population spatial adaptation model
1. | Dpto. de Matemáticas, Universidad de Oviedo, c/ Calvo Sotelo, 33007-Oviedo, Spain, Spain |
References:
[1] |
J. W. Barrett and J. F. Blowey, Finite element approximation of a nonlinear cross-diffusion population model, Numer. Math., 98 (2004), 195-221.
doi: 10.1007/s00211-004-0540-y. |
[2] |
L. Chen and A. Jüngel, Analysis of a multidimensional parabolic population model with strong cross-diffusion, SIAM J. Math. Anal., 36 (2004), 301-322.
doi: 10.1137/S0036141003427798. |
[3] |
L. Chen and A. Jüngel, Analysis of a parabolic cross-diffusion semiconductor model with electron-hole scattering, Commun. Part. Diff. Eqs., 32 (2007), 127-148.
doi: 10.1080/03605300601088815. |
[4] |
P. Deuring, An initial-boundary value problem for a certain density-dependent diffusion system, Math. Z., 194 (1987), 375-396.
doi: 10.1007/BF01162244. |
[5] |
H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114.
doi: 10.1002/mana.19981950106. |
[6] |
G. Galiano, Modeling spatial adaptation of populations by a time non-local convection cross-diffusion evolution problem, Appl. Math. Comput., 218 (2011), 4587-4594.
doi: 10.1016/j.amc.2011.10.041. |
[7] |
G. Galiano, On a cross-diffusion population model deduced from mutation and splitting of a single species, Comput. Math. Appl., 64 (2012), 1927-1936.
doi: 10.1016/j.camwa.2012.03.045. |
[8] |
G. Galiano, M. L. Garzón and A. Jüngel, Analysis and numerical solution of a nonlinear cross-diffusion system arising in population dynamics, RACSAM Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A. Mat., 95 (2001), 281-295. |
[9] |
G. Galiano, M. L. Garzón and A. Jüngel, Semi-discretization in time and numerical convergence of solutions of a nonlinear cross-diffusion population model, Numer. Math., 93 (2003), 655-673.
doi: 10.1007/s002110200406. |
[10] |
G. Galiano and A. Jüngel, Global existence of solutions for a strongly coupled population system, Banach Center Publ., 63 (2004), 209-216. |
[11] |
G. Galiano, A. Jüngel and J. Velasco, A parabolic cross-diffusion system for granular materials, SIAM J. Math. Anal., 35 (2003), 561-578.
doi: 10.1137/S0036141002409386. |
[12] |
G. Galiano and J. Velasco, Competing through altering the environment: A cross-diffusion population model coupled to transport Darcy flow equations, Nonlinear Anal., 12 (2011), 2826-2838.
doi: 10.1016/j.nonrwa.2011.04.009. |
[13] |
G. Gambino, M. C. Lombardo and M. Sammartino, A velocity-diffusion method for a Lotka-Volterra system with nonlinear cross and self-diffusion, Appl. Numer. Math., 59 (2009), 1059-1074.
doi: 10.1016/j.apnum.2008.05.002. |
[14] |
J. U. Kim, Smooth solutions to a quasi-linear system of diffusion equations for a certain population model, Nonlinear Analysis TMA, 8 (1984), 1121-1144.
doi: 10.1016/0362-546X(84)90115-9. |
[15] |
Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Diff. Eqs., 131 (1996), 79-131.
doi: 10.1006/jdeq.1996.0157. |
[16] |
Y. Lou, W.-M. Ni and Y. Wu, The global existence of solutions for a cross-diffusion system, Adv. Math., Beijing, 25 (1996), 283-284. |
[17] |
M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64.
doi: 10.1007/BF00276035. |
[18] |
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. |
[19] |
A. Yagi, Global solution to some quasilinear parabolic system in population dynamics, Nonlinear Analysis TMA, 21 (1993), 603-630.
doi: 10.1016/0362-546X(93)90004-C. |
show all references
References:
[1] |
J. W. Barrett and J. F. Blowey, Finite element approximation of a nonlinear cross-diffusion population model, Numer. Math., 98 (2004), 195-221.
doi: 10.1007/s00211-004-0540-y. |
[2] |
L. Chen and A. Jüngel, Analysis of a multidimensional parabolic population model with strong cross-diffusion, SIAM J. Math. Anal., 36 (2004), 301-322.
doi: 10.1137/S0036141003427798. |
[3] |
L. Chen and A. Jüngel, Analysis of a parabolic cross-diffusion semiconductor model with electron-hole scattering, Commun. Part. Diff. Eqs., 32 (2007), 127-148.
doi: 10.1080/03605300601088815. |
[4] |
P. Deuring, An initial-boundary value problem for a certain density-dependent diffusion system, Math. Z., 194 (1987), 375-396.
doi: 10.1007/BF01162244. |
[5] |
H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114.
doi: 10.1002/mana.19981950106. |
[6] |
G. Galiano, Modeling spatial adaptation of populations by a time non-local convection cross-diffusion evolution problem, Appl. Math. Comput., 218 (2011), 4587-4594.
doi: 10.1016/j.amc.2011.10.041. |
[7] |
G. Galiano, On a cross-diffusion population model deduced from mutation and splitting of a single species, Comput. Math. Appl., 64 (2012), 1927-1936.
doi: 10.1016/j.camwa.2012.03.045. |
[8] |
G. Galiano, M. L. Garzón and A. Jüngel, Analysis and numerical solution of a nonlinear cross-diffusion system arising in population dynamics, RACSAM Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A. Mat., 95 (2001), 281-295. |
[9] |
G. Galiano, M. L. Garzón and A. Jüngel, Semi-discretization in time and numerical convergence of solutions of a nonlinear cross-diffusion population model, Numer. Math., 93 (2003), 655-673.
doi: 10.1007/s002110200406. |
[10] |
G. Galiano and A. Jüngel, Global existence of solutions for a strongly coupled population system, Banach Center Publ., 63 (2004), 209-216. |
[11] |
G. Galiano, A. Jüngel and J. Velasco, A parabolic cross-diffusion system for granular materials, SIAM J. Math. Anal., 35 (2003), 561-578.
doi: 10.1137/S0036141002409386. |
[12] |
G. Galiano and J. Velasco, Competing through altering the environment: A cross-diffusion population model coupled to transport Darcy flow equations, Nonlinear Anal., 12 (2011), 2826-2838.
doi: 10.1016/j.nonrwa.2011.04.009. |
[13] |
G. Gambino, M. C. Lombardo and M. Sammartino, A velocity-diffusion method for a Lotka-Volterra system with nonlinear cross and self-diffusion, Appl. Numer. Math., 59 (2009), 1059-1074.
doi: 10.1016/j.apnum.2008.05.002. |
[14] |
J. U. Kim, Smooth solutions to a quasi-linear system of diffusion equations for a certain population model, Nonlinear Analysis TMA, 8 (1984), 1121-1144.
doi: 10.1016/0362-546X(84)90115-9. |
[15] |
Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Diff. Eqs., 131 (1996), 79-131.
doi: 10.1006/jdeq.1996.0157. |
[16] |
Y. Lou, W.-M. Ni and Y. Wu, The global existence of solutions for a cross-diffusion system, Adv. Math., Beijing, 25 (1996), 283-284. |
[17] |
M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64.
doi: 10.1007/BF00276035. |
[18] |
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. |
[19] |
A. Yagi, Global solution to some quasilinear parabolic system in population dynamics, Nonlinear Analysis TMA, 21 (1993), 603-630.
doi: 10.1016/0362-546X(93)90004-C. |
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