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On the steady state of a shadow system to the SKT competition model
1. | Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130, China |
References:
[1] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, Journal of Functional Analysis, 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[2] |
________, Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. |
[3] |
S. Ei, Two-timing methods with applications to heterogeneous reaction-diffusion systems, Hiroshima Math. J., 18 (1988), 127-160. |
[4] |
P. Fife, Boundary and interior transition layer phenomena for pairs of second-order differential equations, J. Math. Anal. Appl., 54 (1976), 497-521.
doi: 10.1016/0022-247X(76)90218-3. |
[5] |
J. K. Hale and K. Sakamoto, Existence and stability of transition layers, Japan J. Appl. Math., 5 (1988), 367-405.
doi: 10.1007/BF03167908. |
[6] |
M. Iida, T. Muramatsu, H. Ninomiya and E. Yanagida, Diffusion-induced extinction of a superior species in a competition system, Japan J. Indust. Appl. Math., 15 (1998), 233-252.
doi: 10.1007/BF03167402. |
[7] |
Y. Kan-on and E. Yanagida, Existence of non-constant stable equilibria in competition-diffusion equations, Hiroshima Math. J., 23 (1993), 193-221. |
[8] |
T. Kato, Functional Analysis, Springer Classics in Mathematics, 1996. |
[9] |
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. |
[10] |
________, Diffusion vs cross-diffusion: An elliptic approach, J. Differential Equations, 154 (1999), 157-190.
doi: 10.1006/jdeq.1998.3559. |
[11] |
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: 10.3934/dcds.1998.4.193. |
[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. Res. Inst. Math. Sci., 19 (1983), 1049-1079.
doi: 10.2977/prims/1195182020. |
[14] |
M. Mimura, S.-I Ei and Q. Fang, Effect of domain-shape on coexistence problems in a competition-diffusion system, J. Math. Biol., 29 (1991), 219-237.
doi: 10.1007/BF00160536. |
[15] |
M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64.
doi: 10.1007/BF00276035. |
[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] |
J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812.
doi: 10.1016/j.jde.2008.09.009. |
[18] |
N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99.
doi: 10.1016/0022-5193(79)90258-3. |
[19] |
Y. Wu and Q. Xu, The existence and structure of large spiky steady states for SKT competition systems with cross-diffusion, Discrete Contin. Dyn. Syst., 29 (2011), 367-385.
doi: 10.3934/dcds.2011.29.367. |
show all references
References:
[1] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, Journal of Functional Analysis, 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[2] |
________, Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. |
[3] |
S. Ei, Two-timing methods with applications to heterogeneous reaction-diffusion systems, Hiroshima Math. J., 18 (1988), 127-160. |
[4] |
P. Fife, Boundary and interior transition layer phenomena for pairs of second-order differential equations, J. Math. Anal. Appl., 54 (1976), 497-521.
doi: 10.1016/0022-247X(76)90218-3. |
[5] |
J. K. Hale and K. Sakamoto, Existence and stability of transition layers, Japan J. Appl. Math., 5 (1988), 367-405.
doi: 10.1007/BF03167908. |
[6] |
M. Iida, T. Muramatsu, H. Ninomiya and E. Yanagida, Diffusion-induced extinction of a superior species in a competition system, Japan J. Indust. Appl. Math., 15 (1998), 233-252.
doi: 10.1007/BF03167402. |
[7] |
Y. Kan-on and E. Yanagida, Existence of non-constant stable equilibria in competition-diffusion equations, Hiroshima Math. J., 23 (1993), 193-221. |
[8] |
T. Kato, Functional Analysis, Springer Classics in Mathematics, 1996. |
[9] |
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. |
[10] |
________, Diffusion vs cross-diffusion: An elliptic approach, J. Differential Equations, 154 (1999), 157-190.
doi: 10.1006/jdeq.1998.3559. |
[11] |
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: 10.3934/dcds.1998.4.193. |
[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. Res. Inst. Math. Sci., 19 (1983), 1049-1079.
doi: 10.2977/prims/1195182020. |
[14] |
M. Mimura, S.-I Ei and Q. Fang, Effect of domain-shape on coexistence problems in a competition-diffusion system, J. Math. Biol., 29 (1991), 219-237.
doi: 10.1007/BF00160536. |
[15] |
M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64.
doi: 10.1007/BF00276035. |
[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] |
J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812.
doi: 10.1016/j.jde.2008.09.009. |
[18] |
N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99.
doi: 10.1016/0022-5193(79)90258-3. |
[19] |
Y. Wu and Q. Xu, The existence and structure of large spiky steady states for SKT competition systems with cross-diffusion, Discrete Contin. Dyn. Syst., 29 (2011), 367-385.
doi: 10.3934/dcds.2011.29.367. |
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