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1. | CGG, Houston, TX 77072, United States |
2. | Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada |
3. | Department of Mathematics, Tulane University, New Orleans, LA 70118, United States |
References:
[1] |
D. Blatt and H. Comins, Prey-predator models in spatially heterogeneous environments,, J. Theoretical Biology, 48 (1974), 75.
doi: 10.1016/0022-5193(74)90180-5. |
[2] |
L. Chen and A. Jüngel, Analysis of a multidimensional parabolic population model with strong cross-diffusion,, SIAM J. Math. Anal., 36 (2004), 301.
doi: 10.1137/S0036141003427798. |
[3] |
L. Chen and A. Jüngel, Analysis of a parabolic cross-diffusion population model without self-diffusion,, J. Differential Equations, 224 (2006), 39.
doi: 10.1016/j.jde.2005.08.002. |
[4] |
Y. Choi, R. Lui and Y. Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with weak cross-diffusion,, Discrete Contin. Dyn. Syst., 9 (2003), 1193.
doi: 10.3934/dcds.2003.9.1193. |
[5] |
Y. 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.
doi: 10.3934/dcds.2004.10.719. |
[6] |
P. Deuring, An initial-boundary value problem for a certain density-dependent diffusion system,, Math. Z., 194 (1987), 375.
doi: 10.1007/BF01162244. |
[7] |
P. Fife, Asymptotic states for equations of reaction and diffusion,, Bull. Amer. Math. Soc., 84 (1978), 693.
|
[8] |
G. Galiano, M. 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.
doi: 10.1007/s002110200406. |
[9] |
W. Gurney and R. Nisbet, The regulation of inhomogeneous populations,, J. Theoretical Biology, 52 (1975), 441.
doi: 10.1016/0022-5193(75)90011-9. |
[10] |
W. Gurney and R. Nisbet, A note on non-linear population transport,, J. Theoretical Biology, 56 (1976), 249.
doi: 10.1016/S0022-5193(76)80056-2. |
[11] |
G. Hardin, The competitive exclusion principle,, Science, 131 (1960), 1292.
doi: 10.1126/science.131.3409.1292. |
[12] |
J. Jackson and L. Segel, Dissipative structure: An explanation and an ecological example,, J. Theoretical Biology, 37 (1972), 545.
doi: 10.1016/0022-5193(72)90090-2. |
[13] |
E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theoretical Biology, 26 (1970), 399.
doi: 10.1016/0022-5193(70)90092-5. |
[14] |
J. Kim, Smooth solutions to a quasilinear system of diffusion equations for a certain population model,, Nonlinear Analysis, 8 (1984), 1121.
doi: 10.1016/0362-546X(84)90115-9. |
[15] |
D. Le, Global existence for a class of strongly coupled parabolic systems,, Ann. Mat. Pura Appl., 185 (2006), 133.
doi: 10.1007/s10231-004-0131-7. |
[16] |
D. Le and T. Nguyen, Global existence for a class of triangular parabolic systems on domains of arbitrary dimension,, Proc. Amer. Math. Soc., 133 (2005), 1985.
doi: 10.1090/S0002-9939-05-07867-6. |
[17] |
D. Le, L. Nguyen and T. Nguyen, Shigesada-Kawasaki-Teramoto model on higher dimensional domains,, Electron. J. Differential Equations, (2003).
|
[18] |
S. Levin, Dispersion and Population Interactions,, American Naturalist, 108 (1974), 207. Google Scholar |
[19] |
S. Levin, Some mathematical questions in biology - VII,, Lectures on Mathematics in the Life Sciences, 8 (1976).
|
[20] |
S. Levin, Studies in mathematical biology. Part II. Populations and communities,, MAA Studies in Mathematics, 16 (1978).
|
[21] |
Y. Li and C. Zhao, Global existence of solutions to a cross-diffusion system in higher dimensional domains,, Discrete Contin. Dyn. Syst., 12 (2005), 185.
|
[22] |
Y. Lou, S. Martinez and W. Ni, On $3\times 3$ Lotka-Volterra competition systems with cross-diffusion,, Discrete Contin. Dyn. Syst., 6 (2000), 175.
|
[23] |
Y. Lou and W. Ni, Diffusion, self-diffusion and cross-diffusion,, J. Differential Equations, 131 (1996), 79.
doi: 10.1006/jdeq.1996.0157. |
[24] |
Y. Lou, W. Ni and Y. Wu, On the global existence of a cross-diffusion system,, Discrete Contin. Dyn. Syst., 4 (1998), 193.
doi: 10.3934/dcds.1998.4.193. |
[25] |
M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations,, J. Math. Biol., 9 (1980), 49.
doi: 10.1007/BF00276035. |
[26] |
M. Morisita, Habitat preference and evaluation of environment of an animal. Experimental studies on the population density of an ant-lion, Glenuroides japonicus M'L. (I),, Physiol. Ecol. Japan, 5 (1952), 1. Google Scholar |
[27] |
A. Okubo, "Ecology and Diffusion,", Tokyo: Tsukiji Shokan, (1975). Google Scholar |
[28] |
M. Pozio and A. Tesei, Global existence of solutions for a strongly coupled quasilinear parabolic system,, Nonlinear Anal., 14 (1990), 657.
doi: 10.1016/0362-546X(90)90043-G. |
[29] |
R. Redlinger, Existence of the global attractor for a strongly coupled parabolic system arising in population dynamics,, J. Differential Equations, 118 (1995), 219.
doi: 10.1006/jdeq.1995.1073. |
[30] |
G. Rosen, Effects of diffusion on the stability of the equilibrium in multi-species ecological systems,, Bull. Math. Biol., 39 (1977), 373.
|
[31] |
W. Ruan, Positive steady-state solutions of a competing reaction-diffusion system with large cross-diffusion coefficients,, J. Math. Anal. Appl., 197 (1996), 558.
doi: 10.1006/jmaa.1996.0039. |
[32] |
K. Ryu and I. Ahn, Positive steady-states for two interacting species models with linear self-cross diffusions,, Discrete Contin. Dyn. Syst., 9 (2003), 1049.
doi: 10.3934/dcds.2003.9.1049. |
[33] |
N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species,, J. Theo. Biol., 79 (1979), 83.
doi: 10.1016/0022-5193(79)90258-3. |
[34] |
S. Shim, Uniform boundedness and convergence of solutions to cross-diffusion systems,, J. Differential Equations, 185 (2002), 281.
doi: 10.1006/jdeq.2002.4169. |
[35] |
P. Tuoc, Global existence of solutions to Shigesada-Kawasaki-Teramoto cross-diffusion systems on domains of arbitrary dimensions,, Proc. Amer. Math. Soc., 135 (2007), 3933.
doi: 10.1090/S0002-9939-07-08978-2. |
[36] |
P. Tuoc, On global existence of solutions to a cross-diffusion system,, J. Math. Anal. Appl., 343 (2008), 826.
doi: 10.1016/j.jmaa.2008.01.089. |
[37] |
A. Turing, The Chemical Basis of Morphogenesis,, Phil. Transact. Royal Soc. B, 237 (1952), 37.
doi: 10.1098/rstb.1952.0012. |
[38] |
Y. Wu, Qualitative studies of solutions for some cross-diffusion systems,, China-Japan Symposium on Reaction-Diffusion Equations and their Applications and Computational Aspects (Shanghai, (1997), 177.
|
[39] |
A. Yagi, Global solution to some quasilinear parabolic system in population dynamics,, Nonlinear Analysis, 21 (1993), 603.
doi: 10.1016/0362-546X(93)90004-C. |
show all references
References:
[1] |
D. Blatt and H. Comins, Prey-predator models in spatially heterogeneous environments,, J. Theoretical Biology, 48 (1974), 75.
doi: 10.1016/0022-5193(74)90180-5. |
[2] |
L. Chen and A. Jüngel, Analysis of a multidimensional parabolic population model with strong cross-diffusion,, SIAM J. Math. Anal., 36 (2004), 301.
doi: 10.1137/S0036141003427798. |
[3] |
L. Chen and A. Jüngel, Analysis of a parabolic cross-diffusion population model without self-diffusion,, J. Differential Equations, 224 (2006), 39.
doi: 10.1016/j.jde.2005.08.002. |
[4] |
Y. Choi, R. Lui and Y. Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with weak cross-diffusion,, Discrete Contin. Dyn. Syst., 9 (2003), 1193.
doi: 10.3934/dcds.2003.9.1193. |
[5] |
Y. 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.
doi: 10.3934/dcds.2004.10.719. |
[6] |
P. Deuring, An initial-boundary value problem for a certain density-dependent diffusion system,, Math. Z., 194 (1987), 375.
doi: 10.1007/BF01162244. |
[7] |
P. Fife, Asymptotic states for equations of reaction and diffusion,, Bull. Amer. Math. Soc., 84 (1978), 693.
|
[8] |
G. Galiano, M. 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.
doi: 10.1007/s002110200406. |
[9] |
W. Gurney and R. Nisbet, The regulation of inhomogeneous populations,, J. Theoretical Biology, 52 (1975), 441.
doi: 10.1016/0022-5193(75)90011-9. |
[10] |
W. Gurney and R. Nisbet, A note on non-linear population transport,, J. Theoretical Biology, 56 (1976), 249.
doi: 10.1016/S0022-5193(76)80056-2. |
[11] |
G. Hardin, The competitive exclusion principle,, Science, 131 (1960), 1292.
doi: 10.1126/science.131.3409.1292. |
[12] |
J. Jackson and L. Segel, Dissipative structure: An explanation and an ecological example,, J. Theoretical Biology, 37 (1972), 545.
doi: 10.1016/0022-5193(72)90090-2. |
[13] |
E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theoretical Biology, 26 (1970), 399.
doi: 10.1016/0022-5193(70)90092-5. |
[14] |
J. Kim, Smooth solutions to a quasilinear system of diffusion equations for a certain population model,, Nonlinear Analysis, 8 (1984), 1121.
doi: 10.1016/0362-546X(84)90115-9. |
[15] |
D. Le, Global existence for a class of strongly coupled parabolic systems,, Ann. Mat. Pura Appl., 185 (2006), 133.
doi: 10.1007/s10231-004-0131-7. |
[16] |
D. Le and T. Nguyen, Global existence for a class of triangular parabolic systems on domains of arbitrary dimension,, Proc. Amer. Math. Soc., 133 (2005), 1985.
doi: 10.1090/S0002-9939-05-07867-6. |
[17] |
D. Le, L. Nguyen and T. Nguyen, Shigesada-Kawasaki-Teramoto model on higher dimensional domains,, Electron. J. Differential Equations, (2003).
|
[18] |
S. Levin, Dispersion and Population Interactions,, American Naturalist, 108 (1974), 207. Google Scholar |
[19] |
S. Levin, Some mathematical questions in biology - VII,, Lectures on Mathematics in the Life Sciences, 8 (1976).
|
[20] |
S. Levin, Studies in mathematical biology. Part II. Populations and communities,, MAA Studies in Mathematics, 16 (1978).
|
[21] |
Y. Li and C. Zhao, Global existence of solutions to a cross-diffusion system in higher dimensional domains,, Discrete Contin. Dyn. Syst., 12 (2005), 185.
|
[22] |
Y. Lou, S. Martinez and W. Ni, On $3\times 3$ Lotka-Volterra competition systems with cross-diffusion,, Discrete Contin. Dyn. Syst., 6 (2000), 175.
|
[23] |
Y. Lou and W. Ni, Diffusion, self-diffusion and cross-diffusion,, J. Differential Equations, 131 (1996), 79.
doi: 10.1006/jdeq.1996.0157. |
[24] |
Y. Lou, W. Ni and Y. Wu, On the global existence of a cross-diffusion system,, Discrete Contin. Dyn. Syst., 4 (1998), 193.
doi: 10.3934/dcds.1998.4.193. |
[25] |
M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations,, J. Math. Biol., 9 (1980), 49.
doi: 10.1007/BF00276035. |
[26] |
M. Morisita, Habitat preference and evaluation of environment of an animal. Experimental studies on the population density of an ant-lion, Glenuroides japonicus M'L. (I),, Physiol. Ecol. Japan, 5 (1952), 1. Google Scholar |
[27] |
A. Okubo, "Ecology and Diffusion,", Tokyo: Tsukiji Shokan, (1975). Google Scholar |
[28] |
M. Pozio and A. Tesei, Global existence of solutions for a strongly coupled quasilinear parabolic system,, Nonlinear Anal., 14 (1990), 657.
doi: 10.1016/0362-546X(90)90043-G. |
[29] |
R. Redlinger, Existence of the global attractor for a strongly coupled parabolic system arising in population dynamics,, J. Differential Equations, 118 (1995), 219.
doi: 10.1006/jdeq.1995.1073. |
[30] |
G. Rosen, Effects of diffusion on the stability of the equilibrium in multi-species ecological systems,, Bull. Math. Biol., 39 (1977), 373.
|
[31] |
W. Ruan, Positive steady-state solutions of a competing reaction-diffusion system with large cross-diffusion coefficients,, J. Math. Anal. Appl., 197 (1996), 558.
doi: 10.1006/jmaa.1996.0039. |
[32] |
K. Ryu and I. Ahn, Positive steady-states for two interacting species models with linear self-cross diffusions,, Discrete Contin. Dyn. Syst., 9 (2003), 1049.
doi: 10.3934/dcds.2003.9.1049. |
[33] |
N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species,, J. Theo. Biol., 79 (1979), 83.
doi: 10.1016/0022-5193(79)90258-3. |
[34] |
S. Shim, Uniform boundedness and convergence of solutions to cross-diffusion systems,, J. Differential Equations, 185 (2002), 281.
doi: 10.1006/jdeq.2002.4169. |
[35] |
P. Tuoc, Global existence of solutions to Shigesada-Kawasaki-Teramoto cross-diffusion systems on domains of arbitrary dimensions,, Proc. Amer. Math. Soc., 135 (2007), 3933.
doi: 10.1090/S0002-9939-07-08978-2. |
[36] |
P. Tuoc, On global existence of solutions to a cross-diffusion system,, J. Math. Anal. Appl., 343 (2008), 826.
doi: 10.1016/j.jmaa.2008.01.089. |
[37] |
A. Turing, The Chemical Basis of Morphogenesis,, Phil. Transact. Royal Soc. B, 237 (1952), 37.
doi: 10.1098/rstb.1952.0012. |
[38] |
Y. Wu, Qualitative studies of solutions for some cross-diffusion systems,, China-Japan Symposium on Reaction-Diffusion Equations and their Applications and Computational Aspects (Shanghai, (1997), 177.
|
[39] |
A. Yagi, Global solution to some quasilinear parabolic system in population dynamics,, Nonlinear Analysis, 21 (1993), 603.
doi: 10.1016/0362-546X(93)90004-C. |
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