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Qualitative analysis of a Lotka-Volterra competition system with advection
1. | Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130 |
2. | Department of Mathematics and Statistics, Dalhousie University, 6316 Coburg Road, Halifax, Nova Scotia, Canada B3H 4R2, Canada |
3. | Hanqing Advanced Institute of Economics and Finance, Renmin University of China, No. 59 Zhongguancun Street, Haidian District, Beijing 100872, China |
References:
[1] |
N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.
doi: 10.1080/03605307908820113. |
[2] |
H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. |
[3] |
_______, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, differential operators and nonlinear Analysis, Teubner, Stuttgart, Leipzig, 133 (1993), 9-126.
doi: 10.1007/978-3-663-11336-2_1. |
[4] |
A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a Chemotaxis Model with Saturated Chemotactic Flux, Kinet. Relat. Models, 5 (2012), 51-95.
doi: 10.3934/krm.2012.5.51. |
[5] |
Y. S. 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), 719-730.
doi: 10.3934/dcds.2004.10.719. |
[6] |
E. Conway, D. Hoff and J. Smoller, Large time behavior of solutions of systems of nonlinear reaction-diffusion equations, SIAM J. Appl. Math., 35, (1978), 1-16.
doi: 10.1137/0135001. |
[7] |
C. Cosner, Reaction-diffusion-advection models for the effects and evolution of dispersal, Discrete Contin. Dyn. Syst., 34 (2014), 1701-1745.
doi: 10.3934/dcds.2014.34.1701. |
[8] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[9] |
_______, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. and Anal., 52 (1973), 161-180. |
[10] |
P. De Mottoni and F. Rothe, Convergence to homogeneous equilibrium state for generalized Volterra-Lotka systems with diffusion, SIAM J. Appl. Math., 37 (1979), 648-663.
doi: 10.1137/0137048. |
[11] |
S. Ei, Two-timing methods with applications to heterogeneous reaction-diffusion systems, Hiroshima Math. J., 18 (1988), 127-160. |
[12] |
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. |
[13] |
J. K. Hale and K. Sakamoto, Existence and stability of transition layers, Japan J. Appl. Math., 5 (1988), 367-405.
doi: 10.1007/BF03167908. |
[14] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981. |
[15] |
M. A. Herrero and J. J. L. Velazquez, Chemotactic collapse for the Keller-Segel model, J. Math. Biol., 35 (1996), 177-194.
doi: 10.1007/s002850050049. |
[16] |
T. Hillen and K. J. Painter, A user's guidence to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[17] |
D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[18] |
Y. Kan-on and E. Yanagida, Existence of nonconstant stable equilibria in competition-diffusion equations, Hiroshima Math. J., 23 (1993), 193-221. |
[19] |
T. Kato, Functional Analysis, Springer Classics in Mathematics, 1996. |
[20] |
K. Kishimoto and H. Weinberger, The spatial homogeneity of stable equilibria of some reaction-diffusion systems in convex domains, J. Differential Equations, 58 (1985), 15-21.
doi: 10.1016/0022-0396(85)90020-8. |
[21] |
T. Kolokolnikov and J. Wei, Stability of spiky solutions in a competition model with cross-diffusion, SIAM J. Appl. Math., 71 (2011), 1428-1457.
doi: 10.1137/100808381. |
[22] |
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. |
[23] |
________, Diffusion vs cross-diffusion: An elliptic approach, J. Differential Equations, 154 (1999), 157-190.
doi: 10.1006/jdeq.1998.3559. |
[24] |
Y. Lou, W.-M. Ni and Y. Wu, On the global existence of a cross-diffusion system, Discret Contin. Dynam. Systems, 4 (1998), 193-203.
doi: 10.3934/dcds.1998.4.193. |
[25] |
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. |
[26] |
O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Society, (1968), 648 pages. |
[27] |
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. |
[28] |
M. Mimura, Stationary patterns of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. J., 11 (1981), 621-635. |
[29] |
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. |
[30] |
M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64.
doi: 10.1007/BF00276035. |
[31] |
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. |
[32] |
V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, Journal. Theor. Biol., 42 (1973), 63-105.
doi: 10.1016/0022-5193(73)90149-5. |
[33] |
W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, 82 (SIAM), Philadelphia, PA, 2011. xii+110 pp. ISBN: 978-1-611971-96-5
doi: 10.1137/1.9781611971972. |
[34] |
W.-M. Ni, P. Polascik and E. Yanagida, Monotonicity of stable solutions in shadow systems, Trans. Amer. Math. Soc., 353 (2001), 5057-5069.
doi: 10.1090/S0002-9947-01-02880-X. |
[35] |
W.-M. Ni, Y. Wu and Q. Xu, The existence and stability of nontrivial steady states for SKT competition model with cross-diffusion, Discret Cotin Dyn. Syst., 34 (2014), 5271-5298.
doi: 10.3934/dcds.2014.34.5271. |
[36] |
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. |
[37] |
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. |
[38] |
Q. Wang, On the steady state of a shadow system to the SKT competition model, Discrete Contin. Dyn. Syst.-Series B, 19 (2014), 2941-2961.
doi: 10.3934/dcdsb.2014.19.2941. |
[39] |
X. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly's compactness theorem, J. Math. Biol., 66 (2013), 1241-1266.
doi: 10.1007/s00285-012-0533-x. |
[40] |
M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[41] |
M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.
doi: 10.1016/j.jmaa.2011.05.057. |
[42] |
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] |
N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.
doi: 10.1080/03605307908820113. |
[2] |
H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. |
[3] |
_______, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, differential operators and nonlinear Analysis, Teubner, Stuttgart, Leipzig, 133 (1993), 9-126.
doi: 10.1007/978-3-663-11336-2_1. |
[4] |
A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a Chemotaxis Model with Saturated Chemotactic Flux, Kinet. Relat. Models, 5 (2012), 51-95.
doi: 10.3934/krm.2012.5.51. |
[5] |
Y. S. 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), 719-730.
doi: 10.3934/dcds.2004.10.719. |
[6] |
E. Conway, D. Hoff and J. Smoller, Large time behavior of solutions of systems of nonlinear reaction-diffusion equations, SIAM J. Appl. Math., 35, (1978), 1-16.
doi: 10.1137/0135001. |
[7] |
C. Cosner, Reaction-diffusion-advection models for the effects and evolution of dispersal, Discrete Contin. Dyn. Syst., 34 (2014), 1701-1745.
doi: 10.3934/dcds.2014.34.1701. |
[8] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[9] |
_______, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. and Anal., 52 (1973), 161-180. |
[10] |
P. De Mottoni and F. Rothe, Convergence to homogeneous equilibrium state for generalized Volterra-Lotka systems with diffusion, SIAM J. Appl. Math., 37 (1979), 648-663.
doi: 10.1137/0137048. |
[11] |
S. Ei, Two-timing methods with applications to heterogeneous reaction-diffusion systems, Hiroshima Math. J., 18 (1988), 127-160. |
[12] |
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. |
[13] |
J. K. Hale and K. Sakamoto, Existence and stability of transition layers, Japan J. Appl. Math., 5 (1988), 367-405.
doi: 10.1007/BF03167908. |
[14] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981. |
[15] |
M. A. Herrero and J. J. L. Velazquez, Chemotactic collapse for the Keller-Segel model, J. Math. Biol., 35 (1996), 177-194.
doi: 10.1007/s002850050049. |
[16] |
T. Hillen and K. J. Painter, A user's guidence to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[17] |
D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[18] |
Y. Kan-on and E. Yanagida, Existence of nonconstant stable equilibria in competition-diffusion equations, Hiroshima Math. J., 23 (1993), 193-221. |
[19] |
T. Kato, Functional Analysis, Springer Classics in Mathematics, 1996. |
[20] |
K. Kishimoto and H. Weinberger, The spatial homogeneity of stable equilibria of some reaction-diffusion systems in convex domains, J. Differential Equations, 58 (1985), 15-21.
doi: 10.1016/0022-0396(85)90020-8. |
[21] |
T. Kolokolnikov and J. Wei, Stability of spiky solutions in a competition model with cross-diffusion, SIAM J. Appl. Math., 71 (2011), 1428-1457.
doi: 10.1137/100808381. |
[22] |
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. |
[23] |
________, Diffusion vs cross-diffusion: An elliptic approach, J. Differential Equations, 154 (1999), 157-190.
doi: 10.1006/jdeq.1998.3559. |
[24] |
Y. Lou, W.-M. Ni and Y. Wu, On the global existence of a cross-diffusion system, Discret Contin. Dynam. Systems, 4 (1998), 193-203.
doi: 10.3934/dcds.1998.4.193. |
[25] |
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. |
[26] |
O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Society, (1968), 648 pages. |
[27] |
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. |
[28] |
M. Mimura, Stationary patterns of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. J., 11 (1981), 621-635. |
[29] |
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. |
[30] |
M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64.
doi: 10.1007/BF00276035. |
[31] |
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. |
[32] |
V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, Journal. Theor. Biol., 42 (1973), 63-105.
doi: 10.1016/0022-5193(73)90149-5. |
[33] |
W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, 82 (SIAM), Philadelphia, PA, 2011. xii+110 pp. ISBN: 978-1-611971-96-5
doi: 10.1137/1.9781611971972. |
[34] |
W.-M. Ni, P. Polascik and E. Yanagida, Monotonicity of stable solutions in shadow systems, Trans. Amer. Math. Soc., 353 (2001), 5057-5069.
doi: 10.1090/S0002-9947-01-02880-X. |
[35] |
W.-M. Ni, Y. Wu and Q. Xu, The existence and stability of nontrivial steady states for SKT competition model with cross-diffusion, Discret Cotin Dyn. Syst., 34 (2014), 5271-5298.
doi: 10.3934/dcds.2014.34.5271. |
[36] |
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. |
[37] |
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. |
[38] |
Q. Wang, On the steady state of a shadow system to the SKT competition model, Discrete Contin. Dyn. Syst.-Series B, 19 (2014), 2941-2961.
doi: 10.3934/dcdsb.2014.19.2941. |
[39] |
X. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly's compactness theorem, J. Math. Biol., 66 (2013), 1241-1266.
doi: 10.1007/s00285-012-0533-x. |
[40] |
M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[41] |
M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.
doi: 10.1016/j.jmaa.2011.05.057. |
[42] |
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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