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Development of traveling waves in an interacting two-species chemotaxis model
1. | Institute of Applied Mathematical Sciences, NCTS Taipei Office, National Taiwan University, Taipei 106, Taiwan |
2. | Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong |
References:
[1] |
J. Adler, Chemotaxis in bacteria, Science, 153 (1966), 708-716. |
[2] |
R. Bellman, Stability Theory of Differential Equations, McGraw-Hill Book Company, New York-Toronto-London, 1953. |
[3] |
F. Berezovskaya, A. Novozhilov and G. Karev, Families of traveling impulse and fronts in some models with cross-diffusion, Nonlinear Anal. Real World Appl., 9 (2008), 1866-1881.
doi: 10.1016/j.nonrwa.2007.06.001. |
[4] |
C. Conca, E. Espejo and K. Vilches, Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in $\mathbb{R}^2$, Euro. J. Appl. Math., 22 (2011), 553-580.
doi: 10.1017/S0956792511000258. |
[5] |
W. Coppel, Stability and Asymptotic Behavior of Differential Equations, D. C. Heath and Co., Boston, 1965. |
[6] |
Y. Ebihara, Y. Furusho and T. Nagai, Singular solution of traveling waves in a chemotactic model, Bull. Kyushu Inst. Tech. Math. Natur. Sci., 39 (1992), 29-38. |
[7] |
E. Espejo Arenas, A. Stevens and J. Velázques, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis (Munich), 29 (2009), 317-338.
doi: 10.1524/anly.2009.1029. |
[8] |
A. Fasano, A. Mancini and M. Primicerio, Equilibrium of two populations subjected to chemotaxis, Math. Models Methods. Appli. Sci., 14 (2004), 503-533.
doi: 10.1142/S0218202504003337. |
[9] |
D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), 231-270.
doi: 10.1007/s00332-010-9082-x. |
[10] |
D. Horstmann and A. Stevens, A constructive approach to traveling waves in chemotaxis, J. Nonlin. Sci., 14 (2004), 1-25.
doi: 10.1007/s00332-003-0548-y. |
[11] |
H.-Y. Jin, J. Li and Z.-A. Wang, Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity, J. Differential Equations, 255 (2013), 193-219.
doi: 10.1016/j.jde.2013.04.002. |
[12] |
Y. Kalinin, L. Jiang, Y. Tu and M. Wu, Logarithmic sensing in Escherichia coli bacterial chemotaxis, Biophysical Journal, 96 (2009), 2439-2448.
doi: 10.1016/j.bpj.2008.10.027. |
[13] |
E. Keller and L. Segel, Traveling bands of chemotactic bacteria: A theorectical analysis, J. Theor. Biol., 30 (1971), 235-248.
doi: 10.1016/0022-5193(71)90051-8. |
[14] |
F. Kelley, K. Dapsis and D. Lauffenburger, Effects of bacterial chemotaxis on dynamics of microbial competition, Microb. Ecol., 16 (1988), 115-131.
doi: 10.1007/BF02018908. |
[15] |
I. Kiguradse, On the non-negative non-increasing solutions of non-linear second order differential equations, Ann. Mat. Pura Appl. (4), 81 (1969), 169-191.
doi: 10.1007/BF02413502. |
[16] |
D. Lauffenburger, Quantitative studies of bacterial chemotaxis and microbial population dynamics, Microb. Ecol., 22 (1991), 175-185.
doi: 10.1007/BF02540222. |
[17] |
D. Le, Coexistence with chemotaxis, SIAM J. Math. Anal., 32 (2000), 504-521.
doi: 10.1137/S0036141099346779. |
[18] |
T. Li and Z.-A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009/10), 1522-1541.
doi: 10.1137/09075161X. |
[19] |
T. Li and Z.-A. Wang, Nonlinear stability of large amplitude viscous shock waves of a hyperbolic-parabolic system arising in chemotaxis, Math. Models Methods Appl. Sci., 20 (2010), 1967-1998.
doi: 10.1142/S0218202510004830. |
[20] |
T. Li and Z.-A. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Differential Equations, 250 (2011), 1310-1333.
doi: 10.1016/j.jde.2010.09.020. |
[21] |
M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signaling, microglia, and Alzheimer's disease senile plaques: Is there a connection?, Bull. Math. Biol., 65 (2003), 693-730.
doi: 10.1016/S0092-8240(03)00030-2. |
[22] |
R. Lui and Z.-A. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models, J. Math. Biol., 61 (2010), 739-761.
doi: 10.1007/s00285-009-0317-0. |
[23] |
T. Nagai and T. Ikeda, Traveling waves in a chemotaxis model, J. Math. Biol., 30 (1991), 169-184.
doi: 10.1007/BF00160334. |
[24] |
G. Odell and E. Keller, Traveling bands of chemotactic bacteria revisited, J. Theor. Biol., 56 (1976), 243-247.
doi: 10.1016/S0022-5193(76)80055-0. |
[25] |
G. Rosen, Steady-state distribution of bacteria chemotactic toward oxygen, Bull. Math. Biol., 40 (1978), 671-674.
doi: 10.1016/S0092-8240(78)80025-1. |
[26] |
G. Rosen, Theoretical significance of the condition $\delta=2\mu$ in bacterial chemotaxis, Bull. Math. Biol., 45 (1983), 151-153. |
[27] |
G. Rosen and S. Baloga, On the stability of steadily propogating bands of chemotactic bacteria, Math. Biosci., 24 (1975), 273-279.
doi: 10.1016/0025-5564(75)90080-2. |
[28] |
H. Schwetlick, Traveling waves for chemotaxis systems, Proc. Appl. Math. Mech., 3 (2003), 476-478.
doi: 10.1002/pamm.200310508. |
[29] |
Y. Tao and Z.-A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.
doi: 10.1142/S0218202512500443. |
[30] |
Z.-A. Wang, Wavefront of an angiogenesis model, Discrete Contin. Dyn. Syst. Series B, 17 (2012), 2849-2860.
doi: 10.3934/dcdsb.2012.17.2849. |
[31] |
Z.-A. Wang, Mathematics of traveling waves in chemotaxis-review paper, Discrete Contin. Dyn. Syst. Series B, 18 (2013), 601-641.
doi: 10.3934/dcdsb.2013.18.601. |
[32] |
G. Wolansky, Multi-components chemotaxis system in absence of conflict, Eur. J. Appl. Math., 13 (2002), 641-661.
doi: 10.1017/S0956792501004843. |
show all references
References:
[1] |
J. Adler, Chemotaxis in bacteria, Science, 153 (1966), 708-716. |
[2] |
R. Bellman, Stability Theory of Differential Equations, McGraw-Hill Book Company, New York-Toronto-London, 1953. |
[3] |
F. Berezovskaya, A. Novozhilov and G. Karev, Families of traveling impulse and fronts in some models with cross-diffusion, Nonlinear Anal. Real World Appl., 9 (2008), 1866-1881.
doi: 10.1016/j.nonrwa.2007.06.001. |
[4] |
C. Conca, E. Espejo and K. Vilches, Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in $\mathbb{R}^2$, Euro. J. Appl. Math., 22 (2011), 553-580.
doi: 10.1017/S0956792511000258. |
[5] |
W. Coppel, Stability and Asymptotic Behavior of Differential Equations, D. C. Heath and Co., Boston, 1965. |
[6] |
Y. Ebihara, Y. Furusho and T. Nagai, Singular solution of traveling waves in a chemotactic model, Bull. Kyushu Inst. Tech. Math. Natur. Sci., 39 (1992), 29-38. |
[7] |
E. Espejo Arenas, A. Stevens and J. Velázques, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis (Munich), 29 (2009), 317-338.
doi: 10.1524/anly.2009.1029. |
[8] |
A. Fasano, A. Mancini and M. Primicerio, Equilibrium of two populations subjected to chemotaxis, Math. Models Methods. Appli. Sci., 14 (2004), 503-533.
doi: 10.1142/S0218202504003337. |
[9] |
D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), 231-270.
doi: 10.1007/s00332-010-9082-x. |
[10] |
D. Horstmann and A. Stevens, A constructive approach to traveling waves in chemotaxis, J. Nonlin. Sci., 14 (2004), 1-25.
doi: 10.1007/s00332-003-0548-y. |
[11] |
H.-Y. Jin, J. Li and Z.-A. Wang, Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity, J. Differential Equations, 255 (2013), 193-219.
doi: 10.1016/j.jde.2013.04.002. |
[12] |
Y. Kalinin, L. Jiang, Y. Tu and M. Wu, Logarithmic sensing in Escherichia coli bacterial chemotaxis, Biophysical Journal, 96 (2009), 2439-2448.
doi: 10.1016/j.bpj.2008.10.027. |
[13] |
E. Keller and L. Segel, Traveling bands of chemotactic bacteria: A theorectical analysis, J. Theor. Biol., 30 (1971), 235-248.
doi: 10.1016/0022-5193(71)90051-8. |
[14] |
F. Kelley, K. Dapsis and D. Lauffenburger, Effects of bacterial chemotaxis on dynamics of microbial competition, Microb. Ecol., 16 (1988), 115-131.
doi: 10.1007/BF02018908. |
[15] |
I. Kiguradse, On the non-negative non-increasing solutions of non-linear second order differential equations, Ann. Mat. Pura Appl. (4), 81 (1969), 169-191.
doi: 10.1007/BF02413502. |
[16] |
D. Lauffenburger, Quantitative studies of bacterial chemotaxis and microbial population dynamics, Microb. Ecol., 22 (1991), 175-185.
doi: 10.1007/BF02540222. |
[17] |
D. Le, Coexistence with chemotaxis, SIAM J. Math. Anal., 32 (2000), 504-521.
doi: 10.1137/S0036141099346779. |
[18] |
T. Li and Z.-A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009/10), 1522-1541.
doi: 10.1137/09075161X. |
[19] |
T. Li and Z.-A. Wang, Nonlinear stability of large amplitude viscous shock waves of a hyperbolic-parabolic system arising in chemotaxis, Math. Models Methods Appl. Sci., 20 (2010), 1967-1998.
doi: 10.1142/S0218202510004830. |
[20] |
T. Li and Z.-A. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Differential Equations, 250 (2011), 1310-1333.
doi: 10.1016/j.jde.2010.09.020. |
[21] |
M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signaling, microglia, and Alzheimer's disease senile plaques: Is there a connection?, Bull. Math. Biol., 65 (2003), 693-730.
doi: 10.1016/S0092-8240(03)00030-2. |
[22] |
R. Lui and Z.-A. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models, J. Math. Biol., 61 (2010), 739-761.
doi: 10.1007/s00285-009-0317-0. |
[23] |
T. Nagai and T. Ikeda, Traveling waves in a chemotaxis model, J. Math. Biol., 30 (1991), 169-184.
doi: 10.1007/BF00160334. |
[24] |
G. Odell and E. Keller, Traveling bands of chemotactic bacteria revisited, J. Theor. Biol., 56 (1976), 243-247.
doi: 10.1016/S0022-5193(76)80055-0. |
[25] |
G. Rosen, Steady-state distribution of bacteria chemotactic toward oxygen, Bull. Math. Biol., 40 (1978), 671-674.
doi: 10.1016/S0092-8240(78)80025-1. |
[26] |
G. Rosen, Theoretical significance of the condition $\delta=2\mu$ in bacterial chemotaxis, Bull. Math. Biol., 45 (1983), 151-153. |
[27] |
G. Rosen and S. Baloga, On the stability of steadily propogating bands of chemotactic bacteria, Math. Biosci., 24 (1975), 273-279.
doi: 10.1016/0025-5564(75)90080-2. |
[28] |
H. Schwetlick, Traveling waves for chemotaxis systems, Proc. Appl. Math. Mech., 3 (2003), 476-478.
doi: 10.1002/pamm.200310508. |
[29] |
Y. Tao and Z.-A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.
doi: 10.1142/S0218202512500443. |
[30] |
Z.-A. Wang, Wavefront of an angiogenesis model, Discrete Contin. Dyn. Syst. Series B, 17 (2012), 2849-2860.
doi: 10.3934/dcdsb.2012.17.2849. |
[31] |
Z.-A. Wang, Mathematics of traveling waves in chemotaxis-review paper, Discrete Contin. Dyn. Syst. Series B, 18 (2013), 601-641.
doi: 10.3934/dcdsb.2013.18.601. |
[32] |
G. Wolansky, Multi-components chemotaxis system in absence of conflict, Eur. J. Appl. Math., 13 (2002), 641-661.
doi: 10.1017/S0956792501004843. |
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