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July  2014, 34(7): 2907-2927. doi: 10.3934/dcds.2014.34.2907

Development of traveling waves in an interacting two-species chemotaxis model

Tai-Chia Lin 1, and  Zhi-An Wang 2,

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

Received  June 2013 Revised  September 2013 Published  December 2013

By constructing sub and super solutions, we establish the existence of traveling wave solutions to a two-species chemotaxis model, which describes two interacting species chemotactically reacting to a chemical signal that is degraded by the two species. We identify the full parameter regime in which the traveling wave solutions exist, derive the asymptotical decay rates of traveling wave solutions at far field and show that the traveling wave solutions are convergent as the chemical diffusion coefficient goes to zero.
Keywords: Chemotaxis, logarithmic sensitivity, traveling wave solutions, multi-species, maximum principle, sub and super-solutions, asymptotic behavior..
Mathematics Subject Classification: Primary: 35C07, 35K55, 46N60; Secondary: 62P10, 92C1.
Citation: Tai-Chia Lin, Zhi-An Wang. Development of traveling waves in an interacting two-species chemotaxis model. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2907-2927. doi: 10.3934/dcds.2014.34.2907
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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