# American Institute of Mathematical Sciences

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

## 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

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.
Citation: Tai-Chia Lin, Zhi-An Wang. Development of traveling waves in an interacting two-species chemotaxis model. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2907-2927. doi: 10.3934/dcds.2014.34.2907
##### References:
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Velázques, Simultaneous finite time blow-up in a two-species model for chemotaxis,, Analysis (Munich), 29 (2009), 317. doi: 10.1524/anly.2009.1029. Google Scholar [8] A. Fasano, A. Mancini and M. Primicerio, Equilibrium of two populations subjected to chemotaxis,, Math. Models Methods. Appli. Sci., 14 (2004), 503. doi: 10.1142/S0218202504003337. Google Scholar [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. doi: 10.1007/s00332-010-9082-x. Google Scholar [10] D. Horstmann and A. Stevens, A constructive approach to traveling waves in chemotaxis,, J. Nonlin. Sci., 14 (2004), 1. doi: 10.1007/s00332-003-0548-y. Google Scholar [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. doi: 10.1016/j.jde.2013.04.002. Google Scholar [12] Y. Kalinin, L. Jiang, Y. Tu and M. Wu, Logarithmic sensing in Escherichia coli bacterial chemotaxis,, Biophysical Journal, 96 (2009), 2439. doi: 10.1016/j.bpj.2008.10.027. Google Scholar [13] E. Keller and L. Segel, Traveling bands of chemotactic bacteria: A theorectical analysis,, J. Theor. Biol., 30 (1971), 235. doi: 10.1016/0022-5193(71)90051-8. Google Scholar [14] F. Kelley, K. Dapsis and D. Lauffenburger, Effects of bacterial chemotaxis on dynamics of microbial competition,, Microb. Ecol., 16 (1988), 115. doi: 10.1007/BF02018908. Google Scholar [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. doi: 10.1007/BF02413502. Google Scholar [16] D. Lauffenburger, Quantitative studies of bacterial chemotaxis and microbial population dynamics,, Microb. Ecol., 22 (1991), 175. doi: 10.1007/BF02540222. Google Scholar [17] D. Le, Coexistence with chemotaxis,, SIAM J. Math. Anal., 32 (2000), 504. doi: 10.1137/S0036141099346779. Google Scholar [18] T. Li and Z.-A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis,, SIAM J. Appl. Math., 70 (): 1522. doi: 10.1137/09075161X. Google Scholar [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. doi: 10.1142/S0218202510004830. Google Scholar [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. doi: 10.1016/j.jde.2010.09.020. Google Scholar [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. doi: 10.1016/S0092-8240(03)00030-2. Google Scholar [22] R. Lui and Z.-A. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models,, J. Math. Biol., 61 (2010), 739. doi: 10.1007/s00285-009-0317-0. Google Scholar [23] T. Nagai and T. Ikeda, Traveling waves in a chemotaxis model,, J. Math. Biol., 30 (1991), 169. doi: 10.1007/BF00160334. Google Scholar [24] G. Odell and E. Keller, Traveling bands of chemotactic bacteria revisited,, J. Theor. Biol., 56 (1976), 243. doi: 10.1016/S0022-5193(76)80055-0. Google Scholar [25] G. Rosen, Steady-state distribution of bacteria chemotactic toward oxygen,, Bull. Math. Biol., 40 (1978), 671. doi: 10.1016/S0092-8240(78)80025-1. Google Scholar [26] G. Rosen, Theoretical significance of the condition $\delta=2\mu$ in bacterial chemotaxis,, Bull. Math. Biol., 45 (1983), 151. Google Scholar [27] G. Rosen and S. Baloga, On the stability of steadily propogating bands of chemotactic bacteria,, Math. Biosci., 24 (1975), 273. doi: 10.1016/0025-5564(75)90080-2. Google Scholar [28] H. Schwetlick, Traveling waves for chemotaxis systems,, Proc. Appl. Math. Mech., 3 (2003), 476. doi: 10.1002/pamm.200310508. Google Scholar [29] Y. Tao and Z.-A. Wang, Competing effects of attraction vs. repulsion in chemotaxis,, Math. Models Methods Appl. Sci., 23 (2013), 1. doi: 10.1142/S0218202512500443. Google Scholar [30] Z.-A. Wang, Wavefront of an angiogenesis model,, Discrete Contin. Dyn. Syst. Series B, 17 (2012), 2849. doi: 10.3934/dcdsb.2012.17.2849. Google Scholar [31] Z.-A. Wang, Mathematics of traveling waves in chemotaxis-review paper,, Discrete Contin. Dyn. Syst. Series B, 18 (2013), 601. doi: 10.3934/dcdsb.2013.18.601. Google Scholar [32] G. Wolansky, Multi-components chemotaxis system in absence of conflict,, Eur. J. Appl. Math., 13 (2002), 641. doi: 10.1017/S0956792501004843. Google Scholar

show all references

##### References:
 [1] J. Adler, Chemotaxis in bacteria,, Science, 153 (1966), 708. Google Scholar [2] R. Bellman, Stability Theory of Differential Equations,, McGraw-Hill Book Company, (1953). Google Scholar [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. doi: 10.1016/j.nonrwa.2007.06.001. Google Scholar [4] C. Conca, E. Espejo and K. Vilches, Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in $\mathbbR^2$,, Euro. J. Appl. Math., 22 (2011), 553. doi: 10.1017/S0956792511000258. Google Scholar [5] W. Coppel, Stability and Asymptotic Behavior of Differential Equations,, D. C. Heath and Co., (1965). Google Scholar [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. Google Scholar [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. doi: 10.1524/anly.2009.1029. Google Scholar [8] A. Fasano, A. Mancini and M. Primicerio, Equilibrium of two populations subjected to chemotaxis,, Math. Models Methods. Appli. Sci., 14 (2004), 503. doi: 10.1142/S0218202504003337. Google Scholar [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. doi: 10.1007/s00332-010-9082-x. Google Scholar [10] D. Horstmann and A. Stevens, A constructive approach to traveling waves in chemotaxis,, J. Nonlin. Sci., 14 (2004), 1. doi: 10.1007/s00332-003-0548-y. Google Scholar [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. doi: 10.1016/j.jde.2013.04.002. Google Scholar [12] Y. Kalinin, L. Jiang, Y. Tu and M. Wu, Logarithmic sensing in Escherichia coli bacterial chemotaxis,, Biophysical Journal, 96 (2009), 2439. doi: 10.1016/j.bpj.2008.10.027. Google Scholar [13] E. Keller and L. Segel, Traveling bands of chemotactic bacteria: A theorectical analysis,, J. Theor. Biol., 30 (1971), 235. doi: 10.1016/0022-5193(71)90051-8. Google Scholar [14] F. Kelley, K. Dapsis and D. Lauffenburger, Effects of bacterial chemotaxis on dynamics of microbial competition,, Microb. Ecol., 16 (1988), 115. doi: 10.1007/BF02018908. Google Scholar [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. doi: 10.1007/BF02413502. Google Scholar [16] D. Lauffenburger, Quantitative studies of bacterial chemotaxis and microbial population dynamics,, Microb. Ecol., 22 (1991), 175. doi: 10.1007/BF02540222. Google Scholar [17] D. Le, Coexistence with chemotaxis,, SIAM J. Math. Anal., 32 (2000), 504. doi: 10.1137/S0036141099346779. Google Scholar [18] T. Li and Z.-A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis,, SIAM J. Appl. Math., 70 (): 1522. doi: 10.1137/09075161X. Google Scholar [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. doi: 10.1142/S0218202510004830. Google Scholar [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. doi: 10.1016/j.jde.2010.09.020. Google Scholar [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. doi: 10.1016/S0092-8240(03)00030-2. Google Scholar [22] R. Lui and Z.-A. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models,, J. Math. Biol., 61 (2010), 739. doi: 10.1007/s00285-009-0317-0. Google Scholar [23] T. Nagai and T. Ikeda, Traveling waves in a chemotaxis model,, J. Math. Biol., 30 (1991), 169. doi: 10.1007/BF00160334. Google Scholar [24] G. Odell and E. Keller, Traveling bands of chemotactic bacteria revisited,, J. Theor. Biol., 56 (1976), 243. doi: 10.1016/S0022-5193(76)80055-0. Google Scholar [25] G. Rosen, Steady-state distribution of bacteria chemotactic toward oxygen,, Bull. Math. Biol., 40 (1978), 671. doi: 10.1016/S0092-8240(78)80025-1. Google Scholar [26] G. Rosen, Theoretical significance of the condition $\delta=2\mu$ in bacterial chemotaxis,, Bull. Math. Biol., 45 (1983), 151. Google Scholar [27] G. Rosen and S. Baloga, On the stability of steadily propogating bands of chemotactic bacteria,, Math. Biosci., 24 (1975), 273. doi: 10.1016/0025-5564(75)90080-2. Google Scholar [28] H. Schwetlick, Traveling waves for chemotaxis systems,, Proc. Appl. Math. Mech., 3 (2003), 476. doi: 10.1002/pamm.200310508. Google Scholar [29] Y. Tao and Z.-A. Wang, Competing effects of attraction vs. repulsion in chemotaxis,, Math. Models Methods Appl. Sci., 23 (2013), 1. doi: 10.1142/S0218202512500443. Google Scholar [30] Z.-A. Wang, Wavefront of an angiogenesis model,, Discrete Contin. Dyn. Syst. Series B, 17 (2012), 2849. doi: 10.3934/dcdsb.2012.17.2849. Google Scholar [31] Z.-A. Wang, Mathematics of traveling waves in chemotaxis-review paper,, Discrete Contin. Dyn. Syst. Series B, 18 (2013), 601. doi: 10.3934/dcdsb.2013.18.601. Google Scholar [32] G. Wolansky, Multi-components chemotaxis system in absence of conflict,, Eur. J. Appl. Math., 13 (2002), 641. doi: 10.1017/S0956792501004843. Google Scholar
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