January  2011, 10(1): 141-160. doi: 10.3934/cpaa.2011.10.141

Traveling waves and their stability in a coupled reaction diffusion system

1. 

Department of Mathematics and Statistics, University of North Carolina at Wilmington, Wilmington, NC 28403

2. 

Department of Mathematics and Statistics, UNC Wilmington, Wilmington, NC 28403

Received  January 2010 Revised  May 2010 Published  November 2010

We study the traveling wave solutions to a reaction diffusion system modeling the public goods game with altruistic behaviors. The existence of the waves is derived through monotone iteration of a pair of classical upper- and lower solutions. The waves are shown to be unique and strictly monotonic. A similar KPP wave like asymptotic behaviors are obtained by comparison principle and exponential dichotomy. The stability of the traveling waves with non-critical speed is investigated by spectral analysis in the weighted Banach spaces.
Citation: Xiaojie Hou, Wei Feng. Traveling waves and their stability in a coupled reaction diffusion system. Communications on Pure & Applied Analysis, 2011, 10 (1) : 141-160. doi: 10.3934/cpaa.2011.10.141
References:
[1]

S. Ai, S-N. Chow and Y. Yi, Traveling wave solutions in a tissue interaction model for skin pattern formation,, Journal of Dynamics and Differential Equations, 15 (2003), 517. doi: doi:10.1023/B:JODY.0000009746.52357.28. Google Scholar

[2]

J. C. Alexander, R. A. Gardner and C. K. R. T. Jones, A topological invariant arising in the stability analysis of traveling waves,, J. Reine Angew Math., 410 (1990), 167. Google Scholar

[3]

M. Arias, J. Campos and C. Marcelli, Fastness and continuous dependence in front propagation in Fisher-KPP equations,, Discrete and Continuous Dynamical Systems-B, 11 (2009), 11. Google Scholar

[4]

P. W. Bates and F. Chen, Spectral analysis and multidimensional stability of traveling waves for nonlocal Allen-Cahn equation,, J. Math. Anal. Appl., 273 (2002), 45. doi: doi:10.1016/S0022-247X(02)00205-6. Google Scholar

[5]

A. Boumenir and V. Nguyen, Perron theorem in monotone iteration method for traveling waves in delayed reaction-diffusion equations,, Journal of Differential Equations, 244 (2008), 1551. doi: doi:10.1016/j.jde.2008.01.004. Google Scholar

[6]

E. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,", McGraw-Hill, (1955). Google Scholar

[7]

N. Fei and J. Carr, Existence of travelling waves with their minimal speed for a diffusing Lotka-Volterra system,, Nonlinear Analysis: Real World Applications, 4 (2003), 503. doi: doi:10.1016/S1468-1218(02)00077-9. Google Scholar

[8]

Th. Gallay, G. Schneider and H. Uecker, Stable transport of information near essentially unstable localized structures,, Discrete and Continuous Dynamical Systems-B, 4 (2004), 349. doi: doi:10.3934/dcdsb.2004.4.349. Google Scholar

[9]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, 840 (1981). Google Scholar

[10]

Y. Hosono, Travelling waves for a diffusive Lotka-Volterra competition model I: Singular Perturbations,, Discrete Continuous Dynamical Systems-B, 3 (2003), 79. doi: doi:10.3934/dcdsb.2003.3.79. Google Scholar

[11]

X. Hou, W. Feng and X. Lu, A mathematical analysis of a pubilc goods games model,, Nonlinear Analysis: Real World Applications, 10 (2009), 2207. doi: doi:10.1016/j.nonrwa.2008.04.005. Google Scholar

[12]

X. Hou and Y. Li, Local stability of traveling wave solutions of nonlinear reaction diffusion equations,, Discrete and Continuous Dynamical Systems-A, 15 (2006), 681. doi: doi:10.3934/dcds.2006.15.681. Google Scholar

[13]

X. Hou, Y. Li and K. R. Meyer, Traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities,, Discrete and Continuous Dynamical Systems-A, 26 (2010), 265. Google Scholar

[14]

W. Huang, Uniqueness of traveling wave solutions for a biological reaction-diffusion equation,, J. Math. Anal. Appl., 316 (2006), 42. doi: doi:10.1016/j.jmaa.2005.04.084. Google Scholar

[15]

J. I Kanel, On the wave front of a competition-diffusion system in popalation dynamics,, Nonlinear Analysis: Theory, 65 (2006), 301. Google Scholar

[16]

J. I Kanel and Li Zhou, Existence of wave front solutions and estimates of wave speed for a competition-diffusion system,, Nonlinear Analysis: Theory, 27 (1996), 579. Google Scholar

[17]

Y. Kan-on, Note on propagation speed of travelling waves for a weakly coupled parabolic system,, Nonlinear Analysis: Theory, 44 (2001), 239. Google Scholar

[18]

Y. Kan-on, Fisher wave fronts for the lotka-volterra competition model with diffusion,, Nonlinear Analysis: Theory, 28 (1997), 145. Google Scholar

[19]

T. Kapitula, On the stability of Traveling waves in weighted $L^\infty$ spaces,, Journal of Differential Equations, 112 (1994), 179. doi: doi:10.1006/jdeq.1994.1100. Google Scholar

[20]

A. Kolmogorov, A. Petrovskii and N. Piskunov, A study of the equation of diffusion with increase in the quantity of matter,, Bjul. Moskovskovo Gov. Iniv., 17 (1937), 1. Google Scholar

[21]

J. Li, Bifurcations of travelling wave solutions for two generalized Boussinesq systems,, Science in China Series A, 51 (2008), 1577. doi: doi:10.1007/s11425-008-0038-7. Google Scholar

[22]

X. Liao, X. Tang and S. Zhou, Existence of traveling wavefronts in a cooperative systems with discrete delays,, Applied Mathematics and Computation, 215 (2009), 1806. doi: doi:10.1016/j.amc.2009.07.032. Google Scholar

[23]

S. W. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations,, Journal of Differential Equations, 237 (2007), 259. doi: doi:10.1016/j.jde.2007.03.014. Google Scholar

[24]

P. Maini, L. Malaguti, C. Marcelli and S. Matucci, Diffusion-aggregation processes with mono-stable reaction terms,, Discrete and Continuous Dynamical Systems-B, 6 (2006), 1175. doi: doi:10.3934/dcdsb.2006.6.1175. Google Scholar

[25]

Y. Qi, Travelling fronts of reaction diffusion systems modeling auto-catalysis,, Discrete and Continuous Dynamical Systems, (2009), 622. Google Scholar

[26]

B. Sandstede, Stability of traveling waves,, in, (2002), 983. doi: doi:10.1016/S1874-575X(02)80039-X. Google Scholar

[27]

D. Sattinger, On the stability of traveling waves of nonlinear parabolic systems,, Advances in Mathematics, 22 (1976), 312. doi: doi:10.1016/0001-8708(76)90098-0. Google Scholar

[28]

M. M. Tang and P. C. Fife, Propagating fronts for competing species equations with diffusion,, Arch. Rat. Mech. Anal., 73 (1980), 69. doi: doi:10.1007/BF00283257. Google Scholar

[29]

A. Volpert, V. Volpert and V. Volpert, "Traveling Wave Solutions of Parabolic Systems,", Transl. Math. Monograhs \textbf{140}, 140 (1994). Google Scholar

[30]

J. Y. Wakano, A mathematical analysis on public goods games in the continuous space,, Math. Biosciences, 201 (2006), 72. doi: doi:10.1016/j.mbs.2005.12.015. Google Scholar

[31]

Z-C. Wang, W-T. Li and S. Ruan, Existence and Stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay,, Journal of Differential Equations, 238 (2007), 153. doi: doi:10.1016/j.jde.2007.03.025. Google Scholar

[32]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay,, Journal of Dynamics and Differential Equations, 13 (2001), 51. doi: doi:10.1007/s10884-007-9090-1. Google Scholar

[33]

Y. Wu and Y. Li, Stability of travelling waves with noncritical speeds for double degenerate Fisher-type equations,, Discrete Continuous Dynamical Systems-B, 10 (2008), 149. doi: doi:10.3934/dcdsb.2008.10.149. Google Scholar

[34]

D. Xu and X.Q. Zhao, Bistable waves in an epidemic model,, Journal of Dynamics and Differential Equations, 16 (2004), 679. doi: doi:10.1007/s10884-005-6294-0. Google Scholar

[35]

Y.Wu, X. Xing and Q. Ye, Stability of travelling waves with algebraic decay for $n$-degree Fisher-type equations,, Discrete and Continuous Dynamical Systems-B, 16 (2006), 47. doi: doi:10.3934/dcds.2006.16.47. Google Scholar

[36]

X-Q Zhao and W. Wang, Fisher waves in an epidemic model,, Discrete and Continuous Dynamical Systems-B, 4 (2004), 1117. doi: doi:10.3934/dcdsb.2004.4.1117. Google Scholar

show all references

References:
[1]

S. Ai, S-N. Chow and Y. Yi, Traveling wave solutions in a tissue interaction model for skin pattern formation,, Journal of Dynamics and Differential Equations, 15 (2003), 517. doi: doi:10.1023/B:JODY.0000009746.52357.28. Google Scholar

[2]

J. C. Alexander, R. A. Gardner and C. K. R. T. Jones, A topological invariant arising in the stability analysis of traveling waves,, J. Reine Angew Math., 410 (1990), 167. Google Scholar

[3]

M. Arias, J. Campos and C. Marcelli, Fastness and continuous dependence in front propagation in Fisher-KPP equations,, Discrete and Continuous Dynamical Systems-B, 11 (2009), 11. Google Scholar

[4]

P. W. Bates and F. Chen, Spectral analysis and multidimensional stability of traveling waves for nonlocal Allen-Cahn equation,, J. Math. Anal. Appl., 273 (2002), 45. doi: doi:10.1016/S0022-247X(02)00205-6. Google Scholar

[5]

A. Boumenir and V. Nguyen, Perron theorem in monotone iteration method for traveling waves in delayed reaction-diffusion equations,, Journal of Differential Equations, 244 (2008), 1551. doi: doi:10.1016/j.jde.2008.01.004. Google Scholar

[6]

E. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,", McGraw-Hill, (1955). Google Scholar

[7]

N. Fei and J. Carr, Existence of travelling waves with their minimal speed for a diffusing Lotka-Volterra system,, Nonlinear Analysis: Real World Applications, 4 (2003), 503. doi: doi:10.1016/S1468-1218(02)00077-9. Google Scholar

[8]

Th. Gallay, G. Schneider and H. Uecker, Stable transport of information near essentially unstable localized structures,, Discrete and Continuous Dynamical Systems-B, 4 (2004), 349. doi: doi:10.3934/dcdsb.2004.4.349. Google Scholar

[9]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, 840 (1981). Google Scholar

[10]

Y. Hosono, Travelling waves for a diffusive Lotka-Volterra competition model I: Singular Perturbations,, Discrete Continuous Dynamical Systems-B, 3 (2003), 79. doi: doi:10.3934/dcdsb.2003.3.79. Google Scholar

[11]

X. Hou, W. Feng and X. Lu, A mathematical analysis of a pubilc goods games model,, Nonlinear Analysis: Real World Applications, 10 (2009), 2207. doi: doi:10.1016/j.nonrwa.2008.04.005. Google Scholar

[12]

X. Hou and Y. Li, Local stability of traveling wave solutions of nonlinear reaction diffusion equations,, Discrete and Continuous Dynamical Systems-A, 15 (2006), 681. doi: doi:10.3934/dcds.2006.15.681. Google Scholar

[13]

X. Hou, Y. Li and K. R. Meyer, Traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities,, Discrete and Continuous Dynamical Systems-A, 26 (2010), 265. Google Scholar

[14]

W. Huang, Uniqueness of traveling wave solutions for a biological reaction-diffusion equation,, J. Math. Anal. Appl., 316 (2006), 42. doi: doi:10.1016/j.jmaa.2005.04.084. Google Scholar

[15]

J. I Kanel, On the wave front of a competition-diffusion system in popalation dynamics,, Nonlinear Analysis: Theory, 65 (2006), 301. Google Scholar

[16]

J. I Kanel and Li Zhou, Existence of wave front solutions and estimates of wave speed for a competition-diffusion system,, Nonlinear Analysis: Theory, 27 (1996), 579. Google Scholar

[17]

Y. Kan-on, Note on propagation speed of travelling waves for a weakly coupled parabolic system,, Nonlinear Analysis: Theory, 44 (2001), 239. Google Scholar

[18]

Y. Kan-on, Fisher wave fronts for the lotka-volterra competition model with diffusion,, Nonlinear Analysis: Theory, 28 (1997), 145. Google Scholar

[19]

T. Kapitula, On the stability of Traveling waves in weighted $L^\infty$ spaces,, Journal of Differential Equations, 112 (1994), 179. doi: doi:10.1006/jdeq.1994.1100. Google Scholar

[20]

A. Kolmogorov, A. Petrovskii and N. Piskunov, A study of the equation of diffusion with increase in the quantity of matter,, Bjul. Moskovskovo Gov. Iniv., 17 (1937), 1. Google Scholar

[21]

J. Li, Bifurcations of travelling wave solutions for two generalized Boussinesq systems,, Science in China Series A, 51 (2008), 1577. doi: doi:10.1007/s11425-008-0038-7. Google Scholar

[22]

X. Liao, X. Tang and S. Zhou, Existence of traveling wavefronts in a cooperative systems with discrete delays,, Applied Mathematics and Computation, 215 (2009), 1806. doi: doi:10.1016/j.amc.2009.07.032. Google Scholar

[23]

S. W. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations,, Journal of Differential Equations, 237 (2007), 259. doi: doi:10.1016/j.jde.2007.03.014. Google Scholar

[24]

P. Maini, L. Malaguti, C. Marcelli and S. Matucci, Diffusion-aggregation processes with mono-stable reaction terms,, Discrete and Continuous Dynamical Systems-B, 6 (2006), 1175. doi: doi:10.3934/dcdsb.2006.6.1175. Google Scholar

[25]

Y. Qi, Travelling fronts of reaction diffusion systems modeling auto-catalysis,, Discrete and Continuous Dynamical Systems, (2009), 622. Google Scholar

[26]

B. Sandstede, Stability of traveling waves,, in, (2002), 983. doi: doi:10.1016/S1874-575X(02)80039-X. Google Scholar

[27]

D. Sattinger, On the stability of traveling waves of nonlinear parabolic systems,, Advances in Mathematics, 22 (1976), 312. doi: doi:10.1016/0001-8708(76)90098-0. Google Scholar

[28]

M. M. Tang and P. C. Fife, Propagating fronts for competing species equations with diffusion,, Arch. Rat. Mech. Anal., 73 (1980), 69. doi: doi:10.1007/BF00283257. Google Scholar

[29]

A. Volpert, V. Volpert and V. Volpert, "Traveling Wave Solutions of Parabolic Systems,", Transl. Math. Monograhs \textbf{140}, 140 (1994). Google Scholar

[30]

J. Y. Wakano, A mathematical analysis on public goods games in the continuous space,, Math. Biosciences, 201 (2006), 72. doi: doi:10.1016/j.mbs.2005.12.015. Google Scholar

[31]

Z-C. Wang, W-T. Li and S. Ruan, Existence and Stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay,, Journal of Differential Equations, 238 (2007), 153. doi: doi:10.1016/j.jde.2007.03.025. Google Scholar

[32]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay,, Journal of Dynamics and Differential Equations, 13 (2001), 51. doi: doi:10.1007/s10884-007-9090-1. Google Scholar

[33]

Y. Wu and Y. Li, Stability of travelling waves with noncritical speeds for double degenerate Fisher-type equations,, Discrete Continuous Dynamical Systems-B, 10 (2008), 149. doi: doi:10.3934/dcdsb.2008.10.149. Google Scholar

[34]

D. Xu and X.Q. Zhao, Bistable waves in an epidemic model,, Journal of Dynamics and Differential Equations, 16 (2004), 679. doi: doi:10.1007/s10884-005-6294-0. Google Scholar

[35]

Y.Wu, X. Xing and Q. Ye, Stability of travelling waves with algebraic decay for $n$-degree Fisher-type equations,, Discrete and Continuous Dynamical Systems-B, 16 (2006), 47. doi: doi:10.3934/dcds.2006.16.47. Google Scholar

[36]

X-Q Zhao and W. Wang, Fisher waves in an epidemic model,, Discrete and Continuous Dynamical Systems-B, 4 (2004), 1117. doi: doi:10.3934/dcdsb.2004.4.1117. Google Scholar

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