American Institute of Mathematical Sciences

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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