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Traveling wave solutions for Lotka-Volterra system re-visited
1. | Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45219, United States |
2. | Department of Mathematics and Statistics, University of North Carolina at Wilmington, Wilmington, NC 28403 |
3. | Department of Mathematics and Statistics, UNC Wilmington, Wilmington, NC 28403 |
References:
[1] |
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.
|
[2] |
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. |
[3] |
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. |
[4] |
E. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,", McGraw-Hill, (1955). |
[5] |
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. |
[6] |
D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture notes in Mathematics, 840 (1981).
|
[7] |
Y. Hosono, Travelling waves for a diffusive Lotka-Volterra competition model I: Singular perturbations,, Discrete Continuous Dynamical Systems - B, 3 (2003), 79.
|
[8] |
J. I. Kanel, On the wave front of a competition-diffusion system in popalation dynamics,, Nonlinear Analysis: Theory, 65 (2006), 301.
|
[9] |
J. I. Kanel and L. Zhou, Existence of wave front solutions and estimates of wave speed for a competition-diffusion system,, Nonlinear Analysis: Theory, 27 (1996), 579.
|
[10] |
Y. Kan-on, Note on propagation speed of travelling waves for a weakly coupled parabolic system,, Nonlinear Analysis, 44 (2001), 239.
doi: doi:10.1016/S0362-546X(99)00261-8. |
[11] |
Y. Kan-on, Note on propagation speed of travelling waves for a weakly coupled parabolic system,, Nonlinear Analysis: Theory, 44 (2001), 239.
|
[12] |
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. |
[13] |
G. A. Klaasen and W. Troy, The stability of traveling front solutions of a reaction-diffusion system,, SIAM J. Appl-. Math, 41 (1981), 145.
doi: doi:10.1137/0141011. |
[14] |
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. |
[15] |
A. Leung, X. Hou and Y. Li, Exclusive traveling waves for competitive reaction-diffusion systems and their stabilities,, J. Math. Anal. Appl., 338 (2008), 902.
doi: doi:10.1016/j.jmaa.2007.05.066. |
[16] |
A. Leung, "Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering,", MIA, (1989).
|
[17] |
A. Leung, "Nonlinear Systems of Partial Differential Equations: Applications to Life and Physical Sciences,", World Scientific, (2009).
doi: doi:10.1142/9789814277709. |
[18] |
S. Ma, X. Zhao, Global asymptotic stability of minimal fronts in monostable lattice equations,, Discrete Contin. Dyn. Syst., 21 (2008), 259.
doi: doi:10.3934/dcds.2008.21.259. |
[19] |
C. V. Pao, "Nonlinear Parabolic and Elliptic Equations,", Plenum Press, (1992).
|
[20] |
R. Pego and M. Weinstein, Eigenvalues and instabilities of solitary waves,, Phil. Trans. R soc. London A, 340 (1992), 47.
doi: doi:10.1098/rsta.1992.0055. |
[21] |
B. Sandstede, Stability of traveling waves,, in, (2002), 983.
doi: doi:10.1016/S1874-575X(02)80039-X. |
[22] |
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. |
[23] |
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. |
[24] |
T. Gallay, Local stability of critical fronts in nonlinear parabolic partial differential equations,, Nonlinearity, 7 (1994), 741.
doi: doi:10.1088/0951-7715/7/3/003. |
[25] |
A. Volpert, V. Volpert and V. Volpert, "Traveling Wave Solutions of Parabolic Systems,", Transl. Math. Monograhs, 140 (1994).
|
[26] |
J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay,, Journal of Dynamics and Differential Equations, 13 (2001), 651. |
[27] |
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.
|
[28] |
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.
|
[29] |
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. |
show all references
References:
[1] |
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.
|
[2] |
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. |
[3] |
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. |
[4] |
E. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,", McGraw-Hill, (1955). |
[5] |
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. |
[6] |
D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture notes in Mathematics, 840 (1981).
|
[7] |
Y. Hosono, Travelling waves for a diffusive Lotka-Volterra competition model I: Singular perturbations,, Discrete Continuous Dynamical Systems - B, 3 (2003), 79.
|
[8] |
J. I. Kanel, On the wave front of a competition-diffusion system in popalation dynamics,, Nonlinear Analysis: Theory, 65 (2006), 301.
|
[9] |
J. I. Kanel and L. Zhou, Existence of wave front solutions and estimates of wave speed for a competition-diffusion system,, Nonlinear Analysis: Theory, 27 (1996), 579.
|
[10] |
Y. Kan-on, Note on propagation speed of travelling waves for a weakly coupled parabolic system,, Nonlinear Analysis, 44 (2001), 239.
doi: doi:10.1016/S0362-546X(99)00261-8. |
[11] |
Y. Kan-on, Note on propagation speed of travelling waves for a weakly coupled parabolic system,, Nonlinear Analysis: Theory, 44 (2001), 239.
|
[12] |
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. |
[13] |
G. A. Klaasen and W. Troy, The stability of traveling front solutions of a reaction-diffusion system,, SIAM J. Appl-. Math, 41 (1981), 145.
doi: doi:10.1137/0141011. |
[14] |
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. |
[15] |
A. Leung, X. Hou and Y. Li, Exclusive traveling waves for competitive reaction-diffusion systems and their stabilities,, J. Math. Anal. Appl., 338 (2008), 902.
doi: doi:10.1016/j.jmaa.2007.05.066. |
[16] |
A. Leung, "Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering,", MIA, (1989).
|
[17] |
A. Leung, "Nonlinear Systems of Partial Differential Equations: Applications to Life and Physical Sciences,", World Scientific, (2009).
doi: doi:10.1142/9789814277709. |
[18] |
S. Ma, X. Zhao, Global asymptotic stability of minimal fronts in monostable lattice equations,, Discrete Contin. Dyn. Syst., 21 (2008), 259.
doi: doi:10.3934/dcds.2008.21.259. |
[19] |
C. V. Pao, "Nonlinear Parabolic and Elliptic Equations,", Plenum Press, (1992).
|
[20] |
R. Pego and M. Weinstein, Eigenvalues and instabilities of solitary waves,, Phil. Trans. R soc. London A, 340 (1992), 47.
doi: doi:10.1098/rsta.1992.0055. |
[21] |
B. Sandstede, Stability of traveling waves,, in, (2002), 983.
doi: doi:10.1016/S1874-575X(02)80039-X. |
[22] |
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. |
[23] |
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. |
[24] |
T. Gallay, Local stability of critical fronts in nonlinear parabolic partial differential equations,, Nonlinearity, 7 (1994), 741.
doi: doi:10.1088/0951-7715/7/3/003. |
[25] |
A. Volpert, V. Volpert and V. Volpert, "Traveling Wave Solutions of Parabolic Systems,", Transl. Math. Monograhs, 140 (1994).
|
[26] |
J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay,, Journal of Dynamics and Differential Equations, 13 (2001), 651. |
[27] |
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.
|
[28] |
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.
|
[29] |
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. |
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