# American Institute of Mathematical Sciences

January  2011, 15(1): 171-196. doi: 10.3934/dcdsb.2011.15.171

## 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

Received  December 2009 Revised  September 2010 Published  October 2010

Using a new method of monotone iteration of a pair of smooth lower- and upper-solutions, the traveling wave solutions of the classical Lotka-Volterra system are shown to exist for a family of wave speeds. Such constructed upper and lower solution pair enables us to derive the explicit value of the minimal (critical) wave speed as well as the asymptotic decay/growth rates of the wave solutions at infinities. Furthermore, the traveling wave corresponding to each wave speed is unique up to a translation of the origin. The stability of the traveling wave solutions with non-critical wave speed is also studied by spectral analysis of a linearized operator in exponentially weighted Banach spaces.
Citation: Anthony W. Leung, Xiaojie Hou, Wei Feng. Traveling wave solutions for Lotka-Volterra system re-visited. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 171-196. doi: 10.3934/dcdsb.2011.15.171
##### References:
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London A, 340 (1992), 47-94. doi: doi:10.1098/rsta.1992.0055.  Google Scholar [21] B. Sandstede, Stability of traveling waves, in "Handbook of Dynamical Systems II" (B Fiedler, ed.), North-Holland, (2002), 983-1055. doi: doi:10.1016/S1874-575X(02)80039-X.  Google Scholar [22] D. Sattinger, On the stability of traveling waves of nonlinear parabolic systems, Advances in Mathematics, 22 (1976), 312-355. doi: doi:10.1016/0001-8708(76)90098-0.  Google Scholar [23] M. M. Tang and P. C. Fife, Propagating fronts for competing species equations with diffusion, Arch. Rat. Mech. Anal., 73 (1980), 69-77. doi: doi:10.1007/BF00283257.  Google Scholar [24] T. Gallay, Local stability of critical fronts in nonlinear parabolic partial differential equations, Nonlinearity, 7 (1994), 741-764. doi: doi:10.1088/0951-7715/7/3/003.  Google Scholar [25] A. Volpert, V. Volpert and V. Volpert, "Traveling Wave Solutions of Parabolic Systems," Transl. Math. Monograhs, 140, Amer. Math. Soc., Providence, RI. 1994.  Google Scholar [26] J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, Journal of Dynamics and Differential Equations, 13 (2001), 651-687, and Erratum to traveling wave fronts of reaction-diffusion systems with delays, Journal of Dynamics and Differential Equations, 20 (2008), 531-533. Google Scholar [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), 149-170.  Google Scholar [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), 47-66.  Google Scholar [29] D. Xu and X. Q. Zhao, Bistable waves in an epidemic model, Journal of Dynamics and Differential Equations, 16 (2004), 679-707, and Erratum, Journal of Dynamics and Differential Equations, 17 (2005), 219-247. doi: doi:10.1007/s10884-005-6294-0.  Google Scholar

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), 167-212.  Google Scholar [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), 45-57. doi: doi:10.1016/S0022-247X(02)00205-6.  Google Scholar [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), 1551-1570. doi: doi:10.1016/j.jde.2008.01.004.  Google Scholar [4] E. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGraw-Hill, 1955. Google Scholar [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), 503-524. doi: doi:10.1016/S1468-1218(02)00077-9.  Google Scholar [6] D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture notes in Mathematics, 840, Springer-Verlag, 1981.  Google Scholar [7] Y. Hosono, Travelling waves for a diffusive Lotka-Volterra competition model I: Singular perturbations, Discrete Continuous Dynamical Systems - B, 3 (2003), 79-95.  Google Scholar [8] J. I. Kanel, On the wave front of a competition-diffusion system in popalation dynamics, Nonlinear Analysis: Theory, Methods & Applications, 65 (2006), 301-320.  Google Scholar [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, Methods & Applications, 27 (1996), 579-587.  Google Scholar [10] Y. Kan-on, Note on propagation speed of travelling waves for a weakly coupled parabolic system, Nonlinear Analysis, 44 (2001), 239-246. doi: doi:10.1016/S0362-546X(99)00261-8.  Google Scholar [11] Y. Kan-on, Note on propagation speed of travelling waves for a weakly coupled parabolic system, Nonlinear Analysis: Theory, Methods & Applications, 44 (2001), 239-246.  Google Scholar [12] T. Kapitula, On the stability of Traveling waves in weighted $L^{\infty}$ spaces, Journal of Differential Equations, 112 (1994), 179-215. doi: doi:10.1006/jdeq.1994.1100.  Google Scholar [13] G. A. Klaasen and W. Troy, The stability of traveling front solutions of a reaction-diffusion system, SIAM J. Appl-. Math, Vol., 41 (1981), 145-167. doi: doi:10.1137/0141011.  Google Scholar [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), 1-72. Google Scholar [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), 902-924. doi: doi:10.1016/j.jmaa.2007.05.066.  Google Scholar [16] A. Leung, "Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering," MIA, Kluwer, Boston, 1989.  Google Scholar [17] A. Leung, "Nonlinear Systems of Partial Differential Equations: Applications to Life and Physical Sciences," World Scientific, New Jersey, Singapore, London, 2009. doi: doi:10.1142/9789814277709.  Google Scholar [18] S. Ma, X. Zhao, Global asymptotic stability of minimal fronts in monostable lattice equations, Discrete Contin. Dyn. Syst., 21 (2008), 259-275. doi: doi:10.3934/dcds.2008.21.259.  Google Scholar [19] C. V. Pao, "Nonlinear Parabolic and Elliptic Equations," Plenum Press, N. Y., 1992.  Google Scholar [20] R. Pego and M. Weinstein, Eigenvalues and instabilities of solitary waves, Phil. Trans. R soc. London A, 340 (1992), 47-94. doi: doi:10.1098/rsta.1992.0055.  Google Scholar [21] B. Sandstede, Stability of traveling waves, in "Handbook of Dynamical Systems II" (B Fiedler, ed.), North-Holland, (2002), 983-1055. doi: doi:10.1016/S1874-575X(02)80039-X.  Google Scholar [22] D. Sattinger, On the stability of traveling waves of nonlinear parabolic systems, Advances in Mathematics, 22 (1976), 312-355. doi: doi:10.1016/0001-8708(76)90098-0.  Google Scholar [23] M. M. Tang and P. C. Fife, Propagating fronts for competing species equations with diffusion, Arch. Rat. Mech. Anal., 73 (1980), 69-77. doi: doi:10.1007/BF00283257.  Google Scholar [24] T. Gallay, Local stability of critical fronts in nonlinear parabolic partial differential equations, Nonlinearity, 7 (1994), 741-764. doi: doi:10.1088/0951-7715/7/3/003.  Google Scholar [25] A. Volpert, V. Volpert and V. Volpert, "Traveling Wave Solutions of Parabolic Systems," Transl. Math. Monograhs, 140, Amer. Math. Soc., Providence, RI. 1994.  Google Scholar [26] J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, Journal of Dynamics and Differential Equations, 13 (2001), 651-687, and Erratum to traveling wave fronts of reaction-diffusion systems with delays, Journal of Dynamics and Differential Equations, 20 (2008), 531-533. Google Scholar [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), 149-170.  Google Scholar [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), 47-66.  Google Scholar [29] D. Xu and X. Q. Zhao, Bistable waves in an epidemic model, Journal of Dynamics and Differential Equations, 16 (2004), 679-707, and Erratum, Journal of Dynamics and Differential Equations, 17 (2005), 219-247. doi: doi:10.1007/s10884-005-6294-0.  Google Scholar
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