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 and Continuous Dynamical Systems - B, 2011, 15 (1) : 171-196. doi: 10.3934/dcdsb.2011.15.171
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.

[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.

[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.

[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), 503-524. doi: doi:10.1016/S1468-1218(02)00077-9.

[6]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture notes in Mathematics, 840, Springer-Verlag, 1981.

[7]

Y. Hosono, Travelling waves for a diffusive Lotka-Volterra competition model I: Singular perturbations, Discrete Continuous Dynamical Systems - B, 3 (2003), 79-95.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

A. Leung, "Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering," MIA, Kluwer, Boston, 1989.

[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.

[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.

[19]

C. V. Pao, "Nonlinear Parabolic and Elliptic Equations," Plenum Press, N. Y., 1992.

[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.

[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.

[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.

[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.

[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.

[25]

A. Volpert, V. Volpert and V. Volpert, "Traveling Wave Solutions of Parabolic Systems," Transl. Math. Monograhs, 140, Amer. Math. Soc., Providence, RI. 1994.

[26]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, Journal of Dynamics and Differential Equations, 13 (2001), 651-687, and Erratum to traveling wave fronts of reaction-diffusion systems with delays, Journal of Dynamics and Differential Equations, 20 (2008), 531-533.

[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.

[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.

[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.

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.

[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.

[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.

[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), 503-524. doi: doi:10.1016/S1468-1218(02)00077-9.

[6]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture notes in Mathematics, 840, Springer-Verlag, 1981.

[7]

Y. Hosono, Travelling waves for a diffusive Lotka-Volterra competition model I: Singular perturbations, Discrete Continuous Dynamical Systems - B, 3 (2003), 79-95.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

A. Leung, "Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering," MIA, Kluwer, Boston, 1989.

[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.

[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.

[19]

C. V. Pao, "Nonlinear Parabolic and Elliptic Equations," Plenum Press, N. Y., 1992.

[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.

[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.

[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.

[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.

[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.

[25]

A. Volpert, V. Volpert and V. Volpert, "Traveling Wave Solutions of Parabolic Systems," Transl. Math. Monograhs, 140, Amer. Math. Soc., Providence, RI. 1994.

[26]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, Journal of Dynamics and Differential Equations, 13 (2001), 651-687, and Erratum to traveling wave fronts of reaction-diffusion systems with delays, Journal of Dynamics and Differential Equations, 20 (2008), 531-533.

[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.

[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.

[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.

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