American Institute of Mathematical Sciences

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October  2011, 16(3): 973-984. doi: 10.3934/dcdsb.2011.16.973

Existence of traveling wavefront for discrete bistable competition model

Chin-Chin Wu 1,

1. 

Department of Applied Mathematics, National Chung Hsing University, 250, Kuo Kuang Road, Taichung 402, Taiwan

Received  September 2010 Revised  January 2011 Published  June 2011

We study traveling wavefront solutions for a two-component competition system on a one-dimensional lattice. We combine the monotonic iteration method with a truncation to obtain the existence of the traveling wavefront solution.
Keywords: Lattice dynamical system, competition model., bistable, traveling wave.
Mathematics Subject Classification: Primary: 34K05, 34A34; Secondary: 34K60, 34E0.
Citation: Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973
References:
[1]

X. Chen, J. S. Guo and C. C. Wu, Traveling waves in discrete periodic media for bistable dynamics, Arch. Rational Mech. Anal., 189 (2008), 189-236. doi: 10.1007/s00205-007-0103-3.

[2]

C. Conley and R. Gardner, An application of the generalized Morse index to traveling wave solutions of a competitive reaction diffusion model, Indiana Univ. Math. J., 33 (1984), 319-343. doi: 10.1512/iumj.1984.33.33018.

[3]

R. A. Gardner, Existence and stability of traveling wave solutions of competition models: a degree theoretic, J. Differential Equations, 44 (1982), 343-364. doi: 10.1016/0022-0396(82)90001-8.

[4]

J. S. Guo and C. H. Wu, Traveling wave front for a two-component lattice dynamical system arising in competition models, preprint.

[5]

J. S. Guo and C. H. Wu, Wave propagation for a two-component lattice dynamical system arising in strong competition models, J. Differential Equations, 250 (2011), 3504-3533. doi: 10.1016/j.jde.2010.12.004.

[6]

Y. Hosono, Singular perturbation analysis of travelling waves for diffusive Lotka-Volterra competitive models, Numerical and Applied Mathematics, Part II (Paris, 1988), Baltzer, Basel, (1989), 687-692.

[7]

Y. Hosono, The minimal speed of traveling fronts for a diffusive Lotka-Volterra competition model, Bulletin of Math. Biology, 60 (1998), 435-448. doi: 10.1006/bulm.1997.0008.

[8]

Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion, Nonlinear Anal., 28 (1997), 145-164.

[9]

J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572. doi: 10.1137/0147038.

[10]

A. Okubo, P. K. Maini, M. H. Williamson and J. D. Murray, On the spatial spread of the grey squirrel in Britain, Proc. R. Soc. Lond. B, 238 (1989), 113-125. doi: 10.1098/rspb.1989.0070.

[11]

M. M. Tang and P. C. Fife, Propagating fronts for competing species equations with diffusion, Arch. Rational Mech. Anal., 73 (1980), 69-77. doi: 10.1007/BF00283257.

show all references

References:
[1]

X. Chen, J. S. Guo and C. C. Wu, Traveling waves in discrete periodic media for bistable dynamics, Arch. Rational Mech. Anal., 189 (2008), 189-236. doi: 10.1007/s00205-007-0103-3.

[2]

C. Conley and R. Gardner, An application of the generalized Morse index to traveling wave solutions of a competitive reaction diffusion model, Indiana Univ. Math. J., 33 (1984), 319-343. doi: 10.1512/iumj.1984.33.33018.

[3]

R. A. Gardner, Existence and stability of traveling wave solutions of competition models: a degree theoretic, J. Differential Equations, 44 (1982), 343-364. doi: 10.1016/0022-0396(82)90001-8.

[4]

J. S. Guo and C. H. Wu, Traveling wave front for a two-component lattice dynamical system arising in competition models, preprint.

[5]

J. S. Guo and C. H. Wu, Wave propagation for a two-component lattice dynamical system arising in strong competition models, J. Differential Equations, 250 (2011), 3504-3533. doi: 10.1016/j.jde.2010.12.004.

[6]

Y. Hosono, Singular perturbation analysis of travelling waves for diffusive Lotka-Volterra competitive models, Numerical and Applied Mathematics, Part II (Paris, 1988), Baltzer, Basel, (1989), 687-692.

[7]

Y. Hosono, The minimal speed of traveling fronts for a diffusive Lotka-Volterra competition model, Bulletin of Math. Biology, 60 (1998), 435-448. doi: 10.1006/bulm.1997.0008.

[8]

Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion, Nonlinear Anal., 28 (1997), 145-164.

[9]

J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572. doi: 10.1137/0147038.

[10]

A. Okubo, P. K. Maini, M. H. Williamson and J. D. Murray, On the spatial spread of the grey squirrel in Britain, Proc. R. Soc. Lond. B, 238 (1989), 113-125. doi: 10.1098/rspb.1989.0070.

[11]

M. M. Tang and P. C. Fife, Propagating fronts for competing species equations with diffusion, Arch. Rational Mech. Anal., 73 (1980), 69-77. doi: 10.1007/BF00283257.

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