November  2012, 17(8): 2653-2669. doi: 10.3934/dcdsb.2012.17.2653

Exact travelling wave solutions of three-species competition--diffusion systems

1. 

Department of Mathematics, National Taiwan University, and National Center for Theoretical Sciences (Taipei Office), No. 1, Sec. 4, Roosevelt Road, Taipei, 10617, Taiwan

2. 

Department of Mathematics, National Taiwan University, and National Center for Theoretical Sciences (Taipei Office), No. 1, Sec. 4, Roosevelt Road, Taipei, 10617, Taiwan

3. 

Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, 1-1-1 Higashimita, Tamaku, Kawasaki 214-8571, Japan

4. 

Meiji Institute for Advanced Study of Mathematical Sciences, and Graduate School of Advanced Mathematical Sciences, Meiji University, 1-1-1 Higashimita, Tamaku, Kawasaki 214-8571, Japan

Received  March 2011 Revised  September 2011 Published  July 2012

We consider the problem where $W$ invades the $(U,V)$ system in the three species Lotka-Volterra competition-diffusion model. Numerical simulation indicates that the presence of $W$ can dramatically change the competitive interaction between $U$ and $V$ in some parameter range if the invading $W$ is not too small. We also construct exact travelling wave solutions with non-trivial three components and track the bifurcation branches of these solutions by AUTO.
Citation: Chiun-Chuan Chen, Li-Chang Hung, Masayasu Mimura, Daishin Ueyama. Exact travelling wave solutions of three-species competition--diffusion systems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (8) : 2653-2669. doi: 10.3934/dcdsb.2012.17.2653
References:
[1]

E. J. Doedel, B. E. Oldeman, A. R. Champneys, F. Dercole, T. F. Fairgrieve, Y. Kuznetsov, R. C. Paffenroth, B. Sandstede, X. J. Wang and C. H. Zhang, "AUTO-07p: Continuation and Bifurcation Software for Ordinary Differential Equations," Concordia Univ.,, Version 0.7, (2010).   Google Scholar

[2]

S.-I. Ei, R. Ikota and M. Mimura, Segregating partition problem in competition-diffusion systems,, J. Interfaces and Free Boundaries, 1 (1999), 57.   Google Scholar

[3]

M. W. Hirsch, Stability and convergence in strongly monotone dynamical systems,, J. Reine Angew. Math., 383 (1988), 1.   Google Scholar

[4]

J. Hofbauer and K. Sigmund, "Evolutionary Games and Population Dynamics,", Cambridge University Press, (1998).   Google Scholar

[5]

, H. Ikeda,, Unpublished., ().   Google Scholar

[6]

Y. Kannon, Traveling waves in systems of two competing species,, Sūgaku, 49 (1997), 379.   Google Scholar

[7]

K. Kishimoto and H. F. Weinberger, The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains,, J. Differential Equations, 58 (1985), 15.  doi: 10.1016/0022-0396(85)90020-8.  Google Scholar

[8]

M. Rodrigo and M. Mimura, Exact solutions of a competition-diffusion system,, Hiroshima Math. J., 30 (2000), 257.   Google Scholar

[9]

M. Rodrigo and M. Mimura, Exact solutions of reaction-diffusion systems and nonlinear wave equations,, Japan J. Indust. Appl. Math., 18 (2001), 657.  doi: 10.1007/BF03167410.  Google Scholar

[10]

S. Wolfram, "Mathematica: A System for Doing Mathematics by Computer,", Addison-Wesley Longman Publishing Co., (1988).   Google Scholar

[11]

M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems,, Dynam. Stability Systems, 8 (1993), 189.   Google Scholar

show all references

References:
[1]

E. J. Doedel, B. E. Oldeman, A. R. Champneys, F. Dercole, T. F. Fairgrieve, Y. Kuznetsov, R. C. Paffenroth, B. Sandstede, X. J. Wang and C. H. Zhang, "AUTO-07p: Continuation and Bifurcation Software for Ordinary Differential Equations," Concordia Univ.,, Version 0.7, (2010).   Google Scholar

[2]

S.-I. Ei, R. Ikota and M. Mimura, Segregating partition problem in competition-diffusion systems,, J. Interfaces and Free Boundaries, 1 (1999), 57.   Google Scholar

[3]

M. W. Hirsch, Stability and convergence in strongly monotone dynamical systems,, J. Reine Angew. Math., 383 (1988), 1.   Google Scholar

[4]

J. Hofbauer and K. Sigmund, "Evolutionary Games and Population Dynamics,", Cambridge University Press, (1998).   Google Scholar

[5]

, H. Ikeda,, Unpublished., ().   Google Scholar

[6]

Y. Kannon, Traveling waves in systems of two competing species,, Sūgaku, 49 (1997), 379.   Google Scholar

[7]

K. Kishimoto and H. F. Weinberger, The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains,, J. Differential Equations, 58 (1985), 15.  doi: 10.1016/0022-0396(85)90020-8.  Google Scholar

[8]

M. Rodrigo and M. Mimura, Exact solutions of a competition-diffusion system,, Hiroshima Math. J., 30 (2000), 257.   Google Scholar

[9]

M. Rodrigo and M. Mimura, Exact solutions of reaction-diffusion systems and nonlinear wave equations,, Japan J. Indust. Appl. Math., 18 (2001), 657.  doi: 10.1007/BF03167410.  Google Scholar

[10]

S. Wolfram, "Mathematica: A System for Doing Mathematics by Computer,", Addison-Wesley Longman Publishing Co., (1988).   Google Scholar

[11]

M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems,, Dynam. Stability Systems, 8 (1993), 189.   Google Scholar

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