# American Institute of Mathematical Sciences

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 Office), No. 1, Sec. 4, Roosevelt Road, Taipei, 10617, Taiwan 2 Department of Mathematics, National Taiwan University, and National Center for Theoretical Sciences (Taipei Office), 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:
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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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