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

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Preface

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

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

Chiun-Chuan Chen ^1, , Li-Chang Hung ^2, , Masayasu Mimura ^3, and Daishin Ueyama ^4,

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

Abstract
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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.

Keywords: travelling wave, exact solution, bifurcation diagram, Three species competition-diffusion system, competition-mediated coexistence..

Mathematics Subject Classification: Primary: 35C07, 35K57; Secondary: 92B05, 34C23, 35C0.

Citation: Chiun-Chuan Chen, Li-Chang Hung, Masayasu Mimura, Daishin Ueyama. Exact travelling wave solutions of three-species competition--diffusion systems. Discrete and 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. Available from: http://sourceforge.net/projects/auto-07p/files/auto07p/
[2]	S.-I. Ei, R. Ikota and M. Mimura, Segregating partition problem in competition-diffusion systems, J. Interfaces and Free Boundaries, 1 (1999), 57-80.
[3]	M. W. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math., 383 (1988), 1-53.
[4]	J. Hofbauer and K. Sigmund, "Evolutionary Games and Population Dynamics," Cambridge University Press, Cambridge, 1998.
[5]	, H. Ikeda, Unpublished.
[6]	Y. Kannon, Traveling waves in systems of two competing species, Sūgaku, 49 (1997), 379-392.
[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), 15-21. doi: 10.1016/0022-0396(85)90020-8.
[8]	M. Rodrigo and M. Mimura, Exact solutions of a competition-diffusion system, Hiroshima Math. J., 30 (2000), 257-270.
[9]	M. Rodrigo and M. Mimura, Exact solutions of reaction-diffusion systems and nonlinear wave equations, Japan J. Indust. Appl. Math., 18 (2001), 657-696. doi: 10.1007/BF03167410.
[10]	S. Wolfram, "Mathematica: A System for Doing Mathematics by Computer," Addison-Wesley Longman Publishing Co., Inc., Boston, MA, 1988.
[11]	M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems, 8 (1993), 189-217.

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. Available from: http://sourceforge.net/projects/auto-07p/files/auto07p/
[2]	S.-I. Ei, R. Ikota and M. Mimura, Segregating partition problem in competition-diffusion systems, J. Interfaces and Free Boundaries, 1 (1999), 57-80.
[3]	M. W. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math., 383 (1988), 1-53.
[4]	J. Hofbauer and K. Sigmund, "Evolutionary Games and Population Dynamics," Cambridge University Press, Cambridge, 1998.
[5]	, H. Ikeda, Unpublished.
[6]	Y. Kannon, Traveling waves in systems of two competing species, Sūgaku, 49 (1997), 379-392.
[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), 15-21. doi: 10.1016/0022-0396(85)90020-8.
[8]	M. Rodrigo and M. Mimura, Exact solutions of a competition-diffusion system, Hiroshima Math. J., 30 (2000), 257-270.
[9]	M. Rodrigo and M. Mimura, Exact solutions of reaction-diffusion systems and nonlinear wave equations, Japan J. Indust. Appl. Math., 18 (2001), 657-696. doi: 10.1007/BF03167410.
[10]	S. Wolfram, "Mathematica: A System for Doing Mathematics by Computer," Addison-Wesley Longman Publishing Co., Inc., Boston, MA, 1988.
[11]	M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems, 8 (1993), 189-217.

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