# 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 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 [5] , H. Ikeda, Unpublished.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.
 [1] Chaohong Pan, Hongyong Wang, Chunhua Ou. Invasive speed for a competition-diffusion system with three species. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3515-3532. doi: 10.3934/dcdsb.2021194 [2] Daozhou Gao, Xing Liang. A competition-diffusion system with a refuge. Discrete and Continuous Dynamical Systems - B, 2007, 8 (2) : 435-454. doi: 10.3934/dcdsb.2007.8.435 [3] Jong-Shenq Guo, Ying-Chih Lin. The sign of the wave speed for the Lotka-Volterra competition-diffusion system. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2083-2090. doi: 10.3934/cpaa.2013.12.2083 [4] Hirofumi Izuhara, Shunsuke Kobayashi. Spatio-temporal coexistence in the cross-diffusion competition system. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 919-933. doi: 10.3934/dcdss.2020228 [5] Nahla Abdellatif, Radhouane Fekih-Salem, Tewfik Sari. Competition for a single resource and coexistence of several species in the chemostat. Mathematical Biosciences & Engineering, 2016, 13 (4) : 631-652. doi: 10.3934/mbe.2016012 [6] Yuan Lou, Daniel Munther. Dynamics of a three species competition model. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3099-3131. doi: 10.3934/dcds.2012.32.3099 [7] Xiaojie Hou, Yi Li. Traveling waves in a three species competition-cooperation system. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1103-1120. doi: 10.3934/cpaa.2017053 [8] M. Guedda, R. Kersner, M. Klincsik, E. Logak. Exact wavefronts and periodic patterns in a competition system with nonlinear diffusion. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1589-1600. doi: 10.3934/dcdsb.2014.19.1589 [9] Yukio Kan-On. Structure on the set of radially symmetric positive stationary solutions for a competition-diffusion system. Conference Publications, 2013, 2013 (special) : 427-436. doi: 10.3934/proc.2013.2013.427 [10] E. C.M. Crooks, E. N. Dancer, Danielle Hilhorst. Fast reaction limit and long time behavior for a competition-diffusion system with Dirichlet boundary conditions. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 39-44. doi: 10.3934/dcdsb.2007.8.39 [11] Qian Guo, Xiaoqing He, Wei-Ming Ni. Global dynamics of a general Lotka-Volterra competition-diffusion system in heterogeneous environments. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6547-6573. doi: 10.3934/dcds.2020290 [12] Xiongxiong Bao, Wan-Tong Li, Zhi-Cheng Wang. Uniqueness and stability of time-periodic pyramidal fronts for a periodic competition-diffusion system. Communications on Pure and Applied Analysis, 2020, 19 (1) : 253-277. doi: 10.3934/cpaa.2020014 [13] Yoshihisa Morita, Ken-Ichi Nakamura, Toshiko Ogiwara. Front propagation and blocking for the competition-diffusion system in a domain of half-lines with a junction. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022136 [14] Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6185-6205. doi: 10.3934/dcdsb.2021013 [15] Hua Nie, Sze-Bi Hsu, Jianhua Wu. Coexistence solutions of a competition model with two species in a water column. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2691-2714. doi: 10.3934/dcdsb.2015.20.2691 [16] Wei-Ming Ni, Masaharu Taniguchi. Traveling fronts of pyramidal shapes in competition-diffusion systems. Networks and Heterogeneous Media, 2013, 8 (1) : 379-395. doi: 10.3934/nhm.2013.8.379 [17] Benlong Xu, Hongyan Jiang. Invasion and coexistence of competition-diffusion-advection system with heterogeneous vs homogeneous resources. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4255-4266. doi: 10.3934/dcdsb.2018136 [18] Guo-Bao Zhang, Fang-Di Dong, Wan-Tong Li. Uniqueness and stability of traveling waves for a three-species competition system with nonlocal dispersal. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1511-1541. doi: 10.3934/dcdsb.2018218 [19] Anton S. Zadorin. Exact travelling solution for a reaction-diffusion system with a piecewise constant production supported by a codimension-1 subspace. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1567-1580. doi: 10.3934/cpaa.2022030 [20] Bo Duan, Zhengce Zhang. A two-species weak competition system of reaction-diffusion-advection with double free boundaries. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 801-829. doi: 10.3934/dcdsb.2018208

2021 Impact Factor: 1.497