American Institute of Mathematical Sciences

Advanced
March  2013, 8(1): 379-395. doi: 10.3934/nhm.2013.8.379

Traveling fronts of pyramidal shapes in competition-diffusion systems

Wei-Ming Ni 1, and  Masaharu Taniguchi 2,

1. 

Center for Partial Differential Equations, East China Normal University, Minhang, Shanghai, 200241, China
2. 

Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1-W8-38 Ookayama, Meguro-ku, Tokyo 152-8552

Received  January 2012 Revised  March 2013 Published  April 2013

It is well known that a competition-diffusion system has a one-dimensional traveling front. This paper studies traveling front solutions of pyramidal shapes in a competition-diffusion system in $\mathbb{R}^N$ with $N\geq 2$. By using a multi-scale method, we construct a suitable pair of a supersolution and a subsolution, and find a pyramidal traveling front solution between them.
Keywords: pyramidal shapes, Traveling front, competition-diffusion system..
Mathematics Subject Classification: Primary: 35C07; Secondary: 35K5.
Citation: 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
References:
[1]

J.-S. Guo and X. Liang, The minimal speed of traveling fronts for the Lotka-Volterra competition system, J. Dynam. Differential Equations, 23 (2011), 353-363. doi: 10.1007/s10884-011-9214-5.

[2]

J. K. Hale, "Ordinary Differential Equations," Wiley-Interscience, 1969.

[3]

F. Hamel, R. Monneau and J.-M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts, Discrete Contin. Dyn. Syst., 13 (2005), 1069-1096. doi: 10.3934/dcds.2005.13.1069.

[4]

F. Hamel, R. Monneau and J.-M. Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Discrete Contin. Dyn. Syst., 14 (2006), 75-92.

[5]

M. Haragus and A. Scheel, Corner defects in almost planar interface propagation, Ann. I. H. Poincaré, 23 (2006), 283-329. doi: 10.1016/j.anihpc.2005.03.003.

[6]

Y. Hosono, Traveling waves for a diffusive Lotka-Volterra competition model. I. Singular perturbations, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 79-95. doi: 10.3934/dcdsb.2003.3.79.

[7]

Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363. doi: 10.1137/S0036141093244556.

[8]

Y. Kan-on and Q. Fang, Stability of monotone travelling waves for competition-diffusion equations, Japan J. Indust. Appl. Math., 13 (1996), 343-349. doi: 10.1007/BF03167252.

[9]

Y. Kurokawa and M. Taniguchi, Multi-dimensional pyramidal traveling fronts in the Allen-Cahn equations, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 1031-1054. doi: 10.1017/S0308210510001253.

[10]

H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations, J. Differential Equations, 213 (2005), 204-233. doi: 10.1016/j.jde.2004.06.011.

[11]

H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Discrete Contin. Dyn. Syst., 15 (2006), 819-832. doi: 10.3934/dcds.2006.15.819.

[12]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Springer-Verlag, Berlin, 1984. doi: 10.1007/978-1-4612-5282-5.

[13]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1972), 979-1000.

[14]

M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344. doi: 10.1137/060661788.

[15]

M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Differential Equations, 246 (2009), 2103-2130. doi: 10.1016/j.jde.2008.06.037.

[16]

M. Taniguchi, Multi-dimensional traveling fronts in bistable reaction-diffusion equations, Discrete Contin. Dyn. Syst., 32 (2012), 1011-1046. doi: 10.3934/dcds.2012.32.1011.

[17]

A. I. Volpert, V. A. Volpert and V. A. Volpert, "Traveling Wave Solutions of Parabolic Systems," Translations of Mathematical Monographs, 140, American Mathematical Society, Providence, RI}, 1994.

[18]

Z.-C. Wang, Traveling curved fronts in monotone bistable systems, Discrete Contin. Dyn. Syst., 32 (2012), 2339-2374. doi: 10.3934/dcds.2012.32.2339.

show all references

References:
[1]

J.-S. Guo and X. Liang, The minimal speed of traveling fronts for the Lotka-Volterra competition system, J. Dynam. Differential Equations, 23 (2011), 353-363. doi: 10.1007/s10884-011-9214-5.

[2]

J. K. Hale, "Ordinary Differential Equations," Wiley-Interscience, 1969.

[3]

F. Hamel, R. Monneau and J.-M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts, Discrete Contin. Dyn. Syst., 13 (2005), 1069-1096. doi: 10.3934/dcds.2005.13.1069.

[4]

F. Hamel, R. Monneau and J.-M. Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Discrete Contin. Dyn. Syst., 14 (2006), 75-92.

[5]

M. Haragus and A. Scheel, Corner defects in almost planar interface propagation, Ann. I. H. Poincaré, 23 (2006), 283-329. doi: 10.1016/j.anihpc.2005.03.003.

[6]

Y. Hosono, Traveling waves for a diffusive Lotka-Volterra competition model. I. Singular perturbations, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 79-95. doi: 10.3934/dcdsb.2003.3.79.

[7]

Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363. doi: 10.1137/S0036141093244556.

[8]

Y. Kan-on and Q. Fang, Stability of monotone travelling waves for competition-diffusion equations, Japan J. Indust. Appl. Math., 13 (1996), 343-349. doi: 10.1007/BF03167252.

[9]

Y. Kurokawa and M. Taniguchi, Multi-dimensional pyramidal traveling fronts in the Allen-Cahn equations, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 1031-1054. doi: 10.1017/S0308210510001253.

[10]

H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations, J. Differential Equations, 213 (2005), 204-233. doi: 10.1016/j.jde.2004.06.011.

[11]

H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Discrete Contin. Dyn. Syst., 15 (2006), 819-832. doi: 10.3934/dcds.2006.15.819.

[12]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Springer-Verlag, Berlin, 1984. doi: 10.1007/978-1-4612-5282-5.

[13]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1972), 979-1000.

[14]

M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344. doi: 10.1137/060661788.

[15]

M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Differential Equations, 246 (2009), 2103-2130. doi: 10.1016/j.jde.2008.06.037.

[16]

M. Taniguchi, Multi-dimensional traveling fronts in bistable reaction-diffusion equations, Discrete Contin. Dyn. Syst., 32 (2012), 1011-1046. doi: 10.3934/dcds.2012.32.1011.

[17]

A. I. Volpert, V. A. Volpert and V. A. Volpert, "Traveling Wave Solutions of Parabolic Systems," Translations of Mathematical Monographs, 140, American Mathematical Society, Providence, RI}, 1994.

[18]

Z.-C. Wang, Traveling curved fronts in monotone bistable systems, Discrete Contin. Dyn. Syst., 32 (2012), 2339-2374. doi: 10.3934/dcds.2012.32.2339.

[1]

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
[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]

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
[5]

Zhi-Cheng Wang, Hui-Ling Niu, Shigui Ruan. On the existence of axisymmetric traveling fronts in Lotka-Volterra competition-diffusion systems in ℝ3. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 1111-1144. doi: 10.3934/dcdsb.2017055
[6]

Yang Wang, Xiong Li. Uniqueness of traveling front solutions for the Lotka-Volterra system in the weak competition case. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3067-3075. doi: 10.3934/dcdsb.2018300
[7]

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
[8]

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
[9]

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
[10]

Yuri Latushkin, Roland Schnaubelt, Xinyao Yang. Stable foliations near a traveling front for reaction diffusion systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3145-3165. doi: 10.3934/dcdsb.2017168
[11]

Lia Bronsard, Seong-A Shim. Long-time behavior for competition-diffusion systems via viscosity comparison. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 561-581. doi: 10.3934/dcds.2005.13.561
[12]

Shuling Yan, Shangjiang Guo. Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1559-1579. doi: 10.3934/dcdsb.2018059
[13]

Danielle Hilhorst, Masato Iida, Masayasu Mimura, Hirokazu Ninomiya. Relative compactness in $L^p$ of solutions of some 2m components competition-diffusion systems. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 233-244. doi: 10.3934/dcds.2008.21.233
[14]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035
[15]

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
[16]

Hongmei Cheng, Rong Yuan. Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 1015-1029. doi: 10.3934/dcdsb.2015.20.1015
[17]

Zhen-Hui Bu, Zhi-Cheng Wang. Stability of pyramidal traveling fronts in the degenerate monostable and combustion equations Ⅰ. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2395-2430. doi: 10.3934/dcds.2017104
[18]

Xiaojie Hou, Wei Feng. Traveling waves and their stability in a coupled reaction diffusion system. Communications on Pure and Applied Analysis, 2011, 10 (1) : 141-160. doi: 10.3934/cpaa.2011.10.141
[19]

Radosław Kurek, Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1955-1981. doi: 10.3934/dcds.2018079
[20]

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

2021 Impact Factor: 1.41

Tools

Metrics

  • PDF downloads (122)
  • HTML views (0)
  • Cited by (20)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy