-
Previous Article
Individual based models and differential equations models of nosocomial epidemics in hospital intensive care units
- DCDS-B Home
- This Issue
-
Next Article
On carrying-capacity construction, metapopulations and density-dependent mortality
On the existence of axisymmetric traveling fronts in Lotka-Volterra competition-diffusion systems in ℝ3
1. | School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China |
2. | School of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan, Ningxia 750021, China |
3. | Department of Mathematics, University of Miami, Coral Gables, FL 33146, USA |
$\left\{ {\begin{array}{*{20}{l}}{\frac{\partial }{{\partial t}}{u_1}({\bf{x}},t) = \Delta {u_1}({\bf{x}},t) + {u_1}({\bf{x}},t)\left[ {1 - \;{u_1}({\bf{x}},t) - {k_1}{u_2}({\bf{x}},t)} \right],}\\{\frac{\partial }{{\partial t}}{u_2}({\bf{x}},t) = d\Delta {u_2}({\bf{x}},t) + r{u_2}({\bf{x}},t)\left[ {1 - {u_2}({\bf{x}},t) - {k_2}{u_1}({\bf{x}},t)} \right],}\end{array}} \right.$ |
$\mathbf{x}∈ \mathbb{R}^3$ |
$t>0$ |
$k_1,k_2>1$ |
$\mathbf{\Phi}(x+ct)=\left(\Phi_1(x+ct),\Phi_2(x+ct)\right)$ |
$\mathbf{E}_u=(1,0)$ |
$\mathbf{E}_v=(0,1)$ |
$c∈\mathbb{R}$ |
$c>0$ |
$s>c>0$ |
$\mathbf{\Psi}(\mathbf{x}^\prime,x_3+st)=\left(\Phi_1(\mathbf{x}^\prime,x_3+st),\Phi_2(\mathbf{x}^\prime, x_3+st)\right)$ |
$\mathbb{R}^3$ |
$\mathbf{E}_u=(1,0)$ |
$\mathbf{E}_v=(0,1)$ |
$\mathbf{x}^\prime∈\mathbb{R}^2$ |
$x_3$ |
$s$ |
$c$ |
$\mathbb{R}^3$ |
References:
[1] |
E. O. Alcahrani, F. A. Davidson and N. Dodds,
Travelling waves in near-degenerate bistable competition models, Math. Model. Nat. Phenom., 5 (2010), 13-35.
doi: 10.1051/mmnp/20105502. |
[2] |
D. G. Aronson and H. F. Weinberger,
Multidimensional nonlinear diffusion arising in population genetics, Adv. Math.,, 30 (1978), 33-76.
doi: 10.1016/0001-8708(78)90130-5. |
[3] |
A. Bonnet and F. Hamel,
Existence of nonplanar solutions of a simple model of premixed Bunsen flames, SIAM J. Math. Anal., 31 (1999), 80-118.
doi: 10.1137/S0036141097316391. |
[4] |
P. K. Brazhnik and J. J. Tyson,
On traveling wave solutions of Fisher's equation in two spatial dimensions, SIAM J. Appl. Math., 60 (1999), 371-391.
doi: 10.1137/S0036139997325497. |
[5] |
Z.-H. Bu and Z.-C. Wang,
Curved fronts of monostable reaction-advection-diffusion equations in space-time periodic media, Commun. Pure Appl. Anal., 15 (2016), 139-160.
doi: 10.3934/cpaa.2016.15.139. |
[6] |
G. Chapuisat,
Existence and nonexistence of curved front solution of a biological equation, J. Differential Equations, 236 (2007), 237-279.
doi: 10.1016/j.jde.2007.01.021. |
[7] |
X. Chen, J.-S. Guo, F. Hamel, H. Ninomiya and J.-M. Roquejoffre,
Traveling waves with paraboloid like interfaces for balanced bistable dynamics, Ann. Inst. H. Poincaré Anal. Linéaire, 24 (2007), 369-393.
doi: 10.1016/j.anihpc.2006.03.012. |
[8] |
C. Conley and R. Gardner,
An application of the generalized Morse index to travelling wave solutions of a competitive reaction-diffusion model, Indiana Univ. Math. J., 33 (1984), 319-343.
doi: 10.1512/iumj.1984.33.33018. |
[9] |
D. Daners and P. K. McLeod, Abstract Evolution Equations, Periodic Problems and Applications, Pitman Res. Notes Math. Ser. 279, Longman Scientific and Technical, Harlow, 1992. |
[10] |
M. El Smaily, F. Hamel and R. Huang,
Two-dimensional curved fronts in a periodic shear flow, Nonlinear Analysis TMA, 74 (2011), 6469-6486.
doi: 10.1016/j.na.2011.06.030. |
[11] |
P. C. Fife, Dynamics of Internal Layers and Diffusive Interfaces, CBMS-NSF Regional Conference, Series in Applied Mathematics, 53, 1988. |
[12] |
S. A. Gardner,
Existence and stability of travelling wave solutions of competition model: A degree theoretical approach, J. Differential Equations, 44 (1982), 343-364.
doi: 10.1016/0022-0396(82)90001-8. |
[13] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001. |
[14] |
C. Gui,
Symmetry of traveling wave solutions to the Allen-Cahn equation in ℝ2, Arch. Rational Mech. Anal., 203 (2012), 1037-1065.
doi: 10.1007/s00205-011-0480-5. |
[15] |
J.-S. Guo and Y.-C. Lin,
The sign of the wave speed for the Lotka-Volterra competition-diffusion system, Comm. Pure Appl. Anal., 12 (2013), 2083-2090.
doi: 10.3934/cpaa.2013.12.2083. |
[16] |
J.-S. Guo and C.-H. Wu,
Recent developments on wave propagation in 2-species competition systems, Discrete Contin. Dynam. Syst.-B, 17 (2012), 2713-2724.
doi: 10.3934/dcdsb.2012.17.2713. |
[17] |
J.-S. Guo and C.-H. Wu,
Wave propagation for a two-component lattice dynamical system arising in strong competition models, J. Differential Equations, 250 (2011), 3504-3533.
doi: 10.1016/j.jde.2010.12.004. |
[18] |
F. Hamel and R. Monneau,
Solutions of semilinear elliptic equations in ℝN with conicalshaped level sets, Comm. Partial Differential Equations, 25 (2000), 769-819.
doi: 10.1080/03605300008821532. |
[19] |
F. Hamel, R. Monneau and J.-M. Roquejoffre,
Stability of travelling waves in a model for conical flames in two space dimensions, Ann. Sci. Ecole Norm. Sup., 37 (2004), 469-506.
doi: 10.1016/j.ansens.2004.03.001. |
[20] |
F. Hamel, R. Monneau and J.-M. Roquejoffre,
Existence and qualitative properties of multidimensional conical bistable fronts, Discrete Contin. Dynam. Syst., 13 (2005), 1069-1096.
doi: 10.3934/dcds.2005.13.1069. |
[21] |
F. Hamel, R. Monneau and J.-M. Roquejoffre,
Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Discrete Contin. Dynam. Syst., 14 (2006), 75-92.
|
[22] |
F. Hamel and N. Nadirashvili,
Travelling fronts and entire solutions of the Fisher-KPP equation in ℝN, Arch. Rational Mech. Anal., 157 (2001), 91-163.
doi: 10.1007/PL00004238. |
[23] |
F. Hamel and J.-M. Roquejoffre,
Heteroclinic connections for multidimensional bistable reaction-diffusion equations, Discrete Contin. Dynam. Syst.-S, 4 (2011), 101-123.
doi: 10.3934/dcdss.2011.4.101. |
[24] |
M. Haragus and A. Scheel,
A bifurcation approach to non-planar traveling waves in reactiondiffusion systems, GAMM-Mitt., 30 (2007), 75-95.
doi: 10.1002/gamm.200790012. |
[25] |
M. Haragus and A. Scheel,
Almost planar waves in anisotropic media, Comm. Partial Differential Equations, 31 (2006), 791-815.
doi: 10.1080/03605300500361420. |
[26] |
M. Haragus and A. Scheel,
Corner defects in almost planar interface propagation, Ann. Inst. H. Poincaré Anal. Linéaire, 23 (2006), 283-329.
doi: 10.1016/j.anihpc.2005.03.003. |
[27] |
R. Huang,
Stability of travelling fronts of the Fisher-KPP equation in ℝN, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 599-622.
doi: 10.1007/s00030-008-7041-0. |
[28] |
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. |
[29] |
Y. Kan-on,
Existence of standing waves for competition-diffusion equations, Japan J. Indust. Appl. Math., 13 (1996), 117-133.
doi: 10.1007/BF03167302. |
[30] |
Y. Kan-on,
Instability of stationary solutions for a Lotka-Volterra competition model with diffusion, J. Math. Anal. Appl., 208 (1997), 158-170.
doi: 10.1006/jmaa.1997.5309. |
[31] |
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. |
[32] |
Y. Kurokawa and M. Taniguchi,
Multi-dimensional pyramidal travelling fronts in the AllenCahn equations, Proc. Royal Soc. Edinburgh Sect. A: Math., 14 (2011), 1031-1054.
doi: 10.1017/S0308210510001253. |
[33] |
W.-T. Li, G. Lin and S. Ruan,
Existence of travelling wave solutions in delayed reactiondiffusion systems with applications to diffusion-competition systems, Nonlinearity, 19 (2006), 1253-1273.
doi: 10.1088/0951-7715/19/6/003. |
[34] |
X. Liang and X.-Q. Zhao,
Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.
doi: 10.1002/cpa.20154. |
[35] |
G. Lin and W.-T. Li,
Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays, J. Differential Equations, 244 (2008), 487-513.
doi: 10.1016/j.jde.2007.10.019. |
[36] |
R. H. Martin and H. L. Smith,
Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.
doi: 10.2307/2001590. |
[37] |
Y. Morita and H. Ninomiya,
Monostable-type traveling waves of bistable reaction-diffusion equations in the multi-dimensional space, Bull. Inst. Math. Acad. Sinica, 3 (2008), 567-584.
|
[38] |
Y. Morita and K. Tachibana,
An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240.
doi: 10.1137/080723715. |
[39] |
W.-M. Ni and M. Taniguchi,
Traveling fronts of pyramidal shapes in competition-diffusion systems, Netw. Heterog. Media, 8 (2013), 379-395.
doi: 10.3934/nhm.2013.8.379. |
[40] |
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. |
[41] |
H. Ninomiya and M. Taniguchi,
Global stability of traveling curved fronts in the Allen-Cahn equations, Discrete Contin. Dynam. Syst., 15 (2006), 819-832.
doi: 10.3934/dcds.2006.15.819. |
[42] |
M. del Pino, M. Kowalczyk and J. Wei,
A counterexample to a conjecture by De Giorgi in large dimensions, C. R. Math. Acad. Sci. Paris, 346 (2008), 1261-1266.
doi: 10.1016/j.crma.2008.10.010. |
[43] |
M. del Pino, M. Kowalczyk and J. Wei,
Traveling waves with multiple and nonconvex fronts for a bistable semilinear parabolic equation, Comm. Pure Appl. Math., 66 (2013), 481-547.
doi: 10.1002/cpa.21438. |
[44] |
M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, PrenticeHall, Inc. , Englewood Cliffs, N. J. , 1967. 1144 |
[45] |
W.-J. Sheng, W.-T. Li and Z.-C. Wang,
Periodic pyramidal traveling fronts of bistable reaction-diffusion equations with time-periodic nonlinearity, J. Differential Equations, 252 (2012), 2388-2424.
doi: 10.1016/j.jde.2011.09.01. |
[46] |
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, Vol. 41, Amer. Math. Soc. , Providence, RI, 1995. |
[47] |
M. Taniguchi,
Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344.
doi: 10.1137/060661788. |
[48] |
M. Taniguchi,
The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Differential Euqations, 246 (2009), 2103-2130.
doi: 10.1016/j.jde.2008.06.037. |
[49] |
M. Taniguchi,
Multi-Dimensional traveling fronts in bistable reaction-diffusion equations, Discrete Contin. Dynam. Syst., 32 (2012), 1011-1046.
doi: 10.3934/dcds.2012.32.1011. |
[50] |
H. R. Thieme and X.-Q. Zhao,
Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470.
doi: 10.1016/S0022-0396(03)00175-X. |
[51] |
A. I. Volpert, V. A. Volpert and V. A. Volpert, Travelling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, Vol. 140, Amer. Math. Soc. , Providence, RI, 1994. |
[52] |
Z.-C. Wang,
Traveling curved fronts in monotone bistable systems, Discrete Contin. Dynam. Syst., 32 (2012), 2339-2374.
doi: 10.3934/dcds.2012.32.2339. |
[53] |
Z.-C. Wang,
Cylindrically symmetric traveling fronts in periodic reaction-diffusion equation with bistable nonlinearity, Proc. Royal Soc. Edinburgh Sect. A: Math., 145 (2015), 1053-1090.
doi: 10.1017/S0308210515000268. |
[54] |
Z.-C. Wang and Z.-H. Bu,
Nonplanar traveling fronts in reaction-diffusion equations with combustion and degenerate Fisher-KPP nonlinearities, J. Differential Equations, 260 (2016), 6405-6450.
doi: 10.1016/j.jde.2015.12.045. |
[55] |
Z.-C. Wang, W.-T. Li and S. Ruan,
Existence, uniqueness and stability of pyramidal traveling fronts in reaction-diffusion systems, Sci. China Math., 59 (2016), 1868-1908.
doi: 10.1007/s11425-016-0015-x. |
[56] |
Z.-C. Wang and J. Wu,
Periodic traveling curved fronts in reaction-diffusion equation with bistable time-periodic nonlinearity, J. Differential Equations, 250 (2011), 3196-3229.
doi: 10.1016/j.jde.2011.01.017. |
[57] |
T. P. Witelski, K. Ono and T. J. Kaper,
On axisymmetric traveling waves and radial solutions of semi-linear elliptic equations, Nat. Resource Model., 13 (2000), 339-388.
doi: 10.1111/j.1939-7445.2000.tb00039.x. |
[58] |
G. Zhao and S. Ruan,
Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion, J. Math. Pures Appl., 95 (2011), 627-671.
doi: 10.1016/j.matpur.2010.11.005. |
show all references
References:
[1] |
E. O. Alcahrani, F. A. Davidson and N. Dodds,
Travelling waves in near-degenerate bistable competition models, Math. Model. Nat. Phenom., 5 (2010), 13-35.
doi: 10.1051/mmnp/20105502. |
[2] |
D. G. Aronson and H. F. Weinberger,
Multidimensional nonlinear diffusion arising in population genetics, Adv. Math.,, 30 (1978), 33-76.
doi: 10.1016/0001-8708(78)90130-5. |
[3] |
A. Bonnet and F. Hamel,
Existence of nonplanar solutions of a simple model of premixed Bunsen flames, SIAM J. Math. Anal., 31 (1999), 80-118.
doi: 10.1137/S0036141097316391. |
[4] |
P. K. Brazhnik and J. J. Tyson,
On traveling wave solutions of Fisher's equation in two spatial dimensions, SIAM J. Appl. Math., 60 (1999), 371-391.
doi: 10.1137/S0036139997325497. |
[5] |
Z.-H. Bu and Z.-C. Wang,
Curved fronts of monostable reaction-advection-diffusion equations in space-time periodic media, Commun. Pure Appl. Anal., 15 (2016), 139-160.
doi: 10.3934/cpaa.2016.15.139. |
[6] |
G. Chapuisat,
Existence and nonexistence of curved front solution of a biological equation, J. Differential Equations, 236 (2007), 237-279.
doi: 10.1016/j.jde.2007.01.021. |
[7] |
X. Chen, J.-S. Guo, F. Hamel, H. Ninomiya and J.-M. Roquejoffre,
Traveling waves with paraboloid like interfaces for balanced bistable dynamics, Ann. Inst. H. Poincaré Anal. Linéaire, 24 (2007), 369-393.
doi: 10.1016/j.anihpc.2006.03.012. |
[8] |
C. Conley and R. Gardner,
An application of the generalized Morse index to travelling wave solutions of a competitive reaction-diffusion model, Indiana Univ. Math. J., 33 (1984), 319-343.
doi: 10.1512/iumj.1984.33.33018. |
[9] |
D. Daners and P. K. McLeod, Abstract Evolution Equations, Periodic Problems and Applications, Pitman Res. Notes Math. Ser. 279, Longman Scientific and Technical, Harlow, 1992. |
[10] |
M. El Smaily, F. Hamel and R. Huang,
Two-dimensional curved fronts in a periodic shear flow, Nonlinear Analysis TMA, 74 (2011), 6469-6486.
doi: 10.1016/j.na.2011.06.030. |
[11] |
P. C. Fife, Dynamics of Internal Layers and Diffusive Interfaces, CBMS-NSF Regional Conference, Series in Applied Mathematics, 53, 1988. |
[12] |
S. A. Gardner,
Existence and stability of travelling wave solutions of competition model: A degree theoretical approach, J. Differential Equations, 44 (1982), 343-364.
doi: 10.1016/0022-0396(82)90001-8. |
[13] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001. |
[14] |
C. Gui,
Symmetry of traveling wave solutions to the Allen-Cahn equation in ℝ2, Arch. Rational Mech. Anal., 203 (2012), 1037-1065.
doi: 10.1007/s00205-011-0480-5. |
[15] |
J.-S. Guo and Y.-C. Lin,
The sign of the wave speed for the Lotka-Volterra competition-diffusion system, Comm. Pure Appl. Anal., 12 (2013), 2083-2090.
doi: 10.3934/cpaa.2013.12.2083. |
[16] |
J.-S. Guo and C.-H. Wu,
Recent developments on wave propagation in 2-species competition systems, Discrete Contin. Dynam. Syst.-B, 17 (2012), 2713-2724.
doi: 10.3934/dcdsb.2012.17.2713. |
[17] |
J.-S. Guo and C.-H. Wu,
Wave propagation for a two-component lattice dynamical system arising in strong competition models, J. Differential Equations, 250 (2011), 3504-3533.
doi: 10.1016/j.jde.2010.12.004. |
[18] |
F. Hamel and R. Monneau,
Solutions of semilinear elliptic equations in ℝN with conicalshaped level sets, Comm. Partial Differential Equations, 25 (2000), 769-819.
doi: 10.1080/03605300008821532. |
[19] |
F. Hamel, R. Monneau and J.-M. Roquejoffre,
Stability of travelling waves in a model for conical flames in two space dimensions, Ann. Sci. Ecole Norm. Sup., 37 (2004), 469-506.
doi: 10.1016/j.ansens.2004.03.001. |
[20] |
F. Hamel, R. Monneau and J.-M. Roquejoffre,
Existence and qualitative properties of multidimensional conical bistable fronts, Discrete Contin. Dynam. Syst., 13 (2005), 1069-1096.
doi: 10.3934/dcds.2005.13.1069. |
[21] |
F. Hamel, R. Monneau and J.-M. Roquejoffre,
Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Discrete Contin. Dynam. Syst., 14 (2006), 75-92.
|
[22] |
F. Hamel and N. Nadirashvili,
Travelling fronts and entire solutions of the Fisher-KPP equation in ℝN, Arch. Rational Mech. Anal., 157 (2001), 91-163.
doi: 10.1007/PL00004238. |
[23] |
F. Hamel and J.-M. Roquejoffre,
Heteroclinic connections for multidimensional bistable reaction-diffusion equations, Discrete Contin. Dynam. Syst.-S, 4 (2011), 101-123.
doi: 10.3934/dcdss.2011.4.101. |
[24] |
M. Haragus and A. Scheel,
A bifurcation approach to non-planar traveling waves in reactiondiffusion systems, GAMM-Mitt., 30 (2007), 75-95.
doi: 10.1002/gamm.200790012. |
[25] |
M. Haragus and A. Scheel,
Almost planar waves in anisotropic media, Comm. Partial Differential Equations, 31 (2006), 791-815.
doi: 10.1080/03605300500361420. |
[26] |
M. Haragus and A. Scheel,
Corner defects in almost planar interface propagation, Ann. Inst. H. Poincaré Anal. Linéaire, 23 (2006), 283-329.
doi: 10.1016/j.anihpc.2005.03.003. |
[27] |
R. Huang,
Stability of travelling fronts of the Fisher-KPP equation in ℝN, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 599-622.
doi: 10.1007/s00030-008-7041-0. |
[28] |
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. |
[29] |
Y. Kan-on,
Existence of standing waves for competition-diffusion equations, Japan J. Indust. Appl. Math., 13 (1996), 117-133.
doi: 10.1007/BF03167302. |
[30] |
Y. Kan-on,
Instability of stationary solutions for a Lotka-Volterra competition model with diffusion, J. Math. Anal. Appl., 208 (1997), 158-170.
doi: 10.1006/jmaa.1997.5309. |
[31] |
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. |
[32] |
Y. Kurokawa and M. Taniguchi,
Multi-dimensional pyramidal travelling fronts in the AllenCahn equations, Proc. Royal Soc. Edinburgh Sect. A: Math., 14 (2011), 1031-1054.
doi: 10.1017/S0308210510001253. |
[33] |
W.-T. Li, G. Lin and S. Ruan,
Existence of travelling wave solutions in delayed reactiondiffusion systems with applications to diffusion-competition systems, Nonlinearity, 19 (2006), 1253-1273.
doi: 10.1088/0951-7715/19/6/003. |
[34] |
X. Liang and X.-Q. Zhao,
Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.
doi: 10.1002/cpa.20154. |
[35] |
G. Lin and W.-T. Li,
Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays, J. Differential Equations, 244 (2008), 487-513.
doi: 10.1016/j.jde.2007.10.019. |
[36] |
R. H. Martin and H. L. Smith,
Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.
doi: 10.2307/2001590. |
[37] |
Y. Morita and H. Ninomiya,
Monostable-type traveling waves of bistable reaction-diffusion equations in the multi-dimensional space, Bull. Inst. Math. Acad. Sinica, 3 (2008), 567-584.
|
[38] |
Y. Morita and K. Tachibana,
An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240.
doi: 10.1137/080723715. |
[39] |
W.-M. Ni and M. Taniguchi,
Traveling fronts of pyramidal shapes in competition-diffusion systems, Netw. Heterog. Media, 8 (2013), 379-395.
doi: 10.3934/nhm.2013.8.379. |
[40] |
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. |
[41] |
H. Ninomiya and M. Taniguchi,
Global stability of traveling curved fronts in the Allen-Cahn equations, Discrete Contin. Dynam. Syst., 15 (2006), 819-832.
doi: 10.3934/dcds.2006.15.819. |
[42] |
M. del Pino, M. Kowalczyk and J. Wei,
A counterexample to a conjecture by De Giorgi in large dimensions, C. R. Math. Acad. Sci. Paris, 346 (2008), 1261-1266.
doi: 10.1016/j.crma.2008.10.010. |
[43] |
M. del Pino, M. Kowalczyk and J. Wei,
Traveling waves with multiple and nonconvex fronts for a bistable semilinear parabolic equation, Comm. Pure Appl. Math., 66 (2013), 481-547.
doi: 10.1002/cpa.21438. |
[44] |
M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, PrenticeHall, Inc. , Englewood Cliffs, N. J. , 1967. 1144 |
[45] |
W.-J. Sheng, W.-T. Li and Z.-C. Wang,
Periodic pyramidal traveling fronts of bistable reaction-diffusion equations with time-periodic nonlinearity, J. Differential Equations, 252 (2012), 2388-2424.
doi: 10.1016/j.jde.2011.09.01. |
[46] |
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, Vol. 41, Amer. Math. Soc. , Providence, RI, 1995. |
[47] |
M. Taniguchi,
Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344.
doi: 10.1137/060661788. |
[48] |
M. Taniguchi,
The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Differential Euqations, 246 (2009), 2103-2130.
doi: 10.1016/j.jde.2008.06.037. |
[49] |
M. Taniguchi,
Multi-Dimensional traveling fronts in bistable reaction-diffusion equations, Discrete Contin. Dynam. Syst., 32 (2012), 1011-1046.
doi: 10.3934/dcds.2012.32.1011. |
[50] |
H. R. Thieme and X.-Q. Zhao,
Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470.
doi: 10.1016/S0022-0396(03)00175-X. |
[51] |
A. I. Volpert, V. A. Volpert and V. A. Volpert, Travelling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, Vol. 140, Amer. Math. Soc. , Providence, RI, 1994. |
[52] |
Z.-C. Wang,
Traveling curved fronts in monotone bistable systems, Discrete Contin. Dynam. Syst., 32 (2012), 2339-2374.
doi: 10.3934/dcds.2012.32.2339. |
[53] |
Z.-C. Wang,
Cylindrically symmetric traveling fronts in periodic reaction-diffusion equation with bistable nonlinearity, Proc. Royal Soc. Edinburgh Sect. A: Math., 145 (2015), 1053-1090.
doi: 10.1017/S0308210515000268. |
[54] |
Z.-C. Wang and Z.-H. Bu,
Nonplanar traveling fronts in reaction-diffusion equations with combustion and degenerate Fisher-KPP nonlinearities, J. Differential Equations, 260 (2016), 6405-6450.
doi: 10.1016/j.jde.2015.12.045. |
[55] |
Z.-C. Wang, W.-T. Li and S. Ruan,
Existence, uniqueness and stability of pyramidal traveling fronts in reaction-diffusion systems, Sci. China Math., 59 (2016), 1868-1908.
doi: 10.1007/s11425-016-0015-x. |
[56] |
Z.-C. Wang and J. Wu,
Periodic traveling curved fronts in reaction-diffusion equation with bistable time-periodic nonlinearity, J. Differential Equations, 250 (2011), 3196-3229.
doi: 10.1016/j.jde.2011.01.017. |
[57] |
T. P. Witelski, K. Ono and T. J. Kaper,
On axisymmetric traveling waves and radial solutions of semi-linear elliptic equations, Nat. Resource Model., 13 (2000), 339-388.
doi: 10.1111/j.1939-7445.2000.tb00039.x. |
[58] |
G. Zhao and S. Ruan,
Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion, J. Math. Pures Appl., 95 (2011), 627-671.
doi: 10.1016/j.matpur.2010.11.005. |
[1] |
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 |
[2] |
Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2599-2623. doi: 10.3934/dcdsb.2020197 |
[3] |
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 |
[4] |
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 |
[5] |
Yuan Lou, Dongmei Xiao, Peng Zhou. Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 953-969. doi: 10.3934/dcds.2016.36.953 |
[6] |
Qi Wang, Chunyi Gai, Jingda Yan. Qualitative analysis of a Lotka-Volterra competition system with advection. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1239-1284. doi: 10.3934/dcds.2015.35.1239 |
[7] |
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 |
[8] |
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 |
[9] |
Bang-Sheng Han, Zhi-Cheng Wang, Zengji Du. Traveling waves for nonlocal Lotka-Volterra competition systems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1959-1983. doi: 10.3934/dcdsb.2020011 |
[10] |
Jian Fang, Jianhong Wu. Monotone traveling waves for delayed Lotka-Volterra competition systems. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3043-3058. doi: 10.3934/dcds.2012.32.3043 |
[11] |
Qi Wang. Some global dynamics of a Lotka-Volterra competition-diffusion-advection system. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3245-3255. doi: 10.3934/cpaa.2020142 |
[12] |
Li-Jun Du, Wan-Tong Li, Jia-Bing Wang. Invasion entire solutions in a time periodic Lotka-Volterra competition system with diffusion. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1187-1213. doi: 10.3934/mbe.2017061 |
[13] |
Yuzo Hosono. Traveling waves for the Lotka-Volterra predator-prey system without diffusion of the predator. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 161-171. doi: 10.3934/dcdsb.2015.20.161 |
[14] |
Qi Wang, Yang Song, Lingjie Shao. Boundedness and persistence of populations in advective Lotka-Volterra competition system. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2245-2263. doi: 10.3934/dcdsb.2018195 |
[15] |
Anthony W. Leung, Xiaojie Hou, Wei Feng. Traveling wave solutions for Lotka-Volterra system re-visited. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 171-196. doi: 10.3934/dcdsb.2011.15.171 |
[16] |
Zengji Du, Shuling Yan, Kaige Zhuang. Traveling wave fronts in a diffusive and competitive Lotka-Volterra system. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3097-3111. doi: 10.3934/dcdss.2021010 |
[17] |
Yuzo Hosono. Traveling waves for a diffusive Lotka-Volterra competition model I: singular perturbations. Discrete and Continuous Dynamical Systems - B, 2003, 3 (1) : 79-95. doi: 10.3934/dcdsb.2003.3.79 |
[18] |
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 |
[19] |
Chiun-Chuan Chen, Li-Chang Hung. Nonexistence of traveling wave solutions, exact and semi-exact traveling wave solutions for diffusive Lotka-Volterra systems of three competing species. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1451-1469. doi: 10.3934/cpaa.2016.15.1451 |
[20] |
Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. On a limiting system in the Lotka--Volterra competition with cross-diffusion. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 435-458. doi: 10.3934/dcds.2004.10.435 |
2021 Impact Factor: 1.497
Tools
Metrics
Other articles
by authors
[Back to Top]