August  2009, 24(3): 763-780. doi: 10.3934/dcds.2009.24.763

Traveling wave solutions in parametric forms for a diffusion model with a nonlinear rate of growth

1. 

Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78541

2. 

Department of Mathematics, Texas A&M University, College Station, TX 77843

Received  December 2007 Revised  June 2008 Published  April 2009

In this paper, we study a model of insect and animal dispersal where both density-dependent diffusion and nonlinear rate of growth are present. We analyze the existence of bounded traveling wave solution under certain parametric conditions by using the qualitative theory of dynamical systems. An explicit traveling wave solution is obtained by means of the first integral method. Traveling wave solutions in parametric forms for three particular cases are established by the Lie symmetry method.
Citation: Zhaosheng Feng, Goong Chen. Traveling wave solutions in parametric forms for a diffusion model with a nonlinear rate of growth. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 763-780. doi: 10.3934/dcds.2009.24.763
[1]

Stefano Bianchini, Alberto Bressan. A center manifold technique for tracing viscous waves. Communications on Pure & Applied Analysis, 2002, 1 (2) : 161-190. doi: 10.3934/cpaa.2002.1.161

[2]

María Rosa, María de los Santos Bruzón, María de la Luz Gandarias. Lie symmetries and conservation laws of a Fisher equation with nonlinear convection term. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1331-1339. doi: 10.3934/dcdss.2015.8.1331

[3]

Camillo De Lellis, Emanuele Spadaro. Center manifold: A case study. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1249-1272. doi: 10.3934/dcds.2011.31.1249

[4]

Lina Wang, Xueli Bai, Yang Cao. Exponential stability of the traveling fronts for a viscous Fisher-KPP equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 801-815. doi: 10.3934/dcdsb.2014.19.801

[5]

Judith R. Miller, Huihui Zeng. Stability of traveling waves for systems of nonlinear integral recursions in spatial population biology. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 895-925. doi: 10.3934/dcdsb.2011.16.895

[6]

Claudia Valls. The Boussinesq system:dynamics on the center manifold. Communications on Pure & Applied Analysis, 2005, 4 (4) : 839-860. doi: 10.3934/cpaa.2005.4.839

[7]

Richard A. Norton, G. R. W. Quispel. Discrete gradient methods for preserving a first integral of an ordinary differential equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1147-1170. doi: 10.3934/dcds.2014.34.1147

[8]

Yaping Wu, Xiuxia Xing. Stability of traveling waves with critical speeds for $P$-degree Fisher-type equations. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1123-1139. doi: 10.3934/dcds.2008.20.1123

[9]

Xiao-Qiang Zhao, Wendi Wang. Fisher waves in an epidemic model. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1117-1128. doi: 10.3934/dcdsb.2004.4.1117

[10]

Zhaosheng Feng. Traveling waves to a reaction-diffusion equation. Conference Publications, 2007, 2007 (Special) : 382-390. doi: 10.3934/proc.2007.2007.382

[11]

Joseph Thirouin. Classification of traveling waves for a quadratic Szegő equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3099-3122. doi: 10.3934/dcds.2019128

[12]

Sergey V. Bolotin, Piero Negrini. Global regularization for the $n$-center problem on a manifold. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 873-892. doi: 10.3934/dcds.2002.8.873

[13]

Xiao-Biao Lin, Stephen Schecter. Traveling waves and shock waves. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : i-ii. doi: 10.3934/dcds.2004.10.4i

[14]

Elena Celledoni, Brynjulf Owren. Preserving first integrals with symmetric Lie group methods. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 977-990. doi: 10.3934/dcds.2014.34.977

[15]

Changlu Liu, Shuangli Qiao. Symmetry and monotonicity for a system of integral equations. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1925-1932. doi: 10.3934/cpaa.2009.8.1925

[16]

Yingshu Lü, Chunqin Zhou. Symmetry for an integral system with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1533-1543. doi: 10.3934/dcds.2018121

[17]

Yuqian Zhou, Qian Liu. Reduction and bifurcation of traveling waves of the KdV-Burgers-Kuramoto equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 2057-2071. doi: 10.3934/dcdsb.2016036

[18]

Rui Huang, Ming Mei, Yong Wang. Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3621-3649. doi: 10.3934/dcds.2012.32.3621

[19]

Emile Franc Doungmo Goufo, Abdon Atangana. Dynamics of traveling waves of variable order hyperbolic Liouville equation: Regulation and control. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 645-662. doi: 10.3934/dcdss.2020035

[20]

A. Carati. Center manifold of unstable periodic orbits of helium atom: numerical evidence. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 97-104. doi: 10.3934/dcdsb.2003.3.97

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]