Asymptotic stability of traveling wavefronts in a delayed population model with stage structure on a two-dimensional spatial lattice

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May 2010, 13(3): 559-575. doi: 10.3934/dcdsb.2010.13.559

Asymptotic stability of traveling wavefronts in a delayed population model with stage structure on a two-dimensional spatial lattice

Cui-Ping Cheng ^1, , Wan-Tong Li ^2, and Zhi-Cheng Wang ^1,

1.	School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China, China
2.	School of Mathematic and Statistics, Lanzhou University, Lanzhou, Gansu 730000

Received December 2008 Revised January 2010 Published February 2010

Abstract
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Recently, we derived a lattice model for a single species with stage structure in a two-dimensional patchy environment with infinite number of patches connected locally by diffusion and global interaction by delay (IMA J. Appl. Math., 73 (2008), 592-618.). The important feature of the model is the reflection of the joint effect of the diffusion dynamics, the nonlocal delayed effect and the direction of propagation. In this paper we study the asymptotic stability of traveling wavefronts of this model when the immature population is not mobile. Under the assumption that the birth function satisfies monostable condition, we prove that the traveling wavefront is exponentially stable by means of weighted energy method, when the initial perturbation around the wave is suitably small in a weighted norm. The exponential convergent rate is also obtained.

Keywords: asymptotic stability, energy estimates, Lattice differential equation, global interaction., traveling wave.

Mathematics Subject Classification: Primary: 35B40, 35K57; Secondary: 35R20, 92D2.

Citation: Cui-Ping Cheng, Wan-Tong Li, Zhi-Cheng Wang. Asymptotic stability of traveling wavefronts in a delayed population model with stage structure on a two-dimensional spatial lattice. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 559-575. doi: 10.3934/dcdsb.2010.13.559

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