Nonlinear stability of traveling wavefronts in an age-structured reaction-diffusion population model

Guangrui Li; Ming Mei; Yau Shu Wong; Guangrui Li; Ming Mei; Yau Shu Wong

doi:10.3934/mbe.2008.5.85

Mathematical Biosciences and Engineering

2008, Volume 5, Issue 1: 85-100. doi: 10.3934/mbe.2008.5.85

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Nonlinear stability of traveling wavefronts in an age-structured reaction-diffusion population model

1.
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1
2.
Department of Mathematics, Champlain College Saint-Lambert, Saint-Lambert, Quebec, J4P 3P2

Received: 01 January 2007 Accepted: 29 June 2018 Published: 01 January 2008
MSC : Primary: 35K57; Secondary: 34K20, 92D25.

The paper is devoted to the study of a time-delayed reaction- diffusion equation of age-structured single species population. Linear stability for this model was first presented by Gourley [4], when the time delay is small. Here, we extend the previous result to the nonlinear stability by using the technical weighted-energy method, when the initial perturbation around the wavefront decays to zero exponentially as x→-∞, but the initial perturbation can be arbitrarily large on other locations. The exponential convergent rate (in time) of the solution is obtained. Numerical simulations are carried out to confirm the theoretical results, and the traveling wavefronts with a large delay term in the model are reported.
- time-delayed reaction-diffusion equation,
- exponential decay rate.,
- traveling wavefronts,
- non-linear stability
Citation: Guangrui Li, Ming Mei, Yau Shu Wong. Nonlinear stability of traveling wavefronts in an age-structured reaction-diffusion population model[J]. Mathematical Biosciences and Engineering, 2008, 5(1): 85-100. doi: 10.3934/mbe.2008.5.85

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Abstract

The paper is devoted to the study of a time-delayed reaction- diffusion equation of age-structured single species population. Linear stability for this model was first presented by Gourley [4], when the time delay is small. Here, we extend the previous result to the nonlinear stability by using the technical weighted-energy method, when the initial perturbation around the wavefront decays to zero exponentially as x→-∞, but the initial perturbation can be arbitrarily large on other locations. The exponential convergent rate (in time) of the solution is obtained. Numerical simulations are carried out to confirm the theoretical results, and the traveling wavefronts with a large delay term in the model are reported.
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