Nonlinear stability of traveling wavefronts in an age-structured reaction-diffusion population model. (English) Zbl 1148.35038

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

Math. Biosci. Eng. 5, No. 1, 85-100 (2008).

Summary: 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 S. A. Gourley [Q. J. Mech. Appl. Math. 58, No. 2, 257–268 (2005; Zbl 1069.92018)], 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 \(x\to-\infty\), 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.

MSC:

35K57	Reaction-diffusion equations
34K20	Stability theory of functional-differential equations
92D25	Population dynamics (general)
35R10	Functional partial differential equations

Keywords:

time delay; weighted-energy method; large delay term; nonlinear stability; exponential decay rate