Traveling waves of a nonlocal dispersal delayed age-structured population model

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Abstract

We derive from the age-structured model a system of delayed integro-differential equations to describe the interaction of spatial dispersal and time delay arising from the maturation period. We assume that the immature diffuses locally and the mature moves in long-distance random walks. If the mature death and diffusion rates are independent of age, then the total mature population is governed by a nonlocal spatial diffusion model with delays and nonlocal effect. We also consider the existence of traveling waves for this age-structured population model, which is obtained by combining upper and lower solutions for associated integral equations and Schauder’s fixed point theorem, where the birth function is not necessarily monotone. Moreover, the exponential asymptotics at negative infinity is also obtained for the non-monotone birth function, which is attributed to the construction of the wave profile set while the traveling wave either converges to the nontrivial equilibrium or oscillates on it at positive infinity. Lastly, we show that the maturation period τ reduces the wave speed c * while the effect of the dispersal rate of the immature on the matured population α increases it.

Keywords

Traveling wave Nonlocal diffusion Delay Schauder’s fixed point theorem 

Mathematics Subject Classification

35C07 92D25 35R09 
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Copyright information

© The JJIAM Publishing Committee and Springer Japan 2012

Authors and Affiliations

  1. 1.College of ScienceUniversity of Shanghai for Science and TechnologyShanghaiPeople’s Republic of China
  2. 2.School of Mathematical SciencesBeijing Normal UniversityBeijingPeople’s Republic of China

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