Global dynamics of a nonlocal delayed reaction–diffusion equation on a half plane

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Abstract

We consider a delayed reaction–diffusion equation with spatial nonlocality on a half plane that describes population dynamics of a two-stage species living in a semi-infinite environment. A Neumann boundary condition is imposed accounting for an isolated domain. To describe the global dynamics, we first establish some a priori estimate for nontrivial solutions after investigating asymptotic properties of the nonlocal delayed effect and the diffusion operator, which enables us to show the permanence of the equation with respect to the compact open topology. We then employ standard dynamical system arguments to establish the global attractivity of the nontrivial equilibrium. The main results are illustrated by the diffusive Nicholson’s blowfly equation and the diffusive Mackey–Glass equation.

Keywords

Delayed reaction–diffusion equation Neumann boundary condition Spatial nonlocality Compact open topology Half plane Dynamical system approach 

Mathematics Subject Classification

34D23 34K25 35K57 39A30 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.The Journal HouseHunan Normal UniversityChangshaPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsCentral South UniversityChangshaPeople’s Republic of China

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