Global dynamics of a non-local delayed  differential equation  in the half plane

Tao Wang

doi:10.3934/cpaa.2014.13.2475

Article Contents

2014, Volume 13, Issue 6: 2475-2492. Doi: 10.3934/cpaa.2014.13.2475

This issue Previous Article Some eigenvalue problems with non-local boundary conditions and applications Next Article A fourth order elliptic equation with a singular nonlinearity

Global dynamics of a non-local delayed differential equation in the half plane

Tao Wang^1,

1.
School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083

Received: December 31, 2013

Revised: April 30, 2014

Published: October 2014

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Abstract

In this paper, we first derive an equation for a single species population with two age stages and a fixed maturation period living in the half plane such as ocean and big lakes. By adopting the compact open topology, we establish some a priori estimate for nontrivial solutions after describing asymptotic properties of the nonlocal delayed effect, which enables us to show the permanence of the equation. Then we can employ standard dynamical system theoretical arguments to establish the global dynamics of the equation under appropriate conditions. Applying the main results to the model with Ricker's birth function and Mackey-Glass's hematopoiesis function, we obtain threshold results for the global dynamics of these two models.

Keywords:

The half plane,
non-local delayed differential equation,
compact open topology,
a priori estimate,
permanence,
global dynamics.

Mathematics Subject Classification: Primary: 34D23, 34K25; Secondary: 39A30.

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