Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models)

Tarik Mohammed Touaoula

doi:10.3934/dcds.2018191

Article Contents

2018, Volume 38, Issue 9: 4391-4419. Doi: 10.3934/dcds.2018191

This issue Previous Article The Hénon equation with a critical exponent under the Neumann boundary condition Next Article The maximal entropy measure of Fatou boundaries

Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models)

Tarik Mohammed Touaoula

Département de Mathématiques, Faculté des Sciences, Université de Tlemcen, Laboratoire d'Analyse Non Linéaire et Mathématiques Appliquées, Tlemcen, BP 119, 13000, Algeria

Received: August 2017

Revised: April 2018

Published: September 2018

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Abstract

Global asymptotic and exponential stability of equilibria for the following class of functional differential equations with distributed delay is investigated

$ x'(t)=-f(x(t))+\int_{0}^{\tau}h(a)g(x(t-a))da.$

We make our analysis by introducing a new approach, combining a Lyapunov functional and monotone semiflow theory. The relevance of our results is illustrated by studying the well-known integro-differential Nicholson's blowflies and Mackey-Glass equations, where some delay independent stability conditions are provided. Furthermore, new results related to exponential stability region of the positive equilibrium for these both models are established.

Keywords:

Monotone semi-flow,
global asymptotic and exponential stability,
fluctuation method.

Mathematics Subject Classification: Primary: 34K20, 37L15; Secondary: 92B05.

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