On a generalized Yorke condition for scalar delayed population models

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April 2005, 12(3): 481-500. doi: 10.3934/dcds.2005.12.481

On a generalized Yorke condition for scalar delayed population models

Teresa Faria ^1, , Eduardo Liz ^2, , José J. Oliveira ^3, and Sergei Trofimchuk ^4,

1.	Departamento de Matemática, Faculdade de Ciências, and CMAF, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa
2.	Departamento de Matemática Aplicada II, E.T.S.I. Telecomunicación, Universidad de Vigo, Campus Marcosende, 36280 Vigo
3.	Departamento de Matemática, and CMAT, Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal
4.	Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca, Chile

Received October 2003 Revised July 2004 Published December 2004

Abstract
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For a scalar delayed differential equation $\dot x(t)=f(t,x_t)$, we give sufficient conditions for the global attractivity of its zero solution. Some technical assumptions are imposed to insure boundedness of solutions and attractivity of non-oscillatory solutions. For controlling the behaviour of oscillatory solutions, we require a very general condition of Yorke type, together with a 3/2-condition. The results are particularly interesting when applied to scalar differential equations with delays which have served as models in populations dynamics, and can be written in the general form $\dot x(t)=(1+x(t))F(t,x_t)$. Applications to several models are presented, improving known results in the literature.

Keywords: Yorke condition, Delayed population model, global attractivity, 3/2- condition..

Mathematics Subject Classification: 34K20, 34K2.

Citation: Teresa Faria, Eduardo Liz, José J. Oliveira, Sergei Trofimchuk. On a generalized Yorke condition for scalar delayed population models. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 481-500. doi: 10.3934/dcds.2005.12.481

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