# American Institute of Mathematical Sciences

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

## On a generalized Yorke condition for scalar delayed population models

 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

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.
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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