Convergence to equilibria in discrete population models HASSAN A. EL-MORSHEDY† and EDUARDO LIZ‡ * †Department of Mathematics, Damietta Faculty of Science, New Damietta 34517, Egypt ‡Departamento de Matema ´tica Aplicada II, E.T.S.I. Telecomunicacio ´n Campus Marcosende, Universidad de Vigo, 36280 Vigo, Spain (Received 5 April 2004; in final form 23 September 2004) For a family of difference equations x nþ1 ¼ ax n þ f ðx n2k Þ; n ¼ 0; 1; ...; where a [ ð0; 1Þ; k [ {1; 2; ...}; and f : ½0; 1Þ ! ð0; 1Þ is continuous and decreasing, we find sufficient conditions for the convergence of all solutions to the unique positive equilibrium. In particular, we improve, up to our knowledge, all previous results on the global asymptotic stability of the equilibrium in the particular cases of the discrete Mackey – Glass and Lasota–Wazewska models in blood-cells production. Keywords: Difference equations; Global attractor; Discrete population models; Schwarzian derivative Mathematics Subject Classificattion (2000): 39A10; 39A11; 92D25 1. Introduction It is well-known that many discrete models in biology can be described by a difference equation x nþ1 ¼ ax n þ f ðx n2k Þ; n ¼ 0; 1; ...; ð1:1Þ where a [ ð0; 1Þ; k [ {1; 2; ...}; and f : ½0; 1Þ ! ð0; 1Þ is a continuous function. For some examples, see sections 4.5 – 4.7 in the monograph [11]. Equation (1.1) is not only interesting by itself, but also as the discretization of an extensively studied family of delay differential equations, namely x 0 ðtÞ¼ 2dxðtÞþ f ðxðt 2 hÞÞ; d . 0; h . 0: ð1:2Þ Relations between equations (1.1) and (1.2) can be found, for example, in refs. [1,10]. For processes modeled by equation (1.2), the reader can find many examples in the interesting list in [8, p. 78], including models in neurophysiology, metabolic regulation and agricultural commodity markets. We will focus our attention on the case of a decreasing nonlinearity f. In fact, we will assume that f is differentiable and f 0 ðxÞ , 0 for all x . 0: Under this hypothesis, it is clear that equation (1.1) has a unique constant solution ð  xÞ; where  x solves the scalar equation Journal of Difference Equations and Applications ISSN 1023-6198 print/ISSN 1563-5120 online q 2005 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/10236190512331319334 *Corresponding author. Email: eliz@dma.uvigo.es Journal of Difference Equations and Applications, Vol. 11, No. 2, February 2005, 117–131