Phil. Trans. R. Soc. A (2009) 367, 4717–4739 doi:10.1098/rsta.2009.0177 Application of interval iterations to the entrainment problem in respiratory physiology BY JACQUES DEMONGEOT* AND JULES WAKU TIMC-IMAG Laboratory, University J. Fourier of Grenoble, 38706 La Tronche, France We present here some theoretical and numerical results about interval iterations. We consider first an application of the interval iterations theory to the problem of entrainment in respiratory physiology for which the classical point iterations theory fails. Then, after a brief review of some of the main aspects of point iterations, we explain what is meant by the term ‘interval iterations’. It consists essentially in replacing in the point iterations the function to iterate by a set-valued map. We present both theoretical and numerical aspects of this new type of iterations and we observe the dynamical behaviours encountered, such as fixed intervals and interval limit cycles. The comparison between point and interval iterations is carried out with respect to a parameter ε, which determines the thickness of a neighbourhood around the function to iterate. We will finally focus our attention on the Verhulst and Ricker functions largely used in population dynamics, which exhibit various asymptotic behaviours. Keywords: interval iterations; respiratory entrainment; set-valued map; invariant domain; intervals limit cycles; population dynamics 1. Introduction It is well known (May 1976; May & Oster 1976; Demongeot et al. 1997; Murray 2002) that a first-order difference equation (e.g. the logistic one- dimensional equation for a single species) allows the description of complex dynamical behaviours in population growth modelling in many contexts and several disciplines, such as in biology, economy and social sciences (Schaffer et al. 1986; Demongeot & Leitner 1996; Demongeot et al. 1997; Stenseth et al. 1997; Demongeot & Waku 2005). The case of n -dimensional flows (system of difference equations for n species) will not be treated in the following, but could be considered as a natural generalization of the techniques here proposed. Let the basic equation be x t +1 = f (x t ). (1.1) The variable x t can be referred to as the ‘population’ at time t ; f is usually a nonlinear function, containing one or more adjustable parameters, which tune the nonlinear behaviour of the considered system. Population here means people, *Author for correspondence (jacques.demongeot@imag.fr). One contribution of 17 to a Theme Issue ‘From biological and clinical experiments to mathematical models’. This journal is © 2009 The Royal Society 4717