INTRODUCTION Mathematical models describing the long-term evolution of animal populations show a spectrum of complex dynamic behaviors, from stable cycles through to a chaotic regime

Mathematical models describing the long-term evolution of animal populations show a spectrum of complex dynamic behaviors, from stable cycles through to a chaotic regime (May, 1974, 1976; Mackey and Glass, 1977; Mackey, 1985; Romond et al., 1999). In contrast, there are no mathematical analyses and models of proliferation of tumoral cell populations in vitro. The growth of tumoral cell lines has been considered to be linear, and cell functions steady, under stable culture conditions. However, we recently reported that the growth of many long-term cultured cells, of different species and histogenetic origin, displays repeated rises and falls in rate (Wolfrom et al., 1994; Maigné et al., 1998). These oscillations in proliferation rate have important consequences: for most cell types, cell differentiation and expression of ‘household’ functions show negative or positive correlation, respectively, with proliferative activity. Were this temporal evolution to occur in vivo, it would have major implications for essential cell fate bifurcations, such as embryonic cell differentiation, for morphogenesis and fractal growth of certain organs (Khokha et al., 1994), or for tumor cell metabolism. Consequently the nature of these oscillations in proliferation rate is important. Standard growth curves do not allow us to assess whether they are stochastic, i.e. random, or deterministic and therefore regulated. However, this type of temporal pattern can also be described and analyzed mathematically as a nonlinear dynamic system. An impediment to classical non-linear analysis of experimental data about cell proliferation is the small size of data sets. We propose a new graphical approach which can detect, although not appropriately characterize, a deterministic structure in the proliferation pattern. We focused on the proliferation kinetics of the established rat hepatoma cell line Fao. This study is a mathematical analysis of three consecutive series, under three different culture conditions: Fao A (7 day passages), Fao B (6 day passages), and Fao C (5 day passages), maintained for months in culture.