Dynamical Behaviour on the Parameter Space: New Populational Growth Models Proportional to Beta Densities Sandra M. Aleixo Mathematics Unit, DEC and CEAUL, Instituto Superior de Engenharia de Lisboa Rua Conselheiro Emídio Navarro, 1,1949-014 Lisboa, Portugal E-mail: sandra.aleixo@dec.isel.ipl.pt J. Leonel Rocha Mathematics Unit, DEQ, Instituto Superior de Engenharia de Lisboa Rua Conselheiro Emídio Navarro, 1,1949-014 Lisboa, Portugal E-mail: jrocha@deq.isel.ipl.pt Dinis D. Pestana FCUL, DEIO and CEAUL, Universidade de Lisboa Campo Grande, Edifício C4, 1749-016 Lisboa, Portugal E-mail: dinis.pestana@fc.ul.pt Abstract. We present new populational growth models, generalized logistic models which are proportional to beta densities with shape parameters p and 2, where p > 1, with Malthusian parameter r. The complex dynamical behaviour of these models is investigated in the parameter space (r, p), in terms of topological entropy, using explicit methods, when the Malthusian parameter r increases. This parameter space is split into different regions, according to the chaotic behaviour of the models. Keywords. Beta densities, population dynamics, dynamical behaviour. 1. Introduction To forecast the future evolution of the dynamic of populations is an important issue in several areas, such as biological, ecological, social or economical sciences. Sophisticated random models are now available in population dynamics, but in many instances their mean functions are well known deterministic models, useful as a first approach in applied problems. The logistic (or Verhulst) model, [8], which incorporates in its parameters both the Malthusian growth rate r and the retroaction due to the limitation of natural resources, is a natural candidate to model the dynamic of non-overlapping generations, namely when the unit of time is related to the life span of the individuals in the population. The logistic map has been used with success to model the population growth for some species, but is inadequate for other. In several numerical studies, the families of unimodal maps have been used, allowing for exhaustive investigations in terms of symbolic dynamics, [1]. The unimodal maps theory can be used in many branches of science, namely in the ones mentioned above. In population dynamics, aiming to model the growth of a certain species, the use of the Verhulst Model, which is proportional to the Beta(2, 2) density, [7], has been a standard, although in same cases it doesn’t fit properly the observations. In what follows, we introduce more general models, related to X  Beta(p, q) (p> 0 and q> 0), with probability density function f X (x)= 1 B(p, q) x p−1 (1 − x) q−1 I (0,1) (x), where B(p, q)=  1 0 t p−1 (1 − t) q−1 dt, ∀p, q > 0 is the Euler’s beta function. The generalized logistic models we shall use are proportional to the Beta(p, 2) densities, with p ∈ ]1, +∞[, [2]. This family provides more flexible models to theorize 213 Proceedings of the ITI 2009 31 st Int. Conf. on Information Technology Interfaces, June 22-25, 2009, Cavtat, Croatia