This is a project in Mathematics with applications to Biology. There are three main goals in this project: a) To develop the understanding of hyperbolic dynamics; b) To develop the applications of the renormalization theory in hyperbolic dynamics, Physics and Biology; c) To analyse real data coming from Biology and to develop techniques of Dynamical Systems, Renormalization, Quantum Physics and Game Theory to study the data. The members of the team are currently working with researchers from University of Warwick, Imperial College of London, Instituto de Matemática Pura e Aplicada (Rio de Janeiro), Universidade de São Paulo, University of Stony Brook, Forschungszentrum Julich GmbH (Julich), Fraunhofer-Institut Schloss Birlinghoven (Sankt Augustin) and School of Biological Sciences, Royal Holloway University London. In particular with David Rand, Stefano Luzzatto, Maurício Peixoto, Welington de Melo, Edson de Faria, Zaqueu Coelho, Dennis Sullivan, R. Engbert, F. Pasemann, J. V. Sánchez-Andrés, L. Menéndez de la Prida, K. M. Briggs, F. Drepper, H. Siegel, V. A. A. Jansen, M. E. Ramsay, W. J. Edmunds, C. J. Rhodes, M. C. J. Maiden. This project will be essential for the members of the team to be able to continue colaborating with these international researchers interchanging ideas and knowledge. The international researchers will visit Porto University and the members of the team will visit the International Institutions mainly to work with each other and also to present seminars and give advanced short courses. Another important point of this project is that the knowledge obtained from its execution will allow the Ph.D members of this team to supervise M.Sc and Ph.D students in Portugal.