This GeoGebraBook provides a short tour on the iteration of maps starting from the discussion of a case study: population dynamics. The case study was chosen for its relevance to reality and for actuality of the topic (linked to the history of science, to biodiversity, to the caring for the environment, to the preservation of species, ...). The material can be availed by students of upper secondary school, even in the early years of the course with regard to the first two chapters: the best educational use is to use it as a motivator to the introduction and theoretical study of the various mathematical tools (algebra, polynomials , functions, composition of functions, graphs, equations, inequalities, tangent lines, approximation, linearisation, complex numbers, ...) and as a valuable accompanying teaching tool.