We introduce an algorithmic method for obtaining a database of global dynamical behaviours encountered in a multi-parameter family of discrete dynamical systems on a bounded set in IR^n. The dynamics is represented by means of a Conley-Morse decomposition, which includes the homological Conley indices of the Morse sets computed in an algorithmic way. Morse decompositions for adjacent parameter boxes are matched and provide rigorous continuation results. A nonlinear overcompensatory Leslie population model is used as a sample application of this method. This is joint work with Zin Arai, William Kalies, Hiroshi Kokubu, Konstantin Mischaikow, and Hiroe Oka.