Abstract: An algorithmic method will be introduced for obtaining a database of global dynamical behaviours encountered in a multi-parameter family of discrete dynamical systems on a bounded set in R^n. The dynamics is represented by means of the Conley-Morse decomposition, which includes the homological Conley indices of the Morse sets computed in an algorithmic way using the CHomP software (http://chomp.rutgers.edu/). Morse decompositions for adjacent parameter boxes are matched and provide rigorous continuation results, as well as help finding possible bifurcations. 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.