Chaos and population disappearances in simple ecological models

A class of truncated unimodal discrete-time single species models for which low or high densities result in extinction in the following generation are considered. A classification of the dynamics of these maps into five types is proven: (i) extinction in finite time for all initial densities, (ii) s emistability in which all orbits tend toward the origin or a semistable fix ed point, (iii) bistability for which the origin and an interval bounded aw ay from the origin are attracting, (iv) chaotic semistability in which ther e is an interval of chaotic dynamics whose compliment lies in the origin's basin of attraction and (v) essential extinction in which almost every (but not every) initial population density leads to extinction in finite time. Applying these results to the Logistic, Ricker and generalized Beverton-Hol t maps with constant harvesting rates, two birfurcations are shown to lead to sudden population disappearances: a saddle node bifurcation correspondin g to a transition from bistability to extinction and a chaotic blue sky cat astrophe corresponding to a transition from bistability to essential extinc tion.