Chaos and population disappearances in simple ecological models.

*(English)*Zbl 0977.92032Summary: 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) semistability in which all orbits tend toward the origin or a semistable fixed point, (iii) bistability for which the origin and an interval bounded away from the origin are attracting, (iv) chaotic semistability in which there 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 [W.E. Ricker, J. Fish. Res. Board. Can. 11, 559-623 (1954)], and generalized Beverton-Holt maps [R.J.H. Beverton and S.J. Holt, On the dynamics of exploited fish populations (1957)], with constant harvesting rates, two bifurcations are shown to lead to sudden population disappearances: a saddle node bifurcation corresponding to a transition from bistability to extinction and a chaotic blue sky catastrophe corresponding to a transition from bistability to essential extinction.

(i) extinction in finite time for all initial densities, (ii) semistability in which all orbits tend toward the origin or a semistable fixed point, (iii) bistability for which the origin and an interval bounded away from the origin are attracting, (iv) chaotic semistability in which there 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 [W.E. Ricker, J. Fish. Res. Board. Can. 11, 559-623 (1954)], and generalized Beverton-Holt maps [R.J.H. Beverton and S.J. Holt, On the dynamics of exploited fish populations (1957)], with constant harvesting rates, two bifurcations are shown to lead to sudden population disappearances: a saddle node bifurcation corresponding to a transition from bistability to extinction and a chaotic blue sky catastrophe corresponding to a transition from bistability to essential extinction.