Chaos and population disappearances in simple ecological models |
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Authors: | Sebastian J Schreiber |
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Institution: | (1) Department of Mathematics, Western Washington University, Bellingham, Washington 98225, USA. e-mail:sschreib@cc.wwu.edu, US |
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Abstract: | 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 semi-stable 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 and generalized Beverton-Holt maps with constant harvesting rates, two birfurcations 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.
Received: 14 February 2000 / Revised version: 15 August 2000 / Published online: 16 February 2001 |
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Keywords: | or phrases: Population dynamics – Extinction – Population disappearance – Unimodal maps – Chaos – Bifurcations |
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