05/02/2013 - 3:00pm
05/02/2013 - 4:00pm
Robert Sacker, University of Southern California

In this paper we investigate the long term behavior of solutions of the periodic Sigmoid Beverton-Holt difference equation,
    $ x_{n+1}=\frac{a_n x_n^{\de_n}}{1 + x_n^{\de_n}},\quad x_0 $  > $ 0,\quad n = 0,1,2,\dots, $
where the $ a_n $ and $ \de_n $ are $ p $-periodic positive sequences. Under certain conditions there are shown to exist an asymptotically stable $ p $-periodic state and a $ p $-periodic Allee state with the property that populations smaller than the Allee state are driven to extinction while populations greater than the Allee state approach the stable state thus accounting for the long term behavior of all initial states.

This is of paramount importance in the management of fisheries and establishment of safeguards against overfishing. Stephens and Sutherland described several scenarios that cause the Allee effect in animals. For example, cod and many freshwater fish species have high juvenile mortality when there are fewer adults. Fewer red sea urchin give rise to worsening feeding conditions of their young and less protection from predation. In some mast flowering trees, such as smooth cordgrass, Spartina alterniflora, low population density results in lower probability of seed production and germination.

Roberts South 105, Claremont McKenna College
Analysis Seminar-Sacker.pdf498.4 KB

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