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Local stability implies global stability for the 2-dimensional Ricker map

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arxiv 1209.2406 v1 pith:ZXPNN7DA submitted 2012-09-11 math.DS cs.NAmath.NA

Ferenc A. Bartha (1 , 2) , Ábel Garab (2) , Tibor Krisztin (2 , 3) ((1) CAPA group , Department of Mathematics , University of Bergen , (2) Bolyai Institute
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University of Szeged (3) Analyis Stochastics Research Group of the Hungarian Academy of Sciences Bolyai Institute University of Szeged)
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classification math.DScs.NAmath.NA
keywords stabilityequationglobalimplieslocalrickeraidedalpha
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abstract

Consider the difference equation $x_{k+1}=x_k e^{\alpha-x_{n-d}}$ where $\alpha$ is a positive parameter and d is a non-negative integer. The case d = 0 was introduced by W.E. Ricker in 1954. For the delayed version d >= 1 of the equation S. Levin and R. May conjectured in 1976 that local stability of the nontrivial equilibrium implies its global stability. Based on rigorous, computer aided calculations and analytical tools, we prove the conjecture for d = 1.

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