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| Consider the second order difference equation $$x_{k+1}=x_k e^{\alpha-x_{k-d}}$$ where $\alpha$ is a positive parameter and $d$ is a nonnegative integer. The case $d=0$ was introduced by W.~E.~Ricker in 1954. For the delayed version $d \geq 1$ 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 $d=1$. We also apply our method to give necessary and sufficient conditions for the global stability of the trivial equilibrium of the difference equation $x_{k+1}=m x_k + \alpha \tanh{x_{k-1}}$, where $m$ and $\alpha$ are real parameters.
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\noindent {\footnotesize This research was supported by the European Union and the State of Hungary, co-financed by the European Social Fund in the framework of T{\'A}MOP 4.2.4.\ A/2-11-1-2012-0001 `National Excellence Program'.} |
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