# American Institute of Mathematical Sciences

January  2007, 7(1): 191-199. doi: 10.3934/dcdsb.2007.7.191

## Local stability implies global stability in some one-dimensional discrete single-species models

 1 Departamento de Matemática Aplicada II, E.T.S.I. Telecomunicación, Universidad de Vigo, Campus Marcosende, 36280 Vigo

Received  December 2005 Revised  August 2006 Published  October 2006

We prove a criterion for the global stability of the positive equilibrium in discrete-time single-species population models of the form $x_{n+1}=x_nF(x_n)$. This allows us to demonstrate analytically (and easily) the conjecture that local stability implies global stability in some well-known models, including the Ricker difference equation and a combination of the models by Hassel and Maynard Smith. Our approach combines the use of linear fractional functions (Möbius transformations) and the Schwarzian derivative.
Citation: Eduardo Liz. Local stability implies global stability in some one-dimensional discrete single-species models. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 191-199. doi: 10.3934/dcdsb.2007.7.191
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