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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

Eduardo Liz 1,

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.
Keywords: discrete-time single-species model, Ricker difference equation, Schwarzian derivative., global stability.
Mathematics Subject Classification: Primary: 39A11; Secondary: 92D25, 92D4.
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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