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Local stability implies global stability in some one-dimensional discrete single-species models. (English) Zbl 1187.39026

Summary: 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.

MSC:

39A30 Stability theory for difference equations
39A20 Multiplicative and other generalized difference equations
92D25 Population dynamics (general)
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