Local stability implies global stability in some one-dimensional discrete single-species models
By Eduardo Liz. Formal Specification and Development in B, 2006
59 citations recorded by the source corpus.
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 (Mobius transformations) and the Schwarzian derivative.
Subfields: Ecology, Genetics, Public Health, Environmental and Occupational Health
View Paper: https://www.semanticscholar.org/paper/c5a6524577a843f093f76ae068d4b2845bbed1ec
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Citations in the corpus
- Local Stability Implies Global Stability for the Planar Ricker Competition Model (2014, unknown): “It is well-known (see, Devaney [5], Elaydi [6], and Liz [16]) that in many well-know unimodal maps, such as the logistic and the Ricker maps, local asymptotic stability of the fixed point implies global asymptotic stability.”
- On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities (2014, unknown): “, [1,3–5,9–13,15–17,20,23,24,27] and the references therein.”
- Map dynamics versus dynamics of associated delay reaction–diffusion equations with a Neumann condition (2010, background): “For example, by theorem 2.3, proposition 2.5 and corollary 2.7 in Liz (2007), together with some straightforward calculations, equations (4.4) and (4.6) can be easily obtained.”