Global Asymptotic Stability of a Generalization of the Pielou Difference Equation

• Published in: Mediterranean journal of mathematics Vol. 16; no. 4; pp. 1 - 18
• Main Authors: Garab, Ábel, Jiménez López, Víctor, Liz, Eduardo
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.08.2019; Springer Nature B.V
• Subjects: Asymptotic properties; Dependence; Difference equations; Mathematics; Mathematics and Statistics; Stability; Primary 39A30; 39A20; Secondary 92D25; Global asymptotic stability; permanence; dominance; higher order difference equations; discrete population models
• ISSN: 1660-5446; 1660-5454
• DOI: 10.1007/s00009-019-1374-2

The Pielou equation is a well-known discrete-time population model in which the per capita growth rate depends on the population size, but the density dependence operates with a delay of d generations. Thus, the between-year dynamics are governed by a difference equation of order d + 1 . The main result in this paper establishes the global stability of the unique positive equilibrium for a generalization of the two-dimensional Pielou equation. Our proof is based on a rather natural combination of two techniques which could be, in principle, applicable to obtain global asymptotic stability in other problems: some dominance conditions and the determination of a first integral for a related equation, which turns out to be a quasi-Lyanupov function for the generalized Pielou equation. We provide additional results on the global dynamics of the generalized Pielou equation for dimensions higher than two, and discuss its relationship with other families of difference equations traditionally employed for modelling population dynamics.