Global attractivity of a nonautonomous discrete logistic model
type: Item
instance of: scholarly article
title: Global attractivity of a nonautonomous discrete logistic model
zbMATH Open document ID: 0965.39012
author: Qinqin Zhang
author: Zhan Zhou
published in: Hokkaido Mathematical Journal
publication date: 2001-07-23T00:00:00Z
review text: The authors consider the discrete nonautonomous logistic model \[ x_{n+1}=x_n\exp[r_n(1-x_n) ], \qquad n \in \mathbb{N},\tag{1} \] where \(\{ r_n \}\) is a sequence of nonnegative numbers. It is easy to see that, for any given initial condition \[ x_0 = a > 0, \tag{2} \] Eq. (1) has a unique solution \(\{x_n \}\) which is positive for all \(n \in\mathbb{N}\) and satisfies \((2).\) It is known that every solution to \((1)\) with \((2)\) tends to \(1\) if \(r_n \leq 3/2\) and \(\sum_{n=0}^{\infty} r_n = \infty.\) Remark that for \(r_n \equiv r >0\), Eq. (1) reduces to \[ x_{n+1}=x_n \exp[r(1-x_n) ], \qquad n \in \mathbb{N},\tag{3} \] which is a discrete population model of a single species with nonoverlapping generations. In this paper the authors obtain the following results. Theorem 1. If \[ \sum_{n=0}^{\infty} r_n = \infty,\quad \text{and}\quad \lim\sup_{n \rightarrow \infty} r_n \leq 2, \] then any solution \(\{x_n \}\) to Eq. (1) with (2) converges to \(1\) as \(n \rightarrow \infty \). Theorem 2. Assume that \(\{r_n \}\) is bounded. If \[ \lim\sup_{n \rightarrow \infty} r_n > 2, \] then a solution \(\{x_n \}\) to Eq. (1) with \((2)\) can not converge to \(1\) as \(n \rightarrow \infty \). Combining the above two theorems they obtain the necessary and sufficient conditions, for the following statement: Assume \(\sum_{n=0}^{\infty} r_n = \infty \) holds and the limit \(\lim_{n \rightarrow \infty} r_n \) exists. Then any solution \(\{x_n \}\) to Eq. (1) with (2) converges to 1 as \(n \rightarrow \infty\) if and only if \[ \lim_{n \rightarrow \infty}r_n\leq 2 . \]
Mathematics Subject Classification ID: 39B05
Mathematics Subject Classification ID: 39A11
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 1448527
zbMATH Keywords: global attractivity
zbMATH Keywords: nonautonomous discrete logistic model
zbMATH Keywords: positive solution
zbMATH Keywords: discrete population model
reviewed by: Akira Tsutsumi
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1350912955
OpenAlex ID: W2093236583
DOI: 10.14492/HOKMJ/1350912955
description: scientific article; zbMATH DE number 1448527