Global attractivity of a nonautonomous discrete logistic model
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 \[ \lim_{n \rightarrow \infty}r_n\leq 2 . \]
scientific article; zbMATH DE number 1448527