Global Attractivity of xn+1 = )1 − α)xn + βe−γxn−k

The Global Attractivity of the Rational Difference

This paper studies the behavior of positive solutions of the recursive equation yn = 1 + yn−k yn−m , n = 0, 1, 2, . . . , with y−s, y−s+1, . . . , y−1 ∈ (0,∞) and k, m ∈ {1, 2, 3, 4, . . .}, where s = max{k, m}. We prove that if gcd(k, m) = 1, with k odd, then yn tends to 2, exponentially. When combined with a recent result in E. A. Grove, and G. Ladas, Periodicities in Nonlinear Difference Equ...

Global Attractivity in a Genotype Selection Model

where k∈N and {βn} is a sequence of positive real numbers. When k= 0 and βn ≡ β for alln∈N, (1.1) was introduced by May [2, pages 513–560] as an example of a map generated by a simple model for frequency-dependent natural selection. The local stability of the equilibrium ȳ = 1/2 of (1.1) was investigated by May [2]. In [1] (see also [3]), Grove further investigated the stability of the equilibr...

Global Attractivity in Nonlinear Delay Difference Equations

We obtain a set of sufficient conditions under which all positive solutions of the nonlinear delay difference equation x„+l = x„f(xn_k), n = 0,1,2,..., are attracted to the positive equilibrium of the equation. Our result applies, for example, to the delay logistic model JVI+i = aN¡/(\ +ßNt_k) and to the delay difference equation xn+i = x„er^~x"-k'1 .

Global Attractivity in a Nonlinear Difference Equation

In this paper, we study the asymptotic behavior of positive solutions of the nonlinear difference equation xn+1 = xnf(xn−k), where f : [0,∞)→ (0,∞) is a unimodal function, and k is a nonnegative integer. Sufficient conditions for the positive equilibrium to be a global attractor of all positive solutions are established. Our results can be applied to to some difference equations derived from ma...

Global Attractivity of a Family of Max-Type Difference Equations