Global attractivity in nonlinear delay difference equations (original) (raw)

Oscillations and global attractivity in a discrete delay logistic model

Quarterly of Applied Mathematics, 1992

Consider the discrete delay logistic model \[ N t + 1 = α N t 1 + β N t − k , ( 1 ) {N_{t + 1}} = \frac {{\alpha {N_t}}}{{1 + \beta {N_{t - k}}}}, \qquad \left ( 1 \right ) \] where α ∈ ( 1 , ∞ ) , β ∈ ( 0 , ∞ ) \alpha \in \left ( {1, \infty } \right ), \beta \in \left ( {0, \infty } \right ) , and k ∈ N = { 0 , 1 , 2 , . . . } k \in \mathbb {N} = \left \{{0, 1, 2,...} \right \} . We obtain conditions for the oscillation and asymptotic stability of all positive solutions of Eq. (1) about its positive equilibrium ( α − 1 ) / β \left ( {\alpha - 1} \right )/\beta . We prove that all positive solutions of Eq. (1) are bounded and that for k = 0 k = 0 and k = 1 k = 1 the positive equilibrium ( α − 1 ) / β \left ( {\alpha - 1} \right )/\beta is a global attractor.

Global attractivity in a rational delay difference equation with quadratic terms

Journal of Difference Equations and Applications, 2011

For the following rational difference equation with arbitrary delay and quadratic terms: x nþ1 ¼ Ax 2 n þ Bx n x n2k þ Cx 2 n2k þ Dx n þ Ex n2k þ F ax n þ bx n2k þ g ; we determine sufficient conditions on the parameter values which guarantee that the unique non-negative fixed point attracts all positive solutions. When the fixed point is the origin (F ¼ 0), we show that it attracts all non-negative solutions of the more general equation x nþ1 ¼ Ax 2 n þ Bx n x n2k þ Cx 2 n2k þ D 1 x n þ D 2 x n21 þ • • • þ D m x n2mþ1 ax n þ bx n2k þ g ; where m [ {1; 2;. .. }. We also show that altering some of the above conditions on parameters causes the origin to not be globally attracting.

Global Attractivity Results for Nonlinear Delay Differential Equations

Journal of Mathematical Analysis and Applications, 1999

The aim of this paper is to give sufficient conditions for global attractivity of the zero solution of the nonlinear delay differential equation x'(t) =-p(t)f(x(t-7)). An example which guarantees that our $ stability condition is the best possible is d S 0 given.

Global Attractivity in a Predator–Prey System with Pure Delays

Proceedings of the Edinburgh Mathematical Society, 2008

We consider a delay predator–prey system without instantaneous negative feedback and establish some conditions for global attractivity of the positive equilibrium of the system which generalize and improve some of the existing ones. When the system is decoupled, one of the main results reduces to the well-known Wright 3/2 stability condition for the delayed logistic equation.

On the global attractor of delay differential equations with unimodal feedback

arXiv preprint arXiv:0807.3022, 2008

We give bounds for the global attractor of the delay differential equationẋ(t) = −µx(t) + f (x(t − τ )), where f is unimodal and has negative Schwarzian derivative. If f and µ satisfy certain condition, then, regardless of the delay, all solutions enter the domain where f is monotone decreasing and the powerful results for delayed monotone feedback can be applied to describe the asymptotic behaviour of solutions. In this situation we determine the sharpest interval that contains the global attractor for any delay. In the absence of that condition, improving earlier results, we show that if the delay is sufficiently small, then all solutions enter the domain where f ′ is negative. Our theorems then are illustrated by numerical examples using Nicholson's blowflies equation and the Mackey-Glass equation.

Global stability of a class of scalar nonlinear delay differential equations

2003

The problem of global stability in scalar delay differential equations of the form x (t) = f 1 (x(t − h))g 2 (x(t)) − f 2 (x(t − h))g 1 (x(t)) is studied. Functions f i and g i , i = 1, 2, are continuous and such that the equation assumes a unique positive equilibrium. Two types of sufficient conditions for the global asymptotic stability of the unique equilibrium are established: (i) delay independent, and (ii) conditions involving the size h of the delay. Delay independent conditions make use of the global stability in the limiting (as h → ∞) difference equation f 1 (x n)g 2 (x n+1) = f 2 (x n)g 1 (x n+1): the latter always implying the global stability in the differential equation for all values of the delay h ≥ 0. The delay dependent conditions involve the global attractivity property in specially constructed one-dimensional maps (difference equations) that include the nonlinearities f i and g i , and the delay h.

A note on asymptotic stability conditions for delay difference equations

International Journal of Mathematics and Mathematical Sciences, 2005

We obtain necessary and sufficient conditions for the asymptotic stability of the linear delay difference equation x n+1 + p N j=1 x n−k+( j−1)l = 0, where n = 0,1,2,..., p is a real number, and k, l, and N are positive integers such that k > (N − 1)l.

Global attractivity for scalar differential equations with small delay

Journal of Mathematical Analysis and Applications, 2007

For scalar functional differential equationsẋ(t) = f (t, x t), we refine the method of Yorke and 3/2-type conditions to prove the global attractivity of the trivial solution. The results are applied to establish sufficient conditions for the global attractivity of the positive equilibrium of scalar delayed population models of the formẋ(t) = x(t)f (t, x t), and illustrated with the study of two food-limited population models with delay, for which several criteria for their global attractivity are given.

The asymptotic behavior of a class of nonlinear delay difference equations

Proceedings of the American Mathematical Society, 2000

The asymptotic behavior of difference equations of type xn = x p n−1 [1 + g(m i=1 f i (x n−i))], p > 0, is studied, where g and each f i are continuous real functions with g decreasing and f i increasing. Results include sufficient conditions for permanence, oscillations and global attractivity.