Mean-square stability of the zero equilibrium of the nonlinear delay differential equation: Nicholson’s blowflies application

Time Derivative ◽

Sufficient Conditions ◽

Deterministic System ◽

Mean Square ◽

Mean Square Stability ◽

Delay Differential ◽

Nonlinear Delay

Abstract We are concerned about the stochastic nonlinear delay differential equation. The stochasticity arises from the white Gaussian noise which is the time derivative of the standard Brownian motion. The main objective of this paper is to introduce a new technique using Lyapunov functional for the study of stability of the zero solution of the stochastic delay differential system. Constructing a new appropriate deterministic system in the neighborhood of the origin is an effective way to investigate the necessary and sufficient conditions of stability in the sense of the mean square. Nicholson's blowflies equation is one of the major problems in ecology, necessary conditions for the possible extinction of the Nicholson's blowflies population are investigated. We support our theoretical results by providing areas of stability and some numerical simulations of the solution of the system using the Euler-Maruyama scheme which is mean square stable \cite{Maruyama1955,Cao2004}.

Subcritical Hopf bifurcation in dynamical systems described by a scalar nonlinear delay differential equation

Physical Review E ◽

10.1103/physreve.69.036210 ◽

2004 ◽

Vol 69 (3) ◽

Cited By ~ 47

Author(s):

Laurent Larger ◽

Jean-Pierre Goedgebuer ◽

Thomas Erneux

Keyword(s):

Differential Equation ◽

Hopf Bifurcation ◽

Dynamical Systems ◽

Subcritical Hopf Bifurcation ◽

Delay Differential ◽

Nonlinear Delay

Numerical Dynamics of Nonstandard Finite Difference Method for Nonlinear Delay Differential Equation

International Journal of Bifurcation and Chaos ◽

10.1142/s021812741850133x ◽

2018 ◽

Vol 28 (11) ◽

pp. 1850133 ◽

Cited By ~ 1

Author(s):

Xiaolan Zhuang ◽

Qi Wang ◽

Jiechang Wen

Keyword(s):

Differential Equation ◽

Finite Difference ◽

Finite Difference Method ◽

Difference Method ◽

Discrete System ◽

Delay Differential ◽

Nonlinear Delay ◽

Nonstandard Finite Difference

In this paper, we study the dynamics of a nonlinear delay differential equation applied in a nonstandard finite difference method. By analyzing the numerical discrete system, we show that a sequence of Neimark–Sacker bifurcations occur at the equilibrium as the delay increases. Moreover, the existence of local Neimark–Sacker bifurcations is considered, and the direction and stability of periodic solutions bifurcating from the Neimark–Sacker bifurcation of the discrete model are determined by the Neimark–Sacker bifurcation theory of discrete system. Finally, some numerical simulations are adopted to illustrate the corresponding theoretical results.

A Necessary and Sufficient Condition for the Oscillation of an Even Order Nonlinear Delay Differential Equation

Canadian Journal of Mathematics ◽

10.4153/cjm-1973-115-0 ◽

1973 ◽

Vol 25 (5) ◽

pp. 1078-1089 ◽

Cited By ~ 10

Author(s):

Bhagat Singh

Keyword(s):

Differential Equation ◽

Oscillatory Behavior ◽

Necessary And Sufficient Condition ◽

Even Order ◽

Image Position ◽

Delay Differential ◽

Necessary And Sufficient ◽

Nonlinear Delay

In this paper we study the oscillatory behavior of the even order nonlinear delay differential equation(1)where(i) denotes the order of differentiation with respect to t. The delay terms τi σi are assumed to be real-valued, continuous, non-negative, non-decreasing and bounded by a common constant M on the half line (t0, + ∞ ) for some t0 ≧ 0.