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 the 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 Maruyama (Rendiconti del Circolo Matematico di Palermo 4(1):48, 1955), Cao et al. (Appl Math Comput 159(1):127–135, 2004).
Highlights
Functional differential equations (FDEs) have motivated many mathematical and applied statistical research
Realistic models must include some of the past history of the state of the system and this, in turn, leads to the delay differential equations (DDEs)
Θ(t, Xt) = (2(p − δ) + 2pτ |p − δ| + λ2)E X2(t) is negative definite if pτ |p − δ| +. This delay-dependent condition is necessary for the mean square stability
Summary
Functional differential equations (FDEs) have motivated many mathematical and applied statistical research. Stochastic models are proposed to capture the uncertainty and variations in the mathematical models by perturbing the deterministic system with a white Gaussian noise which is ill-defined, and change it to a Brownian motion which is well-defined by ζ(t)dt = dB(t) Without solving these systems, we study the stability of the equilibria which is a very effective way to have a good insight of the solutions and their properties. We shall determine the mean square stability conditions of the stochastic nonlinear delay differential equation and its general form using a new way of constructing a delayed-deterministic system by Lyapunov functional in the presence of the white noise term.
Sign-in/Register to view highlights & summary 
Submitted Version (
Free)
Published Version
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call