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

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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).

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The data sets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.

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This work was supported by the Mathematics Department—Mansoura University of Egypt.

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Authors and Affiliations

  1. Mathematics Department, Faculty of Science, Mansoura University, Mansoura, Egypt

    H. El-Metwally, M. A. Sohaly & I. M. Elbaz

  2. Basic Sciences Department, Faculty of Engineering, The British University in Egypt, El Sherouk City, Cairo, Egypt

    I. M. Elbaz

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  1. H. El-Metwally
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  2. M. A. Sohaly
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  3. I. M. Elbaz
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El-Metwally, H., Sohaly, M.A. & Elbaz, I.M. Mean-square stability of the zero equilibrium of the nonlinear delay differential equation: Nicholson’s blowflies application. Nonlinear Dyn 105, 1713–1722 (2021). https://doi.org/10.1007/s11071-021-06696-6

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