Hostname: page-component-6bf8c574d5-b4m5d Total loading time: 0 Render date: 2025-02-16T17:46:45.259Z Has data issue: false hasContentIssue false
  • English
  • Français

NOVEL STABILITY CONDITIONS FOR SOME GENERALIZATION OF NICHOLSON’S BLOWFLIES MODEL WITH STOCHASTIC PERTURBATIONS

Part of: Applications - Dynamical systems and ergodic theory Functional-differential and differential-difference equations

Published online by Cambridge University Press:  23 August 2023

LEONID SHAIKHET Open the ORCID record for LEONID SHAIKHET [Opens in a new window]  and
SYED ABBAS Open the ORCID record for SYED ABBAS [Opens in a new window]
LEONID SHAIKHET
Affiliation:
Department of Mathematics, Ariel University, Ariel 40700, Israel; e-mail: [email protected]
SYED ABBAS*
Affiliation:
School of Mathematical and Statistical Sciences, Indian Institute of Technology Mandi, Mandi, H.P. 175005, India
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a generalization of the well-known nonlinear Nicholson blowflies model with stochastic perturbations. Stability in probability of the positive equilibrium of the considered equation is studied. Two types of stability conditions: delay-dependent and delay-independent conditions are obtained, using the method of Lyapunov functionals and the method of linear matrix inequalities. The obtained results are illustrated by numerical simulations by means of some examples. The results are new, and complement the existing ones.

Keywords

positive equilibriumstochastic perturbationswhite noisestability in probabilityLyapunov functionalslinear matrix inequality (LMI)numerical simulations

MSC classification

Secondary: 37N25: Dynamical systems in biology 34K20: Stability theory
Type
Research Article
Information
The ANZIAM Journal , Volume 64 , Issue 4 , October 2022 , pp. 394 - 405
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abbas, S. and Shaikhet, L., “Stability of three species symbiosis model with delay and stochastic perturbations”, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 29 (2022) 241–251; https://online.watsci.org/abstract_pdf//2022v29/v29n4b-pdf/2.pdf.Google Scholar
Berezansky, L., Braverman, E. and Idels, L., “Nicholson’s blowflies differential equations revised: main results and open problems”, Appl. Math. Model. 34 (2010) 1405–1417; doi:10.1016/j.apm.2009.08.027.CrossRefGoogle Scholar
Bradul, N. and Shaikhet, L., “Stability of the positive point of equilibrium of Nicholson’s blowflies equation with stochastic perturbations: numerical analysis”, Discrete Dyn. Nat. Soc. 2007 (2007) Article ID 92959, 25 pages; doi:10.1155/2007/92959.CrossRefGoogle Scholar
Burgos, C., Cortes, J. C., Shaikhet, L. and Villanueva, R. J., “A nonlinear dynamic age-structured model of e-commerce in Spain: stability analysis of the equilibrium by delay and stochastic perturbations”, Commun. Nonlinear Sci. Numer. Simul. 64 (2018) 149–158; doi:10.1016/j.cnsns.2018.04.022.CrossRefGoogle Scholar
Ding, H. S. and Alzabut, J. O., “Existence of positive almost periodic solutions for Nicholson’s blowflies model”, Electron. J. Differential Equations 2015 (2015) 1–6; https://ejde.math.txstate.edu/Volumes/2015/180/ding.pdf.Google Scholar
Gikhman, I. I. and Skorokhod, A. V., Stochastic differential equations (Springer, Berlin, 1972).10.1007/978-3-642-88264-7CrossRefGoogle Scholar
Gurney, W., Blythe, S. and Nisbet, R., “Nicholson’s blowflies revised”, Nature 287 (1980) 17–21; doi:10.1038/287017a0.CrossRefGoogle Scholar
Kocic, V. L. and Ladas, G., “Oscillation and global attractivity in discrete model of Nicholson’s blowflies”, Appl. Anal. 38 (1990) 21–31; doi:10.1080/00036819008839952.CrossRefGoogle Scholar
Lin, C. K., Lin, C. T., Lin, Y. and Mei, M., “Exponential stability of nonmonotone traveling waves for Nicholson’s blowflies equation”, SIAM J. Math. Anal. 46 (2014) 1053–1084; doi:10.1137/12090439.CrossRefGoogle Scholar
Liu, B., “Global exponential stability of positive periodic solutions for delayed Nicholson’s blowflies model”, J. Math. Anal. Appl. 412 (2014) 212–221; doi:10.1016/j.jmaa.2013.10.049.CrossRefGoogle Scholar
Maruyama, G., “Continuous Markov processes and stochastic equations”, Rend. Circ. Mat. Palermo (2) 4 (1955) 48–90; doi:10.1007/BF02846028.CrossRefGoogle Scholar
Nicholson, A. J., “An outline of the dynamics of animal populations”, Aust. J. Zoology 2 (1954) 9–65; doi:10.1071/ZO9540009.CrossRefGoogle Scholar
Pang, S., Deng, F. and Mao, X., “Almost sure and moment exponential stability of Euler–Maruyama discretizations for hybrid stochastic differential equations”, J. Comput. Appl. Math. 213 (2008) 127–141; doi:10.1016/j.cam.2007.01.003.CrossRefGoogle Scholar
Rudin, W., Principles of mathematical analysis, 3rd edn (McGraw Hill Education, New York, 2017).Google Scholar
Shaikhet, L., Lyapunov functionals and stability of stochastic difference equations (Springer Science & Business Media, London, 2011); doi:10.1007/978-0-85729-685-6.CrossRefGoogle Scholar
Shaikhet, L., Lyapunov functionals and stability of stochastic functional differential equations (Springer Science & Business Media, Cham, 2013); doi:10.1007/978-3-319-00101-2.CrossRefGoogle Scholar
So, J. W. H. and Yu, J. S., “On the stability and uniform persistence of a discrete model of Nicholson’s blowflies”, J. Math. Anal. Appl. 193 (1995) 233–244; doi:10.1006/jmaa.1995.1231.CrossRefGoogle Scholar