Journal of Dynamics and Differential Equations

, Volume 31, Issue 4, pp 2109–2125 | Cite as

Stability and Moment Boundedness of the Stochastic Linear Age-structured Model

  • Zhen Wang
  • Xiong LiEmail author


In this paper we study the stability and moment boundedness of the solutions to the stochastic linear age-structured model. For the linear age-structured model with general noise, the stability of the first moment is identical to that of the corresponding deterministic age-structured model. However, the stability and boundedness of the second moment are complicated and depend on the stochastic terms. For the linear age-structured model with the additive noise, we first give the explicit expression of the second moment by the Laplace transform in Itô-Doob integral, and then establish the sufficient conditions for boundedness and unboundedness of the second moment through the supremum of the real parts of all characteristic roots.


Stochastic age-structured model Stochastic stability Moment boundedness 

Mathematics Subject Classification

37H10 34F05 34K30 34D20 



We would like to thank the referee for his/her valuable comments and suggestions that greatly improve the presentation of this work.


  1. 1.
    Anita, S.: Analysis and Control of Age-Dependent Population Dynamics. Kluwer Academic Publishers, Netherlands (2000)CrossRefGoogle Scholar
  2. 2.
    Block, G.L., Allen, L.J.S.: Population extinction and quasi-stationary behavior in stochastic density-dependent structured models. Bull. Math. Biol. 62, 199–228 (2000)CrossRefGoogle Scholar
  3. 3.
    Brauer, F., van den Driessche, P., Wu, J.: Mathematical Epidemiology, Part of the Lecture Notes in Mathematics book series (LNM, volume 1945), Springer, (2008)Google Scholar
  4. 4.
    Chowdhury, M., Allen, E.J.: A stochastic continuous-time age-structured population model. Nonlinear Anal. 47, 1477–1488 (2001)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cushing, J.M.: An introduction to structured population dynamics. In: CMB-NSF Regional Conference Series in Applied Mathematics, SIAM, (1998)Google Scholar
  6. 6.
    Gurtin, M.E., MacCamy, R.C.: Nonlinear age-dependent population dynamics. Arch. Ration. Mech. Anal. 54, 281–300 (1974)CrossRefGoogle Scholar
  7. 7.
    Hossain, Md.J., Islam, Md.S.: Non-linear age-time-population dependent stochastic populatiuon model. Bangladesh J. Sci. Res. 25(1), 73–86 (2012)CrossRefGoogle Scholar
  8. 8.
    Iannelli, M.: Mathematical Theory of Age-structured Population Dynamics. Applied Mathematics Monographs. C.N.R, Giadini Editori e stampatori in Pisa (1994)Google Scholar
  9. 9.
    Kirby, R.D., Ladde, A.G., Ladde, G.S.: Stochastic Laplace transform with application. Commun. Appl. Anal. 14, 373–392 (2010)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Li, R., Pang, W., Leung, P.: Convergence of numerical solutions to stochastic age-structured population equations with diffusions and Markovian switching. Appl. Math. Comput. 216, 744–752 (2010)MathSciNetzbMATHGoogle Scholar
  11. 11.
    McKendrick, A.G.: Applications of the mathematics to medical problems. Proc. Edinb Math. Soc. 44, 98–130 (1926)CrossRefGoogle Scholar
  12. 12.
    Metz, J.A.J., Diekmann, E.O. (Eds.) The Dynamics of Physiologically Structured Populations, Springer Lecture Notes in Biomathematics, 68 Springer, Heildelberg, (1986)Google Scholar
  13. 13.
    Pang, W.K., Li, R., Liu, M.: Exponential stability of numerical solutions to stochastic age-dependent population equations. Appl. Math. Comput. 183, 152–159 (2006)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Pollard, J.H.: On the use of the direct matrix product in analyzing certain stochastic population model. Biometrika 53, 397–415 (1966)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Pollard, J.H.: Mathematical Models for the Growth of Human Populations. Cambridge University Press, Cambridge (1973)zbMATHGoogle Scholar
  16. 16.
    Sharpe, F.R., Lotka, A.J.: A problem in age distribution. Philos. Mag. 21, 435–438 (1911)CrossRefGoogle Scholar
  17. 17.
    Webb, G.F.: Theory of Nonlinear Age-Dependent Population Dynamics. Marcel Dekker, New York (1985)zbMATHGoogle Scholar
  18. 18.
    Wang, Z., Li, X.: Stability of non-densely defined semilinear stochastic evolution equations with application to the stochastic age-structured model. J. Dyn. Differ. Equ. 27, 261–281 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Wang, Z., Li, X., Lei, J.: Second moment boundedness of linear stochastic delay differential equations. Discrete Contin. Dyn. Syst. Ser. B 19, 2963–2991 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Wang, Z., Li, X., Lei, J.: Moment boundedness of linear stochastic differential equation with distributed delay. Stoch. Proc. Appl. 124, 586–612 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Zhang, Q., Liu, W., Nie, Z.: Existence, uniqueness and exponential stability for stochastic age-dependent population. Appl. Math. Comput. 154, 183–201 (2004)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Zhang, Q.: Exponential stability of numerical solutions to a stochastic age-structured population system with diffusion. J Comput. Appl. Math. 220, 22–33 (2008)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsHefei University of TechnologyHefeiPeople’s Republic of China
  2. 2.School of Mathematical SciencesBeijing Normal UniversityBeijingPeople’s Republic of China

Personalised recommendations