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

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Abstract

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.

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Notes

  1. But we do not know whether the conclusion is right or not for other stochastic differential equations (see [19, 20]). In [19], the authors gave a conjecture about the characteristic equations for the second moment of stochastic delay differential equations, but they have not proved it.

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Acknowledgements

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

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Authors and Affiliations

  1. School of Mathematics, Hefei University of Technology, Hefei, 230601, Anhui, People’s Republic of China

    Zhen Wang

  2. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Xiong Li

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  1. Zhen Wang
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  2. Xiong Li
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Correspondence to Xiong Li.

Additional information

Zhen Wang: Partially supported by the NSFC (11501158) and AHNSF(1608085QA13).

Xiong Li: Partially supported by the NSFC (11571041) and the Fundamental Research Funds for the Central Universities.

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Wang, Z., Li, X. Stability and Moment Boundedness of the Stochastic Linear Age-structured Model. J Dyn Diff Equat 31, 2109–2125 (2019). https://doi.org/10.1007/s10884-018-9671-1

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  • DOI: https://doi.org/10.1007/s10884-018-9671-1

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