Stability and moment boundedness of an age-structured model with randomly-varying immigration or harvesting

In this paper we study the stability and moment boundedness of the solutions to the linear age-structured model with randomly-varying immigration or harvesting. For this model, we study stability and the asymptotic behavior of the first moment and the results are identical to that of the corresponding deterministic age-structured model and the linear age-structured model with the white noise. However, the stability and boundedness of the second moment are complicated and depend on the randomly-varying immigration or harvesting. For the linear age-structured model with randomly-varying immigration or harvesting, we directly prove that the second moment M(t, a) is bounded for 0 <= t <= a. When t > a >= 0, using the Laplace transform in the framework of Ito-Doob integral, we give the explicit expressions of the second moments M(t, 0) and M(t, a) and then establish the sufficient conditions for the second moments to be bounded and unbounded, respectively, through the supreme of the real parts of all characteristic roots. We also study the asymptotic behaviors of the second moments M(t, 0) and M(t, a) and give the sufficient condition for stochastically ultimately boundedness of the model. (C) 2018 Elsevier Inc. All rights reserved.