First-passage time study of a stochastic growth model

In this paper a stochastic nonlinear growth model is proposed, which can be considered a generalization of the stochastic logistic model. It can be applied to the growth of a generic population as well as to the propagation of the fracture in engineering materials. The excitation is assumed to be a stationary Gaussian white noise stochastic process, which affects the system parametrically. A preliminary study of the stochastic differential equation governing the model reveals that there is a phase transition when the nonlinearity parameter c reaches one: when the systems tends to the stationary state, while it is never stationary when . Then, attention is focused on the first-passage time problem, which is of crucial importance for dynamical systems. The first order stochastic differential equation (SDE) that describes the model, is transformed into an Itô’s SDE by adding the Wong-Zakai-Stratonovich corrective term. For the last equation the backward Kolmogorov (BK) equation is formulated. By solving it with appropriate initial and boundary conditions, the probability of survival is obtained, that is the probability of not exceeding a given threshold. The solution is looked for three cases . In any case the numerical analyses show that the survival probability decays fast.