Abstract
We consider two different time-inhomogeneous diffusion processes useful to model the evolution of a population in a random environment. The first is a Gompertz-type diffusion process with time-dependent growth intensity, carrying capacity and noise intensity, whose conditional median coincides with the deterministic solution. The second is a shifted-restricted Gompertz-type diffusion process with a reflecting condition in zero state and with time-dependent regulation functions. For both processes, we analyze the transient and the asymptotic behavior of the transition probability density functions and their conditional moments. Particular attention is dedicated to the first-passage time, by deriving some closed form for its density through special boundaries. Finally, special cases of periodic regulation functions are discussed.
Highlights
Deterministic and stochastic growth models are been widely used in the literature to study dynamics of a population
We consider two different time-inhomogeneous diffusion processes useful to model the evolution of a population in a random environment
They arise as approximations of the solution of deterministic Gompertz-type growth models
Summary
Deterministic and stochastic growth models are been widely used in the literature to study dynamics of a population Some such models, as the logistic and Gompertz ones, are characterized by an intrinsic rate of growth and by a horizontal asymptote; both can be time-dependent. Sometimes the description of certain phenomena requires the use of time-dependent functions; in these cases, the Gompertz model turns out to be effective to describe the population dynamics. We consider two different time-inhomogeneous diffusion processes useful to model the evolution of a population in a random environment. They are obtained as approximations of the solution of deterministic Gompertz-type growth models.
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