Some time-inhomogeneous diffusion models for population growth in random environments

Deterministic growth laws, expressed by first order differential equations with time-depending intrinsic growth intensity function, are initially introduced. Such equations are then parameterized in a way to allow random fluctuations of the intrinsic growth intensity function. This procedure leads to time-inhomogeneous diffusion processes for which a detailed study of transition probability density functions and of the first-passage time densities through arbitrarily fixed threshold values is performed. Some statistically significant quantities, such as the mean and the variance of the time necessary for the process to attain an assigned state, are obtained in closed form. The behaviors of several diffusion processes, suitable to describe the growth of populations, are finally analyzed and compared. Various numerical computations are performed in the presence of periodic intrinsic intensity function.