Growth with regulation in fluctuating environments

In this paper deterministic growth laws of a logistic-like type are initially introduced. The growth equations are expressed by first order differential equations containing a third order nonlinear term. Such equations are then parameterized in a way to allow for random fluctuations of the intrinsic fertility and of the environmental carrying capacity, thus leading to diffusion processes of new types. Their transition p.d.f. and asymptotic moments are then obtained and a detailed study of the extinction problem is performed within the framework of the first passage time problem through arbitrarily fixed threshold values. Some statistically significant quantities, such as the mean time necessary for the process to attain an assigned state, are obtained in closed form. The behavior of the diffusion processes here derived is finally compared with that of the well known diffusion processes obtained by parameterizing logistic and Gompertz growth equations.Work supported in part by the Group for Mathematical Information Sciences (GNIM) of the National Research Council and by Progetto Finalizzato Sofmat, Contract No. 82.00845.97