Restricted Gompertz-Type Diffusion Processes with Periodic Regulation Functions

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.

A generalized Gompertz growth model with applications and related birth-death processes

“…Taking into account the various behaviours that the curve N GP D (t) may have for different choices of a, we are interested in a generic threshold S > y if the curve grows to infinity, or on a percentage p of the carrying capacity C = ye Ab , or on a percentage of the size of the population at the inflection point t I given in (15). For a generic S, to obtain the crossing time instant θ S of N GP D (t) through S, we solve the equation N (θ S ) = S, recalling (11).…”

“…In this framework, stochastic processes for the modeling of real growth phenomena have been largely considered in the literature. We recall the well-known approach based on diffusion processes for the stochastic model of tumor growth, such as that exploited in Albano and Giorno [1], Giorno et al [17], Giorno and Nobile [15], Hanson and Tier [19], Spina et al [29]. Other studies including Gompertz and logistic growth models based on stochastic diffusions can be found in Campillo et al [8], Himadri Ghosh and Prajneshu [21], and Yoshioka et al [36].…”

A generalized Gompertz growth model with applications and related birth-death processes

“…2, when Ab > a + 1. We consider the maximum specific growth rate, say μ, which is the coefficient of the tangent to the curve N G P D (t) given in (12) in the inflection point t I shown in (16): Moreover, recalling the expressions of t I and N G P D (t I ), the tangent to the curve…”

“…We are interested in the time instant in which N (t) reaches S, with S > N (0) = y. Taking into account the various behaviours that the curve N G P D (t) may have for different choices of a, we are interested in a generic threshold S > y if the curve grows to infinity, or on a percentage p of the carrying capacity C = ye Ab , or on a percentage of the size of the population at the inflection point t I given in (16). For a generic S, to obtain the crossing time instant θ S of N G P D (t) through S, we solve the equation N (θ S ) = S, recalling (12).…”

Powers of the Stochastic Gompertz and Lognormal Diffusion Processes, Statistical Inference and Simulation

“…With respect to the Kolmogorov equations, it was defined by Nafidi [18], in a general way and for both the univariate and the multivariate cases. In various papers, Gutiérrez et al [19-21], Ferrante et al [22], Román-Román et al [23] and Giorno and Nobile [24], have highlighted the importance of this process, and many subsequent extensions have appeared, especially regarding the non-homogeneous case with exogenous factors (external variables) that affect the drift coefficient. In general, these extensions take one of the following two forms:With external information (when no functional form is available): the exogenous factors are completely determined by the observed data (monthly, annual, etc.)…”