A stage structured demographic model with "no-regression" growth: The case of constant development rate

A stage structured population model where the development of an individual in a stage is described by means of the physiological age subject to a random growth is here considered. The physiological age is supposed to satisfy a stochastic differential equation driven by a Gamma process. This formulation, unlike the classical one based on the Wiener process, allows to obtain a non-decreasing physiological age, in agreement with its definition as percentage of development of an individual. Three different formulations have been considered for the Gamma driven physiological age: drift coefficient equal to the development rate function as for the Wiener driven physiological age; drift coefficient depending on development rate function and chosen to have the same expectations of the physiological age driven by the Wiener process; physiological age equal to a Gamma process with same mean of the Wiener driven physiological age. Considerations on the residence times highlight the difference among the models. Based on the Gamma driven physiological age, the dynamics in the different stages of the structured population are described through a system of generalized Fokker-Planck equations. A suitable discretization of the integro-differential system allows to simulate the dynamics that are compared with those of the Fokker-Planck equations obtained for a Wiener driven physiological age. To simplify the comparison, only the case of constant development rate is here considered. Results suggest that it is possible to obtain similar dynamics for Wiener driven physiological age and Gamma driven physiological age with a suitable choice of the drift coefficient and of the Gamma process parameters.