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

To describe the dynamics of a pest, stage structured demographic models appear suitable tools since they allow to know the abundance in each stage. The growth of an individual is described by its physiological age supposed to be stochastic. The physiological age is conveniently represented by a stochastic differential equation driven by a Gamma process to guarantee its non-negativity. Two different formulations using a Gamma process with drift or a pure time-inhomogeneous Gamma process are here considered and compared with the common Wiener driven model which, however, do not grant the positivity of the physiological age. The population dynamics based on the Gamma processes are represented by a system of generalized Kolmogorov equations, while a system of Fokker-Planck equations describes the dynamics in the case of a Wiener driven physiological age. Development, mortality and fecundity rate functions are supposed time-dependent. The Gamma driven physiological age models have the same expectation of the Wiener driven physiological age and present similar residence times in a stage. Consequently, they also produce similar population dynamics allowing us to state that the population dynamics based on Wiener driven physiological age represents a good approximation of the formally correct dynamics obtained using a Gamma driven physiological age with an appropriate choice of the parameters. Suitable discretizations of the models are presented to simulate the dynamics.