Uncertain Stochastic Hybrid Age-Dependent Population Equation Based on Subadditive Measure: Existence, Uniqueness and Exponential Stability

Abstract

The existing literature lacks a study on age-dependent population equations based on subadditive measures. In this paper, we propose a hybrid age-dependent population dynamic system (referred to as APDS) that incorporates uncertain random perturbations driven by both the well-known Wiener process and the Liu process associated with belief degree, which have similar symmetry in terms of form. Firstly, we redefine the Liu integral in a mean square sense and then extend Liu’s lemma and the Itô-Liu formula. We then utilize the extensions of the Itô-Liu formula, Barkholder-Davis-Gundy (BDG) inequality, the Liu’s lemma, the Gronwall’s lemma and the symmetric nature of calculus itself to establish the uniqueness of a strong solution for the hybrid APDS. Additionally, we prove the existence of the hybrid APDS by combining the proof of uniqueness with some important lemmas. Finally, under appropriate assumptions, we demonstrate the exponential stability of the hybrid system.