Dynamical analysis of a parabolic–hyperbolic hybrid model for species with distinct dispersal and sedentary stages

Abstract

Most marine and plant species exhibit two main life stages: a dispersing stage and a sedentary stage. These stages significantly influence the species’ spatial distribution and abundance patterns. To accurately depict the spatial patterns of these species, this paper investigates a hybrid system that combines parabolic and hyperbolic elements, effectively differentiating between the dispersal and sedentary stages. Further spatiotemporal dynamical analysis is conducted to comprehend the large-scale distribution patterns and geographic ranges. Specifically, the model is reformulated into a time-delayed nonlocal system. The existence of spreading speed and its alignment with the minimal wave speed for monotone traveling waves are confirmed for unbounded spatial domains. Meanwhile, a threshold-type result is observed regarding the global attractiveness of the zero or positive steady state for bounded domains. Conditions for population persistence and extinction under both Neumann and Dirichlet boundary conditions are derived. It is further established that the persistence or extinction of the population may be determined by a critical domain size under Dirichlet boundary conditions. Numerical simulations are conducted to provide additional quantitative results, complementing the theoretical findings.