Global asymptotic stability of barnacle-algae interaction model with delay

This work investigates the dynamical behavior of a three-dimensional patch occupancy model that incorporates a discrete time delay into the barnacle-algae interaction. The model framework extends the non-delayed, periodically forced barnacle-algae-mussel system introduced by Benincà in [2]. In the absence of mussels (M = 0) and seasonal forcing, the system studied in [10] reduces to a three-dimensional system, for which global asymptotic stability was previously established. Under biologically motivated assumptions, this paper provides a complete characterization of the global dynamics for the corresponding three-dimensional delayed system. By leveraging the Poincaré-Bendixson Property applicable to type-K competitive systems with delay, we prove the global asymptotic stability of the positive equilibrium whenever it exists. Our analytical results demonstrate that the introduction of a time delay does not disrupt the global convergence behavior observed in the non-delayed system.