Exploring periodic behavior and dynamical analysis in a harvested discrete-time commensalism system

This paper explores the dynamic properties of a discrete-time commensalism system using the forward Euler method based on a continuous model. It investigates the existence and stability of all possible trivial, boundary, and positive fixed points of the system. Additionally, it demonstrates that the system undergoes period-doubling bifurcation at the positive fixed point. Numerical simulation results are provided to support the theoretical analysis. The study suggests that harvesting plays a crucial role in stabilizing the population sizes of both species. The system can maintain stability when a significant portion of the stock is available for harvesting. However, as the accessible stock decreases, the system becomes more susceptible to changes, ultimately leading to destabilization and the potential collapse of the ecological community.