Dynamical Analysis of a Discrete Amensalism System with Michaelis-Menten Type Harvesting for the Second Species

In the study of continuous amensalism systems, it has been widely accepted that Michaelis-Menten type harvesting has a significant impact on the survival and extinction of species. However, scholars have not yet studied discrete amenslism models that include Michaelis-Menten type harvesting. To characterize such dynamics, a discrete amensalism system with Michaelis-Menten type harvesting for the second species is investigated. Firstly, we study the existence and stability of all possible equilibrium points. Under different parameters, there are two stable equilibria, which means that the model is not always globally stable. Then, the conditions of various types of bifurcations likely: pitchfork bifurcations, transcritical bifurcations, fold bifurcations, and flip bifurcations have been established. In addition, a global dynamics analysis of the model is also conducted. Finally, the significance of Michaelis-Menten type harvesting in species relationships is shown by numerical simulations. Although proper harvesting reduces the density of the second species, it favors the stable coexistence of both species and excessive harvesting leads directly to the extinction of the second species. Therefore, the results of this paper can provide a reference for research on how to maximize harvesting without destroying the ecological balance of the species.