Dynamical Analysis of a Discrete Commensalism Model With Michaelis–menten Type Harvesting

In this work, we formulate and analyze a discrete-time commensalism model in which the first species is subject to Michaelis–Menten type harvesting. We first investigate the existence, stability, and global attractivity of all feasible fixed points. Our results demonstrate that when the two species coexist stably, increasing the harvesting intensity leads to the extinction of the harvested species. We then establish the conditions under which various bifurcations occur, including transcritical, pitchfork, and fold bifurcations, with particular emphasis on the period-doubling route to chaos. In addition, a comprehensive analysis of the global dynamics is conducted. Numerical simulations highlight the critical influence of the commensalism strength: Moderate commensalism ensures stable coexistence of species and increases population density, whereas excessive commensalism destabilizes the system, triggering a period-doubling bifurcation cascade to chaos. Furthermore, the intrinsic stability of the host species is shown to profoundly modulate the dynamical complexity of the commensal species. Under certain conditions, instability of the host species can advance the onset of chaos in the commensal species.