Model of Three Species Commensalism-Amensalism Symbiosis with Beddington-DeAngelis Functional Responses

This study investigates the dynamical behavior of a three species ecological system involving unilateral interactions of commensalism and amensalism with Beddington–DeAngelis functional responses. The positivity, boundedness, existence, and uniqueness of the model solutions are established, and four equilibrium points are identified. Stability analysis shows that the coexistence equilibrium point and the neutral species only equilibrium point are locally asymptotically stable, whereas the other equilibria are always unstable. Numerical simulations are conducted to confirm the analytical findings. Ecologically, the results indicate that stability can be achieved only when all neutral species coexist, even though without the commensal–amensal species. In contrast, the commensal-amensal species cannot persist without the presence of all neutral species.