Global Stability and Bifurcation of a Three-Species Commensalism–Amensalism Model with Beddington–DeAngelis Functional Response

This paper investigates the dynamical behavior of a three-species commensalism–amensalism system with Beddington–DeAngelis functional response. The model describes a novel tripartite interaction: a neutral–commensal species (e.g., sea anemone) simultaneously engages in a commensal relationship with a commensal–amensal species (e.g., clownfish) and an indirect nutritional coupling with a neutral–amensal species (e.g., crustacean), while the commensal–amensal and neutral–amensal species interact amensalistically. This paper makes three principal contributions. First, by constructing a Volterra-type Lyapunov function, we rigorously prove the global asymptotic stability of the unique positive equilibrium of the (x,z)-subsystem in the positive quadrant, and further establish the global asymptotic stability of both the amensal-free equilibrium E3 and the coexistence equilibrium E4 in the positive octant. Second, selecting the commensal benefit coefficient c as the bifurcation parameter, we present a complete and rigorous proof of a transcritical bifurcation. Third, we provide systematic Maple-based numerical verification: the bifurcation diagram exhibits excellent agreement with the theoretical curve, and logarithmic-scale plots confirm exponential convergence rates. Ecologically, our results reveal a sharp threshold phenomenon: when the commensal benefit coefficient lies below a critical value c*, the commensal species inevitably goes extinct and the system collapses to a two-species state; when c exceeds c*, the commensal species can invade and achieve stable three-species coexistence. The explicit formula for this threshold provides a quantitative criterion for determining the minimum mutualistic strength required for persistence in conservation contexts. The results obtained in this paper substantially extend the existing theoretical understanding of three-species commensalism–amensalism systems.