Dynamic Behaviors of a Stage-Structured Commensalism Model with Holling Type II Benefits and Birth-Related Allee Effect

Low adult density can suppress recruitment through mate limitation or cooperative reproductive failure, even when a host improves adult survival. Motivated by this ecological tension, we introduce a fecundity-related component Allee effect into a stage-structured commensalism model with a Holling type II host benefit. The nonlinear recruitment term is f(x2)=αx22/(x2+A), and the independently growing host converges to its carrying capacity, yielding an asymptotically autonomous commensal subsystem. We obtain a complete analytical classification. If the long-term host benefit D exceeds mature mortality δ2, a unique coexistence equilibrium attracts every nontrivial commensal population. If the benefit is insufficient, fecundity limitation can instead create two positive equilibria: a low-density saddle whose stable set is the basin boundary and a stable high-density coexistence state. Thus, extinction and coexistence are both possible, depending on initial stage composition and host abundance. Stronger fecundity limitation enlarges the extinction region, whereas greater recruitment, maturation or host benefit promotes persistence. At the critical boundary, the positive equilibria merge in a non-degenerate saddle-node bifurcation. Rigorous global arguments and numerical comparisons with the linear-recruitment model show how a component Allee effect can induce strong-Allee-type demographic behavior, including bistability and threshold-mediated extinction.