Sustaining pollination while limiting parasite spillover from managed bees requires a quantitative understanding of how supplementation of managed colonies, floral resource competition between managed and wild bees, and cross-host transmission jointly shape community dynamics. We develop and analyse a three-dimensional ODE model for wild bees, managed bees and a brood-targeting parasite, incorporating managed-colony supplementation, asymmetric floral competition acting on wild bees, and brood-mediated transmission with host-specific efficiencies. Using two subsystems, we derive explicit ecological thresholds: an input-loss ratio governing managed-bee persistence and a competition-modified wild-bee persistence threshold (parasite-free subsystem), and closed-form supplementation thresholds for the existence and local stability of managed-parasite coexistence (wild-bee-free subsystem). For the full system, the interior coexistence condition reduces to a single polynomial in managed-bee density, implying a finite number of coexistence equilibria and a simple parity rule for their stability. Bifurcation sweeps in the supplementation rate and in the effective transmission efficiency to managed bees reveal multistability between boundary states and coexistence, as well as interior folds that generate hysteresis. We further find both super- and subcritical Hopf bifurcations on the managed-parasite manifold and on interior branches, producing stable limit cycles whose amplitudes vary continuously between Hopf points. Ecologically, both axes act non-monotonically: intermediate supplementation often stabilizes equilibria or confines community-level oscillations in bee and parasite densities, whereas excessive supplementation risks wild-bee loss. Likewise, increasing effective parasite transmission to managed bees can promote parasite persistence but may also, by depressing managed-bee biomass and easing competition, facilitate wild-bee persistence. From an applied perspective, our analysis delineates supplementation regimes and transmission scenarios that avoid wild-bee exclusion, identifies parameter bands where community-level oscillations in bee and parasite densities are expected, and highlights the risk of abrupt regime shifts near folds.
Feng et al. (Fri,) studied this question.