Dynamic interplay of Allee effect and harvesting in a diseased amensalism model: Unraveling ecological complexity

In this paper, we analyze the dynamics of an eco-epidemiological amensalism model involving three species, where the first species exhibits a weak Allee effect. In contrast, the second and third species are subjected to proportional harvesting. The proposed model assumes that the disease only affects the second species while the first species remains unaffected. We consider harvesting as a control parameter for both disease and amensalism dynamics. The amensalism functional response is modeled using the Holling type II response, while disease transmission follows a saturation incidence rate. Our primary mathematical objectives are to analyze the effects of the Allee parameter and harvesting on the system’s dynamics. By treating key parameters as bifurcation and threshold variables, the stability of all equilibria and the associated bifurcations are analyzed. The model exhibits a degenerate Bogdanov–Takens (BT), two saddle–node and three transcritical bifurcations. Conditions for both extinction and persistence are derived in terms of the harvesting rate. A critical threshold of harvesting is identified, beyond which the diseased species declines and the disease-free equilibrium becomes stable. Our findings reveal an upper limit of harvesting within which all species can coexist. However, the coexistence of all three species depends on initial conditions, leading to bistability. Notably, the Allee effect acts only on the first species, underlies the occurrence of a saddle–node bifurcation and eliminating equilibria for a certain parameter range, while leaving the other two species unaffected. These results provide quantitative insights into the interplay between the Allee effect and harvesting in shaping species coexistence and system dynamics.