Abstract
In this paper, we investigate a cannibalistic predator-prey system with Beddington-DeAngelis type response function when the infection only exists in the predator species. The predator species are subject to cannibalistic interaction and the infection spreads among the predators via cannibalism. The model is studied in presence of multiple delays for the maturation of prey and predator. Local stability analysis of the predator-prey model around the biologically realistic two essential steady states is investigated. We also investigate the bifurcation analysis of the proposed model around the co-existing steady states. Using the normal form method and center manifold theory, the direction of Hopf bifurcation and the Hopf bifurcating periodic solutions are explored. In presence of time delays, we derive the criteria for the permanence of the model. Our findings demonstrate that maturation delays destabilize the system and can produce high periodic oscillations. Cannibalism can act as a self-regulatory scenario controlling the transmission of disease among the predator species and stabilizing the oscillations in the predator-prey system. Our theoretical analysis is well supported with numerical simulation.