Abstract:  In this paper, a model of predator-prey with dual delay in maturation and carrying capacity is discussed, in which the past activity of the prey should have an impact on the carrying capacity, the mature prey initiates defense mechanisms to release toxins when subjected to predation, and a commercial harvest of the prey is performed. The stability of the equilibrium of the system in the absence of delay is examined and the optimal harvesting strategy of the model is proven. By investigating the roots of the characteristic equation and applying normalized theory, the properties of the coexistence equilibrium of the system and the conditions for the occurrence of the Hopf bifurcation in the neighborhood of the positive equilibrium are described for various combinations of delays. In the end, numerical simulations are used to verify theoretical analysis results.

Key words: Predator-prey, Time delay, Hopf bifurcation, Harvesting, Toxicant