Exploration on dynamics in a ratio-dependent predator-prey bioeconomic model with time delay and additional food...

In this manuscript, a novel ratio-dependent predator-prey bioeconomic model with time delay and additional food supply is investigated. We first change the bioeconomic model into a normal version by virtue of the differential-algebraic system theory. The local steady-state of equilibria and Hopf bifurcation could be derived by varying time delay. Later, the formulas of the direction of Hopf bifurcation and the properties of the bifurcating periodic solutions are obtained by the normal form theory and the center manifold theorem. Moreover, employing the Pontryagin's maximum principle and considering the instantaneous annual discount rate, the optimal harvesting problem of the model without time delay is analyzed. Finally, four numeric examples are carried out to verify the rationality of our analytical findings. Our analytical results show that Hopf bifurcation occurs in this model when the value of bifurcation parameter, the time delay of the maturation time of prey, crosses a critical value.