Population dynamics driven by harvesting and prey refuge in the absence and presence of time delays

In this paper, we study the dynamics of a delayed predator-prey model with harvesting and prey refuge. For the model without time delays, we show the stability of boundary and positive equilibria, given the existence of bionomic equilibrium, and provide the optimal harvesting policy. For the model with time delays, we show the local stability of the positive equilibrium for four different time delay cases, give the existence of Hopf bifurcation near the positive equilibrium, and prove the direction and stability of bifurcating periodic solutions. Ecologically, via numerical simulations, we find that the synergistic effects of time delay, refuge and harvesting can have complex effects on population dynamics. One of the most important results indicates that the increase of prey refuge or prey harvesting rate can eliminate the periodic solutions induced by time delay, while the increase in predator harvesting rate can maintain this periodic phenomenon.