Effect of fear and delay on a prey-predator model with predator harvesting

The relationship between prey and predator has been modelled, analysed from the early nineteenth century by considering different types of functional responses and ecological effects. Harvesting of predator in a prey-predator system for controlling their dynamics has been receiving considerable attention from both ecological and economical points of view. A lot of theoretical work has been done on harvesting and reveals that harvesting has a significant impact due to its ecological and economical importance. The fear of prey due to the appearance of predators in the ecological system also plays an important role in drawing the shape of the dynamics of the interacting system. Another important fact, in reality, the reproduction of predator is not instantaneous after the consumption of prey. A constant time lag is necessary for each organism. In this paper, we have investigated the dynamics of a delayed prey-predator model with Holling type III functional response and fear of prey in the natural birth rate. The predator is assumed to be economically significant and harvested linearly. The positivity, boundedness, local stability of equilibrium points and local bifurcation (Transcritical, Hopf bifurcation) of the non-delayed system is established here. We have discussed the local and global stability of the interior equilibrium point in presence of delay. In terms of the delay parameter, the system undergoes through a Hopf bifurcation and we determine in which direction the Hopf bifurcation will go. We carried out some numerical simulations illustrate the theoretical findings using the MAPLE and MATLAB software. Finally, we have drawn some concluding remarks.