Exploring predator–prey dynamics: Integrating competitor predators, harvesting delay and fear effect on prey with a modified Beddington–DeAngelis functional response

Selective harvesting is a vital strategy for controlling over-exploitation and protecting the incidental killing of juvenile species. To ensure that the individuals reach a suitable age or size before being harvested, a specific time delay, known as harvesting-induced delay, should be maintained. Also, predators can indirectly slow down the growth rate of prey by inducing fear in them, which affects their behaviour and reproduction. In this work, we study a predator–prey model to examine the impact of predator selective (delayed) harvesting in the presence of fear in prey species and the interspecific competition of predator with the competitive predator. It is assumed that the density of the competitive predator is constant and both predators induce fear in prey. We determine the conditions related to the existence of transcritical and Hopf bifurcations for the non-delay case, and various delay-induced scenarios via Hopf bifurcation. We also numerically explore the impact of delayed harvesting on the system dynamics by simultaneously varying the harvesting effort, fear level, and the interspecific competition with competitive predator in bi-parametric planes. It is observed that the harvesting delay can result in stability invariance, instability invariance, stability change, instability switching, and stability switching phenomena under varied parametric conditions. Further, when the harvesting delay is fixed, the equilibrium point experiences instability invariance and instability change, instability change and instability switching, and instability invariance, stability change and stability switching with respect to the harvesting effort, the interspecific competition, and the fear level, respectively. Our findings indicate that regulated selective harvesting of predator, and a moderate level of fear in prey and interspecific competition with the competitive predator are essential for maintaining stability and coexistence in the ecosystem. The rich and complex dynamics holds significance from a biological viewpoint and may potentially impact the population management strategies.