Effect of non-linear harvesting and delay on a predator-prey model with Beddington-Deangelis functional response

Abstract

The interaction of prey and predator is a critical issue in population dynamics modeling. In this paper, we investigate the dynamics of a delayed predator-prey model with Beddington-DeAngelis functional responses and non-linear harvesting of predators. Also, the fear of predators in prey species is introduced. In the absence of a time delay, the positivity, boundedness, stability of equilibrium points and local bifurcation of system are studied. From the analysis of the non-delayed model, we find that when the birth rate of prey is selected as the bifurcation parameter, the system undergoes a transcritical bifurcation at the trivial equilibrium point. Similarly, setting the bifurcation parameter to the maximum predation rate p resulted in a transcritical bifurcation at the predator-free equilibrium point. Furthermore, when the harvest effort value E is used as the bifurcation parameter, the system will have two Hopf bifurcation points close to the positive equilibrium point. In addition, the stability of the limit cycle generated by Hopf bifurcation is determined by calculating the first Lyapunov number. Our results show that fear of predation risk and harvest effort values can have stable and unstable effects. In addition, the predator mutual interference coefficient b may be responsible for the stability of the system. In the presence of a time delay, the time delay can also cause the system to generate limit cycles near the positive equilibrium point. Finally, some intriguing numerical simulation findings is provided in order to study the model’s dynamics.