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Recently, by studying some field experimental data and research articles it has been observed that due to predation, predators not only affect prey by direct killing, but they also induce the predation fear in prey which reduces its reproduction rate. The grouping behaviour of animals is one of the most important and interesting phenomena in the field of population dynamics. As a result of prey-predator forming groups, many authors have considered different functional responses in their proposed models. In reality, the amount of biomass converted into the predator due to the prey consumption is not instantaneous. To incorporate the gestation delay in this paper we have proposed and analysed a delayed prey-predator model with square root functional response and fear effect in the prey. We have discussed the qualitative analysis of solutions of the considered model in presence and absence of delay. Moreover, the transcritical bifurcation, Hopf bifurcation with it's direction and the impact of the fear effect on both populations are studied here. Finally, some numerical simulations through graphical illustrations are carried out to support the analytical results.. i [ABSTRACT FROM AUTHOR]

In this paper, we have studied the dynamics of two species, ecological model, considering two economically beneficial marine species herring (Clupea harengus) as predator and mysida (Hemimysis anomala) as its prey with Beddington–DeAngelis functional response, the effect of prey refuge and linear harvesting in both populations. The local and global stability of the equilibrium points has been established here. The axial equilibrium point goes through transcritical bifurcation depending on the harvesting effort of both species. The interior equilibrium point goes through super critical Hopf bifurcation depending on prey intrinsic growth rate. Additionally, we have determined the suitable value of the MSY policy for the harvesting and find suitable harvesting level using the optimal control theory, such that both the species persist simultaneously. Finally, numerical simulations have been performed to justify the theoretical findings. It is clear from the numerical simulations that the prey refuge and mutual interference between predators stabilized the unstable system to a stable one. This finding could give some clue to the experimental biologist and business personnel for making management decisions in controlling the harvesting strategies of the two economic beneficial fish populations, mysida and herring. [ABSTRACT FROM AUTHOR]

In the present paper we have investigated the impact of predator harvesting in a two-dimensional prey–predator model with Holling type III functional response. The main objective of this paper is to study the change of dynamical behaviour of the prey–predator model in the presence of non-linear predator harvesting. The model system shows complex dynamics with the change of different system parameters. We have established the positivity and boundedness of the solutions under a certain parametric condition with non-negative initial conditions. The existence and stability criterion of different equilibrium points are investigated in terms of system parameters. We have shown that the system undergoes through saddle-node bifurcation, transcritical bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation under different parametric conditions. The structural changes or the system bifurcations predict the global dynamics of the model system. We have computed the first Lyapunov number to find the direction of the Hopf-bifurcating periodic solution of the system. Using numerical simulation we have studied here the effect of conversion efficiency, protection of the environment to the prey population and non-linear predator harvesting on the model dynamics. Finally the paper is ended with some conclusions. [ABSTRACT FROM AUTHOR]