Complex dynamical study of a delayed prey-predator model with fear in prey and square root harvesting of both species

In the current study, the dynamics of predator-prey systems under the influence of fear effect on the reproduction of prey population and harvesting on both species has been proposed. Assessing the dynamics of the system with the combined influence of fear and harvesting for various values of n is our central objective. We present comprehensive mathematical findings that cover fundamental dynamical features, the presence of positive equilibria, and the stability of all equilibria. Hopf-bifurcating periodic solutions have been demonstrated to emerge around the positive equilibrium point, and the direction of the Hopf-bifurcating limit cycle is determined using the first Lyapunov coefficient. Furthermore, in two-parameter space, we have seen that the system experiences the Bogdanov-Takens bifurcation. Moreover, we have included predator gestation delay and noticed some chaotic dynamics in the system. In addition, we run through numerical simulations to numerically validate our mathematical findings. The article is concluded with a conclusion at the end.