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

This article focuses on the mathematical modeling and analysis of a predator–prey system incorporating the fear effect on prey reproduction, nonlinear harvesting on both species, and predator gestation delay. The model uses a Holling type II functional response with square root harvesting and introduces a fear function that reduces prey reproduction depending on predator density. Key dynamical behaviors such as positivity, boundedness, existence and stability of equilibria, Hopf bifurcation, and Bogdanov–Takens bifurcation are rigorously analyzed, supported by numerical simulations. The study reveals that increasing fear levels can stabilize the system, while gestation delay in predators may induce complex dynamics including quasi-periodic oscillations and chaos, as confirmed by positive Lyapunov exponents. These findings highlight the significant impact of fear and delay on predator–prey interactions and resource harvesting dynamics.