Dynamics and optimal harvesting of a fractional predator-prey system with fear and anti-predator mechanisms.

Wild prey are shaped by predators long before they are caught. Fear alone suppresses reproduction, physical refuges reduce exposure, and many species strike back directly at their attackers, imposing real costs on the hunter. To investigate the combined effects of these behavioral mechanisms, we develop a Caputo fractional-order predator-prey model incorporating fear-induced reproductive suppression, proportional prey refuge, active anti-predator retaliation, a Holling type II functional response, and nonlinear saturation harvesting. The fractional memory order accounts for hereditary effects in ecological interactions, with the classical integer-order system recovered as a special case. Fundamental dynamical properties of the model, including non-negativity, uniform boundedness, and existence and uniqueness of solutions, are established. All biologically feasible equilibria are characterized, and local and global asymptotic stability conditions are derived using Matignon's criterion and Lyapunov function techniques. The analysis further reveals that the coexistence equilibrium can lose stability through a Hopf bifurcation induced by the fractional memory order, and a critical threshold for the onset of this bifurcation is derived. Numerical evaluation of the first Lyapunov coefficient provides evidence that the bifurcation is supercritical, leading to the emergence of stable periodic oscillations. Applying Pontryagin's Maximum Principle, we derive a bang-singular-bang optimal harvesting strategy that improves discounted net revenue while maintaining ecologically sustainable population levels. Numerical simulations support the theoretical results and illustrate the combined influence of fear, refuge, anti-predator behavior, harvesting, and memory on ecosystem dynamics. The findings demonstrate that the fractional memory order acts as an effective regulator of ecological stability, providing a biologically interpretable mechanism for controlling oscillatory behavior that has no counterpart in classical integer-order predator-prey models.