Delay-induced multiple stability scenarios on a fractional-order predator-prey model with Cosner functional response

This study proposed a dimensionally consistent fractional-order predator-prey model incorporating the Cosner functional response and a gestation delay. We examine the critical mathematical aspects, including the fractional-order system's existence and uniqueness, non-negativity, and boundedness. Later, the existing equilibrium's local and global asymptotic behavior for the non-delayed model is studied using the Matignon conditions and the Lyapunov function method, respectively. Stability analysis of the existing equilibrium for the delayed model confirms the existence of a Hopf bifurcation. Analytically, the effect of harvesting on prey and predator populations is examined under specific parametric conditions. Theoretically, we prove that time delay can induce seven stability scenarios, including (i) stability change, (ii) stability switching, (iii) instability switching, (iv) instability invariance, and (v) stability invariance. Numerical simulations validate all the stability scenarios.