Dynamical Behavior and HOPF Bifurcation in a Fractional-Order Delayed Holling-Tanner Predator-Prey Model

We examine a fractional-order delayed predator-prey model with a rational-type functional response, combining memory effect through a fractional derivative of order [Formula: see text], and predator reaction time through a discrete delay [Formula: see text]. We determine the existence and positivity conditions of the boundary and coexistence equilibria and develop explicit formulation for them. The proposed system is linearized to produce the characteristic equation related to the coexistence equilibrium. We identify the critical frequency [Formula: see text], and related critical delay [Formula: see text] at which stability may be lost by using the fractional Laplace transform to separate the real and imaginary parts of the characteristic equation. The coexistence equilibrium is asymptotically stable for [Formula: see text], but periodic oscillations start when a pair of complex conjugate eigenvalues cross the imaginary axis at [Formula: see text]. To confirm that a Hopf bifurcation occurs, a transversality requirement is further established. In the numerical simulation, the graphical analysis indicates that how the saturation parameter [Formula: see text], and fractional-order [Formula: see text] impact the amplitude, and stability of the model oscillations. The obtain results show that how fractional-order memory, and delayed predator response are important in determining the dynamics, and stability of predator-prey model.