Application of a class of difference equations in population dynamics

Difference equations have far-reaching implications across various disciplines, particularly in biology. Recent studies have revealed that discrete biological mathematical systems exhibit intricate and complex dynamic behaviors. This paper investigates the stability and bifurcation dynamics of a discrete predator-prey model with a Holling-II type functional response and a nonlinear Michaelis-Menten type harvesting. We employed the semi-discretization method to derive the discrete system and analyzed the existence and local stability of the fixed points. By employing the center manifold theorem and bifurcation theory, we deduced the transcritical bifurcation conditions at the boundary fixed points E1, E2, and E3, as well as the Neimark-Sacker bifurcation conditions at the positive fixed point E4. Numerical simulation validated the correctness of our theoretical analysis and further elucidated the system's dynamic behaviors, including the transitions in the stability of fixed points and the Neimark-Sacker bifurcation phenomena.