Neimark-Sacker bifurcation and dynamical behavior in a discrete predator-prey system with exponential terms

The present research explores the complex dynamics within a two-dimensional discrete-time predator-prey framework incorporating exponential nonlinearities. The interactions between species are formulated through the corresponding difference equations

$ y_{n+1} = \frac{a z_n}{1+p y_n} e^{-y_n}, \ z_{n+1} = \frac{b z_n}{1+q z_n} e^{-\alpha y_n} , \ $

where the discrete time step is $ n \ge 0 $. All associated parameters $ a, b, p, q, $ $ \alpha $ represent strictly positive real numbers operating under nonnegative initial states. Essential mathematical features of this formulated model undergo rigorous examination. These analytical steps include verifying the boundedness of all trajectories alongside determining the local asymptotic stability (LAS) criteria for the positive fixed point and estimating its convergence speed. A major objective of this work involves deriving the exact parameter thresholds required for the emergence of an Neimark-Sacker (NS) bifurcation. Applying normal form theory for discrete-time systems allows us to ascertain both the bifurcation direction and the topological stability of the newly generated invariant closed curve. Extensive numerical experiments subsequently corroborate all derived theoretical results.