Codimension-1 and Codimension-2 Bifurcations Analysis of Discrete Predator–Prey Model with Herd Behavior

The herd behavior of prey plays an important role in the predator–prey system. In this paper, we analyze the dynamical properties of a discrete predator–prey model with herd behavior. We found that due to the introduction of herd effects, this model exhibits complex dynamical behavior such as codimension-1 bifurcation (Flip and Neimark–Sacker bifurcations) and codimension-2 bifurcation (1:2, 1:3, and 1:4 strong resonance). Using the center manifold theorem and normal form theory, we obtained the critical conditions that may lead to changes in the stability of the system and even the emergence of new stable states or oscillating behaviors. Numerical simulations not only validate theoretical proofs but also reveal the complex and rich dynamical behavior of the discrete model.