Codimension-two bifurcations and chaos of a discrete predator–prey system with allee effect

We investigate local codimension-two resonances of the coexistence fixed point in a forward-Euler discretization of a predator–prey model with a sigmoid functional response and an Allee effect. Earlier work established the fixed points and codimension-one period-doubling and Neimark–Sacker bifurcations. Here we derive the spectral locations of the $ 1{:}2 $, $ 1{:}3 $, and $ 1{:}4 $ resonance candidates, state the regular two-parameter unfolding and normal-form nondegeneracy requirements, and compute the corresponding normal-form coefficients. For a representative strong-Allee parameter set, the analytical resonance locations agree with MatContM continuation to the displayed precision. We also separate algebraic existence from biological feasibility under strong and weak Allee effects and clarify the local biological scope of the Euler map. The bifurcation and maximal-Lyapunov-exponent diagrams illustrate the dynamical complexity.