Exact Stability Atlases and a Memoryless-Surrogate Failure Theorem for a Caputo Allee Predator-Prey Model

We study a strong-Allee predator–prey system with Holling type-II predation under a commensurate Caputo derivative of order α∈(0,1]. The coexistence equilibrium is given in closed form, and its local fractional stability reduces to the trace and determinant of the Jacobian, both affine in the reciprocal Allee threshold. For every parameter set satisfying one explicit inequality, this yields a stability atlas in the (A,α) plane with three algebraic breakpoints and a critical order α*(A) that decreases strictly from 1 to 0; on a benchmark slice the breakpoints are the rationals 2/23, 2/7, 10/19, 2/3. The second result is a dichotomy about discretization. On the fractional-stabilization region, where the Caputo equilibrium is asymptotically stable despite a right-half-plane spectrum, the memoryless map z↦z+εf(z) is unstable for every step size and every consistent memoryless one-step method for all small steps, whereas the implicit Grünwald–Letnikov scheme, which retains the history, is stable for every step size and reproduces the exact algebraic decay, and the explicit one for all small steps. Positivity, an extinction strip, a uniform ultimate bound, and global existence complete the picture; proof objects audit the benchmark constants and numerical corroboration covers two further parameter sets and 106 random ones.