Distributed-Memory Stabilization in a Fractional Cournot–Bertrand Duopoly

This paper generalizes a fractional Cournot–Bertrand duopoly model with delayed self-feedback by replacing discrete memory effects with equal-mean, distributed Gamma memories. The resulting framework preserves the original economic structure while distinguishing the effect of the shape of the memory distribution from that of its average length. We derive the equilibria, the linearized characteristic equations, and real–imaginary crossing conditions for point, weak-Gamma and strong-Gamma memories. For the reference output-feedback configuration, the point-delay benchmark has the reported Hopf threshold τ0=3.6355 at frequency ψ0=0.5947, whereas the two Gamma kernels remain separated from an imaginary-axis crossing over the tested equal-mean interval. At the reference crossing frequency, their feedback gains are approximately 0.42 and 0.46, respectively, compared with unit gain for the point delay. Numerical root tracking, stability diagnostics, parameter scans, and solver-convergence checks support the conclusion that distributed aggregation can materially enlarge the practically stable operating region.