DELAY-DEPENDENT AND DELAY-INDEPENDENT STABILITY OF COURNOT DUOPOLY MODEL WITH TAX EVASION AND TIME-DELAY

The paper derives the crossing conditions for q(λ,τ)=p1(λ)p2(λ)−e^{−λτ}(aλ−b)(cλ−d), obtains the frequency equation P(ω)=N1(ω)^2+N2(ω)^2−Q(ω)^2=0 and the delay set τ0=(1/ω0)tan^{-1}(N2/N1)+nπ/ω0, and classifies stability by μ: delay-independent for μ∈[3−2√2,3+2√2] and delay-dependent otherwise (Theorem 3.4). The candidate solution uses the same frequency-sweeping framework, matches the N1,N2 and τ0 formulas, and reaches the identical μ-interval, adding a helpful factorization P(ω)=(a^2ω^2+b^2)(c^2ω^2+d^2)·Δ(ω). Hence both are consistent and essentially the same proof strategy, with the model giving a slightly clearer algebraic reduction.