Fractional Lotka-Volterra model with time-delay and delayed controller for a bioreactor

The paper’s Proposition 3 asserts that the crossing direction satisfies sign(Re dλ/dτ|_{λ=iω0}) = sign(κ1^2 − 2κ2), independent of the crossing frequency ω0. However, their own intermediate formula for (dλ/dτ)^{-1} shows the real part is proportional to 2ω0^2 + κ1^2 − 2κ2, i.e., it depends on ω0; the final equality to sign(κ1^2 − 2κ2) drops the +2ω0^2 term without justification. By contrast, the model solution derives Re(dλ/dτ) with an explicit, correct numerator proportional to ω0^2(2ω0^2 + κ1^2 − 2κ2), which indeed implies a positive crossing whenever κ1^2 − 2κ2 > 0. The model also correctly handles parts (a) and (c). One minor flaw in the model’s write-up is a stray claim that κ2 + κ3 > 0 “since κ3 > 0” (not generally true), but this is not used in the final stability argument, which properly relies on initial stability at τ = 0 and continuity of roots.