Persistence criteria for a chemostat with variable nutrient input and variable washout with delayed response in growth

The paper proves a necessary-and-sufficient long-interval integral criterion for persistence (Theorem 2.1) with a complete two-direction proof using well-posed auxiliary quantities ψ, ϕ and technical lemmas; its logic is coherent and internally referenced. The model’s solution captures the right objects and intuition but contains critical flaws: it asserts global uniqueness of ϕ (not true in general per the paper’s remark), relies on an unproven pointwise inequality r_x ≥ ϕ without justifying the order-reversing fixed-point subtleties, and, most importantly, its necessity direction uses a lower bound on ln x that cannot yield decay (a sign-direction mistake). These issues undermine the model’s argument, while the paper’s proof stands.