Asymptotic regimes of an integro-difference equation with discontinuous kernel

The paper studies the IDE u_{n+1}(x)=∫_Ω k(x,y)F(u_n(y))dy under H1–H3 (bounded, uniformly positive, piecewise-continuous kernel; F continuous, bounded, strictly increasing on [0,∞) with KPP-type F(u)/u decreasing) and proves a sharp trichotomy: (i) if F(0)=0 and the principal eigenvalue λ0 of the linearization T0(u)=F'(0)∫k u satisfies λ0≤1, then 0 is the unique stationary state and u_n→0 in L2; (ii) if F(0)=0 and λ0>1, then there is a unique positive stationary state w and u_n→w; (iii) if F(0)>0 then, regardless of λ0, there is a unique positive stationary state and u_n→w. These are exactly Theorems 1–3 in the paper, with the supporting lemmas on positivity, monotone iteration, and L2 convergence (Lemma 6) explicitly given in the text. See the statement of H2–H3 and T, T0 in the setup, the existence of principal eigenpairs for T0 via Krein–Rutman, and the main theorems and their proofs for the three cases (e.g., Theorem 1: λ0≤1 implies extinction; Theorem 2: λ0>1 and F(0)=0 implies unique w≫0 and global convergence; Theorem 3: F(0)>0 implies unique w≫0 and global convergence) . The candidate solution reproduces the same trichotomy and is logically sound, but the proof strategy differs: it emphasizes strict subhomogeneity of T from the KPP ratio F(u)/u, strong positivity (k≥δ>0), a linear functional built from the dual principal eigenfunction ψ of T0*, and a one-dimensional Lyapunov sequence a_n=∫ψ u_n to handle the λ0≤1 extinction case; it also uses a projective-order argument to prove uniqueness when λ0>1. The paper, in contrast, proves uniqueness via a comparison lemma (Lemma 2), constructs sub-/super-solutions (εφ0 and a large constant), and uses monotone iteration plus Lemma 6 to obtain L2 convergence. Thus, the statements coincide, but the proofs are different. Minor paper-level issues include an internal consistency hiccup: H3 states r0=F'(0)>1, yet Corollary 1 assumes r0≤1 alongside (10) to force λ0≤1—this should be framed as a separate regime (or H3 should be stated without fixing the sign of r0); and a cross-reference to “Corollary 2” (used to invoke uniqueness) appears without the corollary being shown in the excerpt, though the uniqueness claim follows from Lemma 2 and the monotonicity arguments in the main text. None of these affect correctness of the main results. Overall: both correct, with different (but compatible) proof techniques .