A predator-prey model with age-structured role reversal

The paper states and proves a local existence–uniqueness–positivity theorem for the mixed ODE–renewal PDE system (3.2a)–(3.2h), with B(x,τ) and μ(x,τ) exactly as specified, under x0>0 and nonnegative, compactly supported u0, and parameter assumptions; see the model definitions and Theorem 1 in Section 3.3, and Appendix A for the proof strategy via discretization along characteristics and a convergence/Grönwall argument . The candidate solution proves the same result via a classical mild formulation and a contraction mapping on (x,β) with β(t)=u(t,0), invoking Banach’s fixed point theorem. Assumptions align with those of the paper, and all key estimates (L∞→L1 bounds, Lipschitz bounds for B,μ in x, and small-time self-mapping/contraction) are standard and sound. Aside from a minor arithmetic slack in one small-time bound (easily tightened), the fixed-point proof is coherent. Hence both are correct, with substantially different proof methods (discretization-versus-Picard).