From the distributions of times of interactions to preys and predators dynamical systems

The paper’s Theorem 3.4 proves that, under Assumptions 3.1–3.3 and the fast–slow scaling λK = K1/K2 → ∞, the scaled prey–predator counts converge to the ODE system x′ = (γ−β)x − y φ(x), y′ = y ψ(x), and the predator age–occupation measures converge to y(t) φ(x(t)) pr(x(t),a) dt da; see the statement of Theorem 3.4 and the definitions of pr, φ, ψ in Section 3.1 and the Introduction . The candidate solution follows the same occupation-measure averaging route: (i) semimartingale decompositions on the accelerated time scale, (ii) tightness using a Lyapunov control (Assumption 3.2), (iii) division of the weak age–measure equations by λK to force the stationary age-balance equations, and (iv) closure to the same ODE via the identified limits. These steps match the paper’s Lemmas 3.5–3.9 and the identification argument (notably the representation of Γ^K in terms of pr and the equality of inflows αSΓS = αMΓM) . A small presentation issue in the paper is that equation (11) in Section 3.1 appears to omit γr in ψ (likely a typo), whereas both the Introduction and Section 4 consistently define ψ with γr−βr, which is what the model solution uses . Aside from this minor typo, the arguments are logically aligned and complete.