Modelling physiologically structured populations: renewal equations and partial differential equations

The paper proves asynchronous exponential growth for renewal equations by: (i) introducing the discounted next–generation operator K_λ and showing meromorphicity of (I−K_λ)^{-1} on Δ, with a simple pole at λ=r (Proposition 4.15 and the p=1 residue analysis) ; (ii) imposing a spectral-gap assumption “if λ∈Σ and λ≠r then Re λ<r” and deriving an ε-gap (Lemma 4.17) via a Riemann–Lebesgue argument ; (iii) performing Laplace inversion to obtain ||e^{−rt}b(t)−cψ_r||_1≤Le^{−vt} (Theorem 4.14) and then transferring this to the birth-measure B with the same exponential rate (Theorem 4.30) . The candidate solution executes the same Laplace–resolvent/Riesz–projection program on X=L^1, derives e^{−rt}U(t)=Π+R(t) with exponentially decaying R(t), and then shows ||e^{−rt}B(t)−cΨ_r||≤Me^{−kt}. Two minor issues: it incorrectly asserts ρ(K_λ)→∞ as Re λ↓z0 (not implied by an upper bound on ||K_λ||) and overstates that (I−K_λ)^{-1} has a single pole on all of Δ; the paper only requires and proves a simple pole at r and allows other poles with Re<r. These do not affect the main argument, which otherwise matches the paper’s approach and conclusions.