Biomass transfer on autocatalytic reaction network: a delay differential equation formulation

The paper’s Theorem 3-3 asserts exactly the renewal-type DDE for Xi(t): dXi/dt = −βi Xi + ∫0^t αi(t−t′) Xi(t′) dt′, with βi = ∑φ∈out(xi) J[φ]/Xi|Y* and αi(τ) = ∑π∈P1[xi]∪P2[xi] cπ fπ(τ) (with fπ a convolution of exponentials determined by Rj = (∑out(uj) J)/Uj|Y*) and cπ = c0π qπ θπ; see Theorem 3-3 and Propositions 3-1 and 3-2 . The candidate solution reproduces this pathwise derivation on the balanced-growth manifold, using the same constants and pathway decomposition, and checks the Malthusian identity λ = −βi + ∫0^∞ αi(τ)e^{−λτ} dτ, which is the paper’s λ + β = A(λ) in Theorem 4-2 . Minor differences are phrasing (the paper frames the DDE as a long-term limit and explicitly discusses neglecting P4-pathways under positive growth, whereas the model takes an exact-on-manifold view and omits P4), but there is no substantive conflict. The waiting-time convolution and rate constants also match the paper’s Markov/linear-compartment argument .