The P* Rule in the Stochastic Holt–Lawton Model of Apparent Competition

The paper formalizes the Holt–Lawton model (eq. (1)) under A1–A3 and proves: (a) a single-host extinction/persistence dichotomy with polynomial small-density bounds and the invariant-measure identity ∫y dμ = E[ln R]/E[a] (Theorem 3.1); and (b) the P* rule for k hosts: if P*1 = E[ln R1]/E[a1] is strictly maximal and each host can persist alone, then hosts 2,…,k go extinct exponentially while host 1 persists with the same α,β bounds (Theorem 3.2), via the Benaïm–Schreiber persistence/exclusion framework and external Lyapunov exponents ri(μ) = E[ln Ri] − E[ai]∫y dμ (Section 4, Theorems 4.1–4.2) . The candidate solution reproduces these steps: computes the single-host invariant-measure identity, computes invasion rates λj(μi) = E[aj](P*j − P*i), verifies positivity/negativity along boundary measures, and applies stochastic persistence/exclusion theorems to conclude host-1 persistence with α,β bounds and exponential extinction of the others, exactly as in the paper’s proofs (cf. Section 6) . Aside from a bibliographic difference (it cites Schreiber–Benaïm–Atchadé 2011 instead of Benaïm–Schreiber 2019 for the general framework), the arguments are materially the same and correct.