Niche differentiation in the light spectrum promotes coexistence of phytoplankton species: a spatial modelling approach

The paper establishes that if (P) holds and both invasion eigenvalues are negative, µu<0 and µv<0, then the two-species semiflow is uniformly persistent and admits at least one locally asymptotically stable coexistence equilibrium (Proposition 3.6) . It then supplies explicit sufficient inequalities (20)–(21), derived via Lemma 3.7’s bound on resident shading, that guarantee µv<0 and µu<0 by a weighted-integral sign test for the principal eigenvalue (Lemma A.1) . The candidate solution reproduces exactly this chain: (i) define the invasion profiles h1,h2 in terms of the semi-trivial equilibria (matching Proposition 3.4’s definitions of µv,µu) , (ii) bound the resident’s cumulative biomass using Lemma 3.7 , (iii) verify positivity of the weighted integrals under (20)–(21) , (iv) infer µv<0 and µu<0 via the weighted sign criterion (Lemma A.1) , and (v) invoke Proposition 3.6 to conclude uniform persistence and a locally asymptotically stable coexistence equilibrium . The only minor issue is a mislabel: the candidate cites “criterion (17)” for the eigenvalue sign test, whereas (17) in the paper denotes the condition “µu<0 and µv<0,” and the eigenvalue sign test itself is Lemma A.1. Substantively, however, the reasoning and conclusions coincide with the paper.