On the competitive exclusion principle for continuously distributed populations

The candidate solution reproduces Theorem 1.2 of the paper for the trait-structured SIS system and its two-case asymptotic alternative. It proves S(t) → 1/α* and derives (i) convergence in absolute variation to 1_{α=α*} e^{τγ} I0 when I0(α^{-1}(α*))>0, and (ii) uniform persistence with concentration in d0 toward M+(α^{-1}(α*)) when I0(α^{-1}(α*))=0—exactly as stated in the paper’s main theorem. The paper’s proof proceeds via averaged quantities, compactness, disintegration, and Lyapunov tools, while the candidate gives a barrier/ODE-based proof using q(t)=α*∫_0^t S−t and a decomposition K=K*(q)+R. Apart from minor presentational gaps (e.g., measurability of a nearest-point projection), the model’s arguments align with the paper’s statements and conclusions, yielding the same end results .