A two-strain reaction-diffusion malaria model with seasonality and vector-bias

The paper proves three threshold results for a periodic two-strain reaction–diffusion malaria model with vector-bias: (A) global extinction if R1<1 and R2<1 (Theorem 4.1), (B) competitive exclusion if exactly one strain has Ri>1 (Theorem 4.2), and (C) uniform persistence plus existence of a strictly positive ω-periodic solution when R1>1, R2>1 and the invasion numbers R̂1>1, R̂2>1 (Theorem 4.3). These are established via a time-varying phase region X(t), an ω-periodic semiflow Q(t), the Poincaré map Q=Q(ω), and an acyclicity argument on boundary invariant sets, together with spectral characterizations of Ri and R̂i via the Poincaré maps of associated linear periodic-parabolic systems (Lemmas 3.1 and 3.2) . The candidate solution reaches the same conclusions using a slightly different blueprint: compactness and strong monotonicity (with respect to a competitive cone) of the Poincaré map, standard periodic next-generation operator theory, and persistence results under an assumed acyclic boundary. Its steps align with the paper’s outcomes and standard literature. However, it omits two technical points proved in the paper: (i) handling the time-dependent state region X(t) (the paper’s Q(t) is an ω-periodic semiflow on X(t) with a global attractor) and (ii) verifying acyclicity on the boundary (the paper gives a nontrivial construction to exclude cycles) . The paper’s arguments appear complete and correct for the stated results, while the model solution’s proof sketch is broadly correct but leaves these justifications implicit.