AbstractWe consider the problem whereaandfare 1‐periodic int, ais positive,fsatisfies appropriate decreasing conditions; smoothness ofa, f, ∂Ω is also assumed. Denote by λ0the principal eigenvalue of Δ with zero Dirichlet boundary conditions, and define . We prove: (a) if ε ≤ 0, then no non‐negative periodic solution exists but zero, and any solution with continuous non‐negative initial datum converges to zero uniformly ast→ ∞; (b) if ε>0, then a unique non trivial non‐negative 1‐periodic solutionu* exists, and any solution with continuous, non‐negative not identically zero initial datum appro