Modeling the Co-evolution of Climate Impact and Population Behavior: A Mean-Field Analysis

For the mean-field system (11) ẋ = x(1−x)(2x+μ ε+α+σ−κ−1), ε̇ = (γ(1−x)−τ)ε under Assumption 2, the paper’s Theorem 1 claims almost all interior trajectories converge to a limit cycle . However, a Bendixson–Dulac certificate with B(x,ε)=1/(x(1−x)ε) on Ω=(0,1)×(0,∞) yields ∂(Bf1)/∂x+∂(Bf2)/∂ε=2/ε>0, excluding any periodic orbit in the interior. The interior equilibrium is indeed an unstable spiral as the paper shows , but the subsequent appeal to the Poincaré–Bendixson theorem misapplies the hypothesis about the limit set’s location and overlooks the Dulac obstruction; consequently, Theorem 1 is false. The model’s solution correctly identifies this via Bendixson–Dulac.