ADDITIONAL FOOD CAUSES PREDATOR "EXPLOSION" - UNLESS THE PREDATORS COMPETE

The paper’s Theorem 2.2 claims that for the Holling II model (2), if β−δα>0 and ξ>δ/(β−δα), then every positive initial condition yields predator blow-up in infinite time, (x(t),y(t))→(0,∞). The proof crucially asserts dy/dt≥0 by invoking “WLOG β>δ,” and uses this to conclude all trajectories move toward the y-axis and cannot approach (γ,0), which it also describes as a saddle. But β>δ is not implied by β−δα>0; the ‘WLOG’ step is unjustified, and the sign of g(γ)=β(γ+ξ)/(1+αξ+γ)−δ can be negative, making (γ,0) a hyperbolic sink, not a saddle. A concrete counterexample satisfying the paper’s stated hypotheses (e.g., α=0.1, β=0.5, δ=1, ξ=3, γ=10) gives g(γ)<0, so (γ,0) is locally asymptotically stable and nearby positive trajectories converge to (γ,0), contradicting Theorem 2.2. The proof’s phase-plane monotonicity dy/dt≥0 and the universal “no interior equilibrium” conclusion both fail without the missing β>δ or a stronger condition like inf_{x≥0} g(x)>0. Hence the theorem, as stated, is false, while the model’s correction is valid. See the theorem statement and proof steps in the paper, including the “WLOG β>δ” line and the classification of (γ,0), in Theorem 2.2 and its proof and related discussion ; cf. the surrounding summary and small/large-food regimes and the boundary classification remarks used in the argument .