The role of harvesting and growth rate for spatially heterogeneous populations

The paper studies the Neumann two-species competition–diffusion model with harvesting proportional to r(x) and classifies the global dynamics across five parameter regimes for (μ, ν), using principal eigenvalue “invasion” tests and a monotone semiflow classification theorem (Appendix Theorem 9, after Hsu–Smith–Waltman) to conclude either global coexistence (for μ, ν < 1 in a neighborhood of the diagonal) or convergence to a semi-trivial/trivial equilibrium when one or both harvesting rates exceed the growth rate. These are stated as Theorems 4–8 and proved via instability of semi-trivial equilibria and the repelling nature of the origin when appropriate , with the abstract classification framed in Appendix A (Theorem 9) . The candidate solution follows the same conceptual route—principal eigenvalues for invasibility and a general strongly monotone, eventually compact semiflow theorem—while adding standard PDE regularity/compactness details and a convenient w_max estimate for the double-harvesting case. Minor presentation gaps in the paper (e.g., some hypotheses implicit when appealing to the abstract theorem and to positivity integrals) do not alter the conclusions. Hence, both are correct and rely on substantially the same proof architecture.