Propagation-Window Bounds in Degenerate Plant-Consumer Reaction-Diffusion Systems
We study traveling waves in high-dimensional plant--consumer reaction--diffusion systems with $N$ competing plants and $N$ associated consumers. We focus on the degenerate case in which consumer growth vanishes when plant populations are zero, so the standard Fisher--KPP linearization does not determine the leading-edge behavior. Under a weak-interaction condition, we identify a plant-driven lower threshold $s_{\mathrm P}$, a constructive upper threshold $s_{\mathrm C}^{\mathrm{ex}}$, and a universal necessary upper threshold $s_{\mathrm C}^{\mathrm{nec}}$, and prove $(s_{\mathrm P},s_{\mathrm C}^{\mathrm{ex}})\subseteq \mathcal S\subseteq[s_{\mathrm P},s_{\mathrm C}^{\mathrm{nec}})$, where $\mathcal S$ is the set of speeds admitting positive extinction-to-coexistence waves. Existence follows from new lower solutions that remove previously imposed diffusion and compatibility restrictions. Nonexistence below $s_{\mathrm P}$ follows from a Sturm argument, while center-manifold analysis and a Riccati crossing argument yield the finite upper-speed obstruction. An explicit two-species wave beyond $s_{\mathrm C}^{\mathrm{ex}}$ shows that this constructive threshold is not the true maximal speed.