Propagation-Window Bounds in Degenerate Plant-Consumer Reaction-Diffusion Systems

We study traveling waves in a class of high-dimensional plant-consumer reaction-diffusion systems with $N$ competing plant species and $N$ associated consumer populations. The consumer equations are linearly degenerate at the extinction state. Under the weak-interaction condition $\|K\|_{\infty}<1$, the system admits a unique positive coexistence equilibrium. Define $s*{\mathrm P}:=\max_{1\le i\le N}2\sqrt{d_i r_i},\, s_{\mathrm C}^{\mathrm{ex}}:=\min_{1\le i\le N}2\sqrt{D_i e_i\sigma_i},\, s_{\mathrm C}^{\mathrm{nec}}:=\min_{1\le i\le N}2\sqrt{D_i e_i}$, where $\sigma_i:=1-\sum_{j=1}^N\kappa_{ij}$, and let $\mathcal S$ be the set of speeds admitting a strictly positive traveling wave connecting extinction to coexistence. If $s_{\mathrm P}<s_{\mathrm C}^{\mathrm{ex}}$, we prove $(s_{\mathrm P},s_{\mathrm C}^{\mathrm{ex}})\subseteq \mathcal S \subseteq [s_{\mathrm P},s_{\mathrm C}^{\mathrm{nec}})$. The existence result is based on a new hybrid lower-solution construction that removes the diffusion-separation and auxiliary compatibility assumptions from the earlier four-component theory. Plant lower profiles combine a polynomial-exponential leading-edge bump with a logistic bridge, while consumer lower profiles are Gaussian near the leading edge and constant behind the front. Nonexistence below $s_{\mathrm P}$ follows from a Sturm-type oscillation argument, whereas leading-edge asymptotics and a Riccati crossing argument yield the universal upper obstruction $s<s_{\mathrm C}^{\mathrm{nec}}$. Finally, for a fixed two-species system, we construct an explicit wave with $s_{\mathrm P}<s_{\mathrm C}^{\mathrm{ex}}<s_{\mathrm{exact}}<s_{\mathrm C}^{\mathrm{nec}}$, showing that $s_{\mathrm C}^{\mathrm{ex}}$ is only a constructive threshold, not an intrinsic maximal wave speed.