Rate-induced tipping in heterogeneous reaction-diffusion systems: An invariant manifold framework and geographically shifting ecosystems

The paper correctly identifies and numerically documents B‑tipping at a critical habitat length and R‑tipping at a critical habitat speed as saddle‑node folds of pulse solutions detected via a quadratic tangency of W^u and W^s in a compactified moving‑frame ODE, and continues these folds in two parameters. However, it does not provide rigorous proofs of existence, uniqueness, or stability of pulses; the results are computational and framework‑oriented. The model’s solution gives a plausible geometric–analytic proof sketch (manifold dimensions, Σ‑section, generic quadratic tangency, order‑preserving PDE stability) but asserts stronger, largely unproved statements (e.g., at most two pulses via a claimed unimodality/zero‑number argument for the shooting map, and global exactness of “two pulses for all L > Lcrit” across parameter ranges). Hence both are incomplete: the paper in analytical rigor, the model in over‑general claims and missing hypotheses.