Thermal phototactic bioconvection in an isotropic porous medium heated from above

The paper’s linearized model includes Beer–Lambert shading feedback, producing nonlocal terms (ℵ1∫Θ, k^2ℵ2Θ, ℵ0) and boundary couplings (via Φ) in the cell equation and boundary conditions; all are retained in Eqs. (43)–(52) and (46)–(57). The candidate solution drops these terms and replaces the cell dynamics by a local drift–diffusion operator, then asserts energy identities (E3–E6) and accretivity that do not hold for the paper’s operator. As a result, the candidate’s “exact” monotonicity proofs (in RT and Le, for every k) do not apply to the paper’s model. Nevertheless, the paper’s numerical findings—RT stabilizes (Rc_B increases) and Le destabilizes (Rc_B decreases), and oscillations are most pronounced when the sublayer sits around z≈3/4 and are suppressed as RT grows—are consistent and explicitly documented in the results and conclusion.