Pattern formation and global analysis of a systematically reduced plant model in dryland environment

The paper linearizes the space-time system (1.2) at the positive equilibrium E11, expands in Neumann modes with θn = n^2/l^2 on (0, lπ), and derives the 2×2 modal characteristic polynomial λ^2 − (r11 + r22 − (d1 + d2)θn)λ + [d1d2θn^2 − (d1r22 + d2r11)θn + r11r22 − r12r21] = 0, exactly as the candidate does. Solving r11 + r22 = 0 gives the spatially homogeneous Hopf curve a = a* in closed form; setting λ = 0 in the dispersion relation yields the Turing loci d1 = [d2 r11 θn − (r11 r22 − r12 r21)]/[θn(d2 θn − r22)], with the positivity/threshold condition n > l sqrt[(r11 r22 − r12 r21)/(d2 r11)]. The paper then shows monotone dependence of d1,T^(n) on a, defines f(y) and proves unimodality with the maximizer y*, hence selects n# via the nearest-integer rule and identifies the codimension-2 (0, n#) Hopf–Turing point at (a*, d1,T^(n#)(a*)) and the (n, n+1) Turing–Turing points where curves intersect. All these statements and formulas coincide with the candidate’s steps, including the transversality checks (paper: dReλ/da|_{a=a*} = −1 and dλ/dd1|_{d1=d1,T} < 0; candidate: ∂aτ0(a*) = −2 implying the same, and ∂d1Δn > 0). Minor differences are expository only (the paper also proves that d1,T^(n) decreases in a). Therefore, the two arguments are essentially the same in substance and correct. See the paper’s equations (3.5)–(3.7) and Theorem 3.1 for the Hopf, Turing, Hopf–Turing, and Turing–Turing claims, the eigenmode setup and r_ij entries, and the f(y), y*, and n# construction (including the tie-break rule), as well as the transversality computations and the segmentation by a_n (all matching the candidate solution).