Existence of localised radial patterns in a model for dryland vegetation

The paper proves existence of three small-amplitude localized radial families near a Turing bifurcation (Spot A, Rings, Spot B) for the reduced 4D r-ODE (3.8), with Bessel-core expansions and scalings O(ε), O(ε^{3/2}), and O(ε^{3/4}); it computes c0 = ω/4 and c3 with threshold ω* = 30/23, and then invokes established radial normal-form/blow-up results to conclude existence and decay, matching expansions (3.27)–(3.29) and the core basis {J0, rJ1, 2J0} stated in the paper . The candidate solution independently constructs a core manifold from the Frobenius/Bessel modes, a far-field stable manifold via roughness of exponential dichotomies, and closes with a Lyapunov–Schmidt reduction on adjoint core modes to obtain the same normal-form scalings, constants (−c1/κ1 = 3/ω), and ω* = 30/23. Minor paper issues include a sign typo for c3 in Lemma 3.5 versus Appendix A.2, but the threshold and theorems are consistent overall .