We study a chemical reaction-diffusion model (the Brusselator) for pattern formation on developing plant tips. A family of spherical cap domains is used to represent tip flattening during development. Applied to conifer embryos, we model the chemical prepatterning underlying cotyledon ("seed leaf") formation, and demonstrate the dependence of patterns on tip flatness, radius, and precursor concentrations. Parameters for the Brusselator in spherical cap domains can be chosen to give supercritical pitchfork bifurcations of patterned solutions of the nonlinear reaction-diffusion system that correspond to the cotyledon patterns that appear on the flattening tips of conifer embryos.

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a b s t r a c tIn this paper we consider the bifurcation of limit cycles of the systemẋ = y(for ε sufficiently small, where a, b ∈ R − {0}, and P, Q are polynomials of degree n, we obtain that up to first order in ε the upper bounds for the number of limit cycles that bifurcate from the period annulus of the quintic center given by ε = 0 are (3/2)(n + sin 2 (nπ /2)) + 1 if a = b and n − 1 if a = b. Moreover, there are systems with at least (3/2)(n + sin 2 (nπ /2)) + 1 if a = b and, n − 1 limit cycles if a = b.

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