BIFURCATION IN A SPHERICAL REACTION DIFFUSION SYSTEM WITH IMPOSED GRADIENT

Bifurcation of steady states is studied for the system ∂φ/∂t = (D + ηΦD′)Hφ + F(φ) + αG(φ), where D, D' are p x p matrices, H is a self-adjoint operator on a Hilbert space H of functions on a d-dimensional domain, Φ is a given function, φε{lunate} Hp, and η, α are real parameters. Considering η as a perturbation parameter, we develop a general method for successive construction of primary, secondary and further bifurcation points and branches. Motivated by a previously studied biological problem connected to cytokinesis (cell division) in early blastulas, we examine the branching of solutions from a homogeneous state in the three-sphere, with H the Laplacian operator and Φ a simple gradient from one pole of the sphere to the other. F and G are nonlinear functions describing autocatalytic biochemical reaction kinetics, and in such systems, spontaneous pattern formation may occur (Turing system of the second kind). We examine the splitting of bifurcation points for parameters, which yields φ {reversed tilde equals} δjn(κnlτ) Ynm(ω) with n = 2 in the unperturbed (η = 0) problem. Previous studie s have shown a bipolar solution as the only one in this case (i.e. m = 0), but with imposed gradient, we show here, that although such a branch still emerges, new branches arise with m ≠ 0, confirming previous numerical studies. © 1990.