Numerical Studies of Bifurcation in Reaction-Diffusion Models Using Pseudo-Spectral and Path-Following Methods

Abstract

A number of authors have studied the numerical calculation of time-dependent and steady-state solutions of reaction-diffusion equations, i.e. equations of the form (c.f. De Dier and Roose 1986, Fijii et al. 1982, Manoranjan 1984, Eilbeck 1986),

$$\underline {\text{u}} _{\text{t}\,} \, = \,{\text{d}}\Delta \underline {\text{u}} \, + \,{\text{G}}(\underline {\text{u}},\,\alpha _1,...,\,\alpha _{\text{m}} \,)$$

(1.1)

where u(x,t) ε ℝ^k; x ε Ω c ⊂ ℝⁿ, n = 1,2 or 3; d is a k × k (usually diagonal) matrix of diffusion coefficients; α₁,…α_m are parameters, and F is a nonlinear function F: ℝ^k × ℝ^m → ℝ^k.