Multiple stationary states, sustained oscillations and transient behaviour in autocatalytic reaction-diffusion equations

Abstract

The behaviour of the prototype cubic autocatalator A + 2B → 3B rate = k 1 ab 2 , B → C rate = k 2 b , when reaction is coupled with diffusion from a surrounding reservoir of constant composition is investigated. For indefinitely stable catalysts (i. e. for k 2 = 0) the model exhibits ignition, extinction (wash-out) and hysteresis. The range of conditions over which multiple stationary states are found decreases as the concentration of the autocatalyst b ex in the reservoir increases. Finally ignition and extinction points merge in a cusp catastrophe with the consequent loss of multiplicity. Away from the critical points, the ultimate approach to the stationary state is, in general, governed by an exponential decay. The characteristic relaxation time for this approach lengthens as ignition or extinction points are approached, thus displaying ‘slowing down’. Eventually, non-exponential time-dependences are also found. With finite catalyst lifetimes ( k 2 > 0), the dependence of the stationary-state composition on the diffusion rate or the size of the reaction zone shows more complex patterns. Five qualitatively different responses can be found: (i) unique, (ii) isola, (iii) breaking wave, (iv) mushroom and (v) breaking wave + isola. The stationary-state profile for the distribution of the autocatalytic species B now allows multiple internal extrema (the onset of dissipative structures). The cubic autocatalator also provides the simplest, yet chemically consistent, example of temporal and spatial oscillations in a reaction-diffusion system.