Temporal Oscillations Induced by Boundary Conditions in Systems with and Without Spatial Patterns

Abstract

We investigate the influence of boundary conditions on a semiinfinite 2-component reaction-diffusion model. These boundary conditions are thought of as arising from a membrane that may be chemically active. Thus we consider the set of equations

$${\partial _t}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{n} = \underline{\underline D} \partial _X^2\underline n + \underline f \left( {\underline n } \right) x < O$$

(1)

$$\underline{\underline D} {\partial _x}\underline n = \underline g \left( {\underline n } \right) x < O$$

(2)

where n is a concentration vector, D a (diagonal) diffusion matrix, and f and g are the reaction rates in the bulk and in the membrane, respectively. For this model, the influence of boundary solutions on bulk patterns — with emphasis on wavelength selection — have been discussed in [1]. Here we concentrate on oscillatory boundary solutions arising from the interplay of bulk and membrane (for details see [2]). We find that they can still occur when neither oscillations in the bulk nor in the membrane alone are possible as is the case when autocatalysis is confined to the bulk and coupling of the two reactants to the membrane or vice versa.