The effects of coupling on pattern formation in a simple autocatalytic system

The spatiotetnporal structures that can arise in two identicalcells, each governed by cubic autocatalator kinetics and coupledvia the diffusive interchange of a reactant, are discussed.The coupling gives rise to five spatially uniform steady states,one of which exists in the uncoupled system. By studying thelinearized equations, it is found that three of these steadystates, including that of the uncoupled system, may give riseto the possibility of bifurcations to spatially nonuniform steadystates. In the case of the steady state corresponding to thatof the uncoupled system, it is seen that the coupling leadsto bifurcations not present in the uncoupled system which giverise to locally stable nonuniform steady states. A weakly nonlinearanalysis is developed for both small and large coupling strengtha, and for parameter values in a neighbourhood of the bifurcationpoints on the new steady states. This clarifies the nature ofthe nonuniform solutions close to bifurcation, which are thenfollowed numerically using a path-following technique. The couplingis found to produce extra nonuniform steady solutions whichare stable close to their bifurcation points.     

A Moving Boundary Problem for the Sphere

A new technique is employed to solve a moving boundary problemin a sphere. A transformation is given which considerably simplifiesthe non-linearities associated with the moving boundary andenables an iterative analytic series solution to be found. Threeterms of the series expansion are obtained analytically andnumerically evaluated for comparison with results of previousauthors. Graphs of the motion of the reaction (freezing) frontand concentration (temperature) profiles behind the front aregiven. Also tabulated are the times to complete reaction (freezing)of the particle (that is the time for the moving boundary toreach the centre of the sphere).     

Production planning towards the end of a product life cycle

Previously published papers have derived optimal policies forreplenishing stock when demand is declining according to somedeterministic law. Here we elaborate on that work, providinga full analysis, and extend the analysis back in time by consideringthe period of relatively level demand which is likely to havepreceded the period of decline in demand. We show how optimalinventory replenishment policies extending back to this periodof level demand can be derived.