Kinetic evolutionary behavior of catalysis-select migration

We propose a catalysis-select migration driven evolution model of two-species (A- and B-species) aggregates, where one unit of species A migrates to species B under the catalysts of species C, while under the catalysts of species D the reaction will become one unit of species B migrating to species A. Meanwhile the catalyst aggregates of species C perform self-coagulation, as do the species D aggregates. We study this catalysis-select migration driven kinetic aggregation phenomena using the generalized Smoluchowski rate equation approach with C species catalysis-select migration rate kernel K(k;i,j)=Kkij and D species catalysis-select migration rate kernel J(k;i,j)=Jkij. The kinetic evolution behaviour is found to be dominated by the competition between the catalysis-select immigration and emigration, in which the competition is between JD₀ and KC₀ (D₀ and C₀ are the initial numbers of the monomers of species D and C, respectively). When JD₀-KC₀>0, the aggregate size distribution of species A satisfies the conventional scaling form and that of species B satisfies a modified scaling form. And in the case of JD₀-KC₀<0, species A and B exchange their aggregate size distributions as in the above JD₀-KC₀>0 case.