Kinetic Behavior of Aggregation-Exchange Growth Process with Catalyzed-Birth

We propose an aggregation model ofa two-species system to mimic thegrowth of cities' population and assets, in which irreversiblecoagulation reactions and exchange reactions occur between any twoaggregates of the same species, and the monomer-birth reactions of one species occur by the catalysis of the other species. In the case with population-catalyzed birth of assets, the rate kernel of an asset aggregate B_k of size k grows to become an aggregate B_k+1 through a monomer-birth catalyzed by a populationaggregate A_j of size j is J(k,j)=Jkj^λ. And inmutually catalyzed birth model, the birth rate kernels of populationand assets are H(k,j)=Hkj^η and J(k,j)=Jkj^λ, respectively. The kinetics of the system is investigated based on the mean-field theory. In the model ofpopulation-catalyzed birth of assets, the long-time asymptotic behavior of the assets aggregate size distribution obeys the conventional or modified scaling form. In mutually catalyzed birth system, the asymptotic behaviors of population and assets obey theconventional scaling form in the case of η=λ=0, and they obey the modified scaling form in the case of η=0,λ=1. In the case of η=λ=1, the total mass of population aggregates and that of asset aggregates both growmuch faster than those in population-catalyzed birth of assetsmodel, and they approaches to infinite values in finite time.