Aggregation Behaviors of a Two-Species System with Lose-Lose Interactions

We propose an aggregation evolution model of two-species （A- and B-species） aggregates to study the prevalent aggregation phenomena in social and economic systems. In this model, A- and B-species aggregates perform self-exchange-driven growths with the exchange rate kernels K（k, l） = Kkl and L（k, l） = Lkl, respectively, and the two species aggregates perform self-birth processes with the rate kernels J1（k） = J1 k and J2（ k ） = J2k, and meanwhile the interaction between the aggregates of different species A and B causes a lose-lose scheme with the rate kernel H（k,l） = Hkl. Based on the mean-field theory, we investigated the evolution behaviors of the two species aggregates to study the competitions among above three aggregate evolution schemes on the distinct initial monomer concentrations A0 and B0 of the two species. The results show that the evolution behaviors of A- and B-species are crucially dominated by the competition between the two self-birth processes, and the initial monomer concentrations Ao and Bo play important roles, while the lose-lose scheme play important roles in some special cases.     

Competing role of catalysis-coagulation and catalysis-fragmentation in kinetic aggregation behaviours

We propose a kinetic aggregation model where species A aggregates evolve by the catalysis-coagulation and the catalysis-fragmentation,while the catalyst aggregates of the same species B or C perform self-coagulation processes.By means of the generalized Smoluchowski rate equation based on the mean-field assumption,we study the kinetic behaviours of the system with the catalysis-coagulation rate kernel K(i,j;l) ∝ l ν and the catalysis-fragmentation rate kernel F(i,j;l) ∝ l μ,where l is the size of the catalyst aggregate,and ν and μ are two parameters reflecting the dependence of the catalysis reaction on the size of the catalyst aggregate.The relation between the values of parameters ν and μ reflects the competing roles between the two catalysis processes in the kinetic evolution of species A.It is found that the competing roles of the catalysis-coagulation and catalysis-fragmentation in the kinetic aggregation behaviours are not determined simply by the relation between the two parameters ν and μ,but also depend on the values of these two parameters.When ν μ and ν≥ 0,the kinetic evolution of species A is dominated by the catalysis-coagulation and its aggregate size distribution a k(t) obeys the conventional or generalized scaling law;when ν μ and ν≥ 0 or ν 0 but μ≥ 0,the catalysis-fragmentation process may play a dominating role and a k(t) approaches the scale-free form;and in other cases,a balance is established between the two competing processes at large times and a k(t) obeys a modified scaling law.