Kinetics of aggregation growth with competition between catalyzed birth and catalyzed death

An aggregation growth model of three species A, B and C with the competition between catalyzed birth and catalyzed death is proposed. Irreversible aggregation occurs between any two aggregates of the like species with the constant rate kernels I_n (n=1,2,3). Meanwhile, a monomer birth of an A species aggregate of size k occurs under the catalysis of a B species aggregate of size j with the catalyzed birth rate kernel K(k,j)=Kkj^\upsilon, and a monomer death of an A species aggregate of size k occurs under the catalysis of a C species aggregate of size j with the catalyzed death rate kernel L(k,j)=Lkj^\upsilon, where \upsilon is a parameter reflecting the dependence of the catalysis reaction rates of birth and death on the size of catalyst aggregate. The kinetic evolution behaviours of the three species are investigated by the rate equation approach based on the mean-field theory. The form of the aggregate size distribution of A species a_k(t) is found to be dependent crucially on the competition between the catalyzed birth and death of A species, as well as the irreversible aggregation processes of the three species: (1) In the \upsilon<0 case, the irreversible aggregation dominates the process, and a_k(t) satisfies the conventional scaling form; (2) In the \upsilon\geq 0 case, the competition between the catalyzed birth and death dominates the process. When the catalyzed birth controls the process, a_k(t) takes the conventional or generalized scaling form. While the catalyzed death controls the process, the scaling description of the aggregate size distribution breaks down completely.