Evolution behavior of catalytically activated replication-decline in a coagulation process

We propose a catalytically activated replication-decline model of three species, in which two aggregates of the same species can coagulate themselves, an A aggregate of any size can replicate itself with the help of B aggregates, and the decline of A aggregate occurs under the catalysis of C aggregates. By means of mean-field rate equations, we derive the asymptotic solutions of the aggregate size distribution ak(t) of species A, which is found to depend strongly on the competition among three mechanisms: the self-coagulation of species A, the replication of species A catalyzed by species B, and the decline of species A catalyzed by species C. When the self-coagulation of species A dominates the system, the aggregate size distribution ak(t) satisfies the conventional scaling form. When the catalyzed replication process dominates the system, ak(t) takes the generalized scaling form. When the catalyzed decline process dominates the system, ak(t) approaches the modified scaling form.