Kinetic Behaviors of Catalysis-Driven Growth of Three-Species Aggregates on Base of Exchange-Driven Aggregations

We propose a solvable aggregation model to mimic the evolution of population A, asset B, and the quantifiable resource C in a society. In this system, the population and asset aggregates themselves grow through self-exchanges with the rate kernels K₁(k, j) = K₁kj and K₂(k, j) = K₂kj, respectively. The actions of the population and asset aggregations on the aggregation evolution of resource aggregates are described by the population-catalyzed monomer death of resource aggregates and asset-catalyzed monomer birth of resource aggregates with the rate kernels J₁(k, j) = J₁k and J₂(k, j) = J₂k, respectively. Meanwhile, the asset and resource aggregates conjunctly catalyze the monomer birth of population aggregates with the rate kernel I₁(k, i, j) = I₁ki^μj^η, and population and resource aggregates conjunctly catalyze the monomer birth of asset aggregates with the rate kernel I₂(k, i, j) = I₂ki^vj^η. The kinetic behaviors of species A, B, and C are investigated by means of the mean-field rate equation approach. The effects of the population-catalyzed death and asset-catalyzed birth on the evolution of resource aggregates based on the self-exchanges of population and asset appear in effective forms. The coefficients of the effective population-catalyzed death and the asset-catalyzed birth are expressed as J_1e = J₁/K₁ and J_2e = J₂/K₂, respectively. The aggregate size distribution of C species is found to be crucially dominated by the competition between the effective death and the effective birth. It satisfies the conventional scaling form, generalized scaling form, and modified scaling form in the cases of J_1e < J_2e, J_1e = J_2e, and J_1e > J_2e, respectively. Meanwhile, we also find the aggregate size distributions of populations and assets both fall into two distinct categories for different parameters μ, ν, and η: (i) When μ = ν = η = 0 and μ = ν = 0, η = 1, the population and asset aggregates obey the generalized scaling forms; and (ii) When μ = ν = 1, η = 0, and μ = ν = η = 1, the population and asset aggregates experience gelation transitions at finite times and the scaling forms break down.