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

我们建议一个可解决的聚集模型在一个社会模仿人口 A，财产 B，和可以计量的资源 C 的进化。在这个系统，人口和财产总数自己与率内核 K 1(k, j 通过自我交换成长)= K 1 kj 和 K 2(k, j )= K 2 kj 分别地。资源总数的聚集进化上的人口和财产聚集的行动被资源总数的催化人口的单体死亡和资源总数的催化财产的单体出生与率核 J 1(k, j 描述)= J 1 k 和 J 2(k, j )= J 2 k ，分别地。同时，财产和资源总数接合用率核催化人口总数的单体出生我1(k, i ， j )=我 1 ki μ j η,和人口和资源总数接合用率核催化财产总数的单体出生我2(k, i ， j )=我 2 ki v j η。种类 A， B，和 C 的运动行为借助于吝啬地的率方程途径被调查。资源总数的进化上的催化人口的死亡和催化财产的出生的效果基于人口和财产的自我交换出现在有效形式。有效催化人口的死亡和催化财产的出生的系数分别地被表示为 J 1e= J 1/K1 和 J 2e= J 2/K2, 。C 种类的总数尺寸分发被发现被在有效死亡和有效出生之间的比赛关键地统治。它在 J 1e【 J 2e, J 1e= J 2e, 和 J 1e 的情况中满足常规可伸缩形式，概括可伸缩的形式，和修改可伸缩的形式 】 J 2e, 分别地。同时，我们也发现总数人口的尺寸分布，财产两个都为不同参数 μ ， ν ，和 η 掉进二个不同范畴：(i) 什么时候 μ =ν =η = 0 并且 μ =ν = 0， η = 1，人口和财产总数服从概括可伸缩形式；并且(i i ) 什么时候 μ =ν = 1， η = 0，并且 μ =ν =η = 1，人口和财产总数经历在有限时间的转变和可伸缩的形式毁坏的冻结。

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 selfexchanges with the rate kernels Kl（k,j） = K1kj and K2（h,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 kerne/s 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^μjη, and population and resource aggregates conjunctly catalyze the monomer birth of asset aggregates with the rate kernel /2（k, i, j） = I2ki^νj^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 J1e = J1/K1 and J2e= 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 J1e〈J2e, J1e=J2e, and J1e〉J2e, 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=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.