Kinetic Behavior of Aggregation-Exchange Growth Process with Catalyzed-Birth

我们建议一个二种类的系统的一个聚集模型模仿城市的生长“人口和财产，和一种的单体出生反应，不可逆的凝结反应和置换反应在发生在一样的种的任何二个总数之间由另外的种的催化作用发生。在有财产的催化人口的出生的情况中，尺寸 k 的总数 B_k 变得通过单体出生成为总数 B_(k+1 ) 的财产的率核由尺寸 j 的人口总数 A_j 催化是 J (k， j )=Jkj~ λ。并且在互相催化的出生模型，，人口和财产的出生率核是 H (k， j )=Hkj~η a nd J (k， j )=Jkj~ λ分别地。系统的动力学基于 themean 地理论被调查。在财产的催化人口的出生的模型，财产总数尺寸分发的长期的 asymptoticbehavior 服从常规或修改的可伸缩的形式。在互相催化的出生系统，人口和财产的 asymptotic 行为在η = λ = 的情况中服从常规可伸缩形式 0，并且他们在η = 的情况中服从修改可伸缩的形式 0，λ = 1。在η = λ = 的情况中 1，人口总数和财产的全部的团聚集两个在财产的催化人口的出生变得比那些快得多模型，和他们在有限时间走近到无限的 vaiues。

We propose an aggregation model of a two-species system to mimic the growth of cities＇ population and assets, in which irreversible coagulation reactions and exchange reactions occur between any two aggregates of the same species, and the monomer-birth reactions of one species occur by the catalysis Of the other species. In the case with population-catalyzed birth of assets, the rate kernel of an asset aggregate Bκ of size k grows to become an aggregate Bκ＋1 through a monomer-birth catalyzed by a population aggregate Aj of size j is J（κ,j） = Jkjλ. And in mutually catalyzed birth model, the birth rate kernels of population and assets are H（k,j）=Hkjη and J（k,j） = Jkjλ, respectively. The kinetics of the system is investigated based on the mean-field theory. In the model of population-catalyzed birth of aseets, the long-time asymptotic behavior of the assets aggregate size distribution obeys the conventional or modified scaling form. In mutually catalyzed birth system, the asymptotic behaviors of population and assets obey the conventional scaling form in the case of η=λ =0, and they obey the modified scaling form in the case of η=0, λ=1. In the case of η = λ= 1, the total mass of population aggregates and that of asset aggregates both grow much faster than those in population-catalyzed birth of assets model, and they approaches to infinite values in finite time.