This paper proposes a controlled particle deposition model for cluster growth on the substrate surface and then presents exact results for the cluster （island） size distribution. In the system, at every time step a fixed number of particles are injected into the system and immediately deposited onto the substrate surface. It investigates the cluster size distribution by employing the generalized rate equation approach. The results exhibit that the evolution behaviour of the system depends crucially on the details of the adsorption rate kernel. The cluster size distribution can take the Poisson distribution or the conventional scaling form in some cases, while it is of a quite complex form in other cases.
We propose a kinetic aggregation model where species A aggregates evolve by the catalysis-coagulation and the catalysis-fragmentation, while the catalyst aggregates of the same species B or C perform self-coagulation processes. By means of the generalized Smoluchowski rate equation based on the mean-field assumption, we study the kinetic behaviours of the system with the catalysis-coagulation rate kernel K（i,j;l） l^v and the catalysis-fragmentation rate kernel F（i,j; l） l^μ, where l is the size of the catalyst aggregate, and v and μ are two parameters reflecting the dependence of the catalysis reaction on the size of the catalyst aggregate. The relation between the values of parameters v and μ reflects the competing roles between the two catalysis processes in the kinetic evolution of species A. It is found that the competing roles of the catalysis-coagulation and catalysis-fragmentation in the kinetic aggregation behaviours are not determined simply by the relation between the two parameters v and μ, but also depend on the values of these two parameters. When v 〉 μ and v ≥0, the kinetic evolution of species A is dominated by the catalysis-coagulation and its aggregate size distribution αk（t） obeys the conventional or generalized scaling law; when v 〈 μ and v ≥ 0 or v 〈 0 but μ≥ 0, the catalysis-fragmentation process may play a dominating role and ak（t） approaches the scale-free form; and in other cases, a balance is established between the two competing processes at large times and ακ（t） obeys a modified scaling law.
This paper proposes a pest propagation model to investigate the evolution behaviours of pest aggregates. A pest aggregate grows by self-monomer birth, and it may fragment into two smaller ones. The kinetic evolution behaviours of pest aggregates are investigated by the rate equation approach based on the mean-field theory. For a system with a self-birth rate kernel I（k） = Ik and a fragmentation rate kernel L（i,j） = L, we find that the total number MoA（t） and the total mass of the pest aggregates MA （t） both increase exponentially with time if L ≠ 0. Furthermore, we introduce two catalysis-driven monomer death mechanisms for the former pest propagation model to study the evolution behaviours of pest aggregates under pesticide and natural enemy controlled pest propagation. In the pesticide controlled model with a catalyzed monomer death rate kernel J1 （k） ---- J1 k, it is found that only when I 〈 J1 B0 （B0 is the concentration of catalyst aggregates） can the pests be killed off. Otherwise, the pest aggregates can survive. In the model of pest control with a natural enemy, a pest aggregate loses one of its individuals and the number of natural enemies increases by one. For this system, we find that no matter how many natural enemies there are at the beginning, pests will be eliminated by them eventually.
We propose a catalysis-select migration driven evolution model of two-species（A-and B-species） aggregates,where one unit of species A migrates to species B under the catalysts of species C,while under the catalysts of species D the reaction will become one unit of species B migrating to species A.Meanwhile the catalyst aggregates of species C perform self-coagulation,as do the species D aggregates.We study this catalysis-select migration driven kinetic aggregation phenomena using the generalized Smoluchowski rate equation approach with C species catalysis-select migration rate kernel K（k;i,j） = Kkij and D species catalysis-select migration rate kernel J（k;i,j） = Jkij.The kinetic evolution behaviour is found to be dominated by the competition between the catalysis-select immigration and emigration,in which the competition is between JD0 and KC0（D0 and C0 are the initial numbers of the monomers of species D and C,respectively）.When JD0 KC0 〉 0,the aggregate size distribution of species A satisfies the conventional scaling form and that of species B satisfies a modified scaling form.And in the case of JD0 KC0 〈 0,species A and B exchange their aggregate size distributions as in the above JD0 KC0 〉 0 case.
