Kinetics of Infection-Driven Growth Model with Birth and Death

We propose a two-species infection model, in which an infected aggregate can gain one monomer from a healthy one due to infection when they meet together. Moreover, both the healthy and infected aggregates may lose one monomer because of self-death, but a healthy aggregate can spontaneously yield a new monomer. Consider a simple system in which the birth/death rates are directly proportional to the aggregate size, namely, the birth and death rates of the healthy aggregate of size k are J1 k and J2k while the self-death rate of the infected aggregate of size k is J3k. We then investigate the kinetics of such a system by means of rate equation approach. For the J1 〉 J2 case, the aggregate size distribution of either species approaches the generalized scaling form and the typical size of either species increases wavily at large times. For the J1 = J2 case, the size distribution of healthy aggregates approaches the generalized scaling form while that of infected aggregates satisfies the modified scaling form. For the J1 〈 J2 case, the size distribution of healthy aggregates satisfies the modified scaling form, but that of infected aggregates does not scale.     

Kinetics of Infection-Driven Growth Model with Birth and Death

We propose a two-species infection model, in which an infected aggregate can gain one monomer from a healthy one due to infection when they meet together. Moreover, both the healthy and infected aggregates may lose one monomer because of self-death, but a healthy aggregate can spontaneously yield a new monomer. Consider a simple system in which the birth/death rates are directly proportional to the aggregate size, namely, the birth and death rates of the healthy aggregate of size k are J₁k and J₂k while the self-death rate of the infected aggregate of size k is J₃k. We then investigate the kinetics of such a system by means of rate equation approach. For the J₁>J₂ case, the aggregate size distribution of either species approaches the generalized scaling form and the typical size of either species increases wavily at large times. For the J₁=J₂ case, the size distribution of healthy aggregates approaches the generalized scaling form while that of infected aggregates satisfies the modified scaling form. For the J₁<J₂ case, the size distribution of healthy aggregates satisfies the modified scaling form, but that of infected aggregates does not scale.     

Aggregation Processes with Catalysis-Driven Decomposition

We propose a three-species aggregation model with catalysis-driven decomposition. Based on the mean-field rate equations, we investigate the evoIution behavior of the system with the size-dependent catalysis-driven decomposition rate J（i; j; k） = Jijk^v and the constant aggregation rates. The results show that the cluster size distribution of the species without decomposition can always obey the conventional scaling law in the case of 0 ≤v ≤ 1, while the kinetic evolution of the decomposed species depends crucially on the index v. Moreover, the total size of the species without decomposition can keep a nonzero value at large times, while the total size of the decomposed species decreases exponentially with time and vanishes finally.     

Aggregation Processes with Catalysis-Driven Decomposition

We propose a three-species aggregation model with catalysis-drivendecomposition. Based on the mean-field rate equations, weinvestigate the evolution behavior of the system with thesize-dependent catalysis-driven decomposition rate J(i;j;k)=J ijk^v and the constant aggregation rates. The results show that the cluster size distribution of the species without decomposition can always obey the conventional scaling law in the case of 0≤ v ≤1, while the kinetic evolution of the decomposed species depends crucially on the index v. Moreover, the total size of the species without decomposition can keep a nonzero value atlarge times, while the total size of the decomposed speciesdecreases exponentially with time and vanishes finally.     

Kinetic Behavior of Exchange-Driven Growth with Catalyzed-Birth Processes

Two catalyzed-birth models of n-species (n≥2) aggregates with exchange-driven growth processes are proposed and compared. In the first one, the exchange reaction occurs between any two aggregates Amk and Amj of the same species with the rate kernels Km (k,j)=Kmkj (m=1, 2,..., n, n≥2), and aggregates of An species catalyze a monomer-birth of Al species (l=1,2,..., n-1) with the catalysis rate kernel Jl(k,j)=Jlkjυ. The kinetic behaviors are investigated by means of the mean-field theory. We find that the evolution behavior of aggregate-size distribution alk(t) of Al species depends crucially on the value of the catalysis rate parameter v: (i) alk(t) obeys the conventional scaling law in the case of υ≤0, (ii) alk (t) satisfies a modified scaling form in the case of υ＞0. In the second model,the mechanism of monomer-birth of An-species catalyzed by Al species is added on the basis of the first model, that is,the aggregates of Al and An species catalyze each other to cause monomer-birth. The kinetic behaviors of Al and Anspecies are found to fall into two categories for the different υ: (i) growth obeying conventional scaling form with υ≤0,(ii) gelling at finite time withυ＞0.     

Scaling Behavior of an Aggregation-Migration Model

We study the kinetic behavior of a two-species aggregation-migration model in which an irreversible aggregation occurs between any two clusters of the same species and a reversible migration occurs simultaneously between two different species. For a simple model with constant aggregation rates and with the migration rates K_A(i;j)=K'_A(i;j) ∝ij^v₁ and K_B(i;j)=K'_B(i;j) ∝ij^v₂, we find that the evolution behavior of the system depends crucially on the values of the indexes v₁ and v₂. The aggregate size distribution of either species obeys a conventional scaling law for most cases. Moreover, we also generalize the two-species system to the multi-species case and analyze its kinetic behavior under the symmetrical conditions.     

Kinetics of Catalysis-Driven Aggregation Processes with Sequential Input of Catalyst

We investigate the kinetic behavior of a two-species catalysis-driven aggregation model, in which coagulation of species A occurs only with the help of species B. We suppose the monomers of species B are stable and cannot selfcoagulate in reaction processes. Meanwhile, the monomers are continuously injected into the system. The model with a constant rate kernel is investigated by means of the mean-field rate equation. We show that the Mneties of the system depends crucially on the details of the input term. The injection rate of species B is assumed to take the given time- dependent form K（t） -t^λ, and the sealing solution of the duster size distribution is then investigated analytically. It is found that the cluster size distribution can satisfy the conventional or modified scaling form in most cases.     

Monomer Migration and Annihilation Processes

We propose a two-species monomer migration-annihilation model, in which monomer migration reactions occur between any two aggregates of the same species and monomer annihilation reactions occur between two different species. Based on the mean-field rate equations, we investigate the evolution behaviors of the processes. For the case with an annihilation rate kernel proportional to the sizes of the reactants, the aggregation size distribution of either species approaches the modified scaling form in the symmetrical initial case, while for the asymmetrical initial case the heavy species with a large initial data scales according to the conventional form and the light one does not scale. Moreover, at most one species can survive finally. For the case with a constant annihilation rate kernel, both species may scale according to the conventional scaling law in the symmetrical case and survive together at the end.     

Exchange-Driven Growth with Birth Rate Less Than Death

We further study the kinetic behavior of the exchange-driven growth with birth and death for the case of birth rate kernel being less than that of death based on the mean-field theory. The symmetric exchange rate kernel is K(k,j)=K'(k,j)=Ikj^υ, and the birth and death rates are proportional to the aggregate's size. The long time asymptotic behavior of the aggregate size distribution a_k(t) is found to obey a much unusual scaling law with an exponentially growing scaling function Φ(x)=exp(x).     

Kinetic Behavior of Exchange-Driven Growth with Catalyzed-Birth Processes

Two catalyzed-birth models of n-species （n ≥ 2） aggregates with exchange-driven growth processes are proposed and compared. In the first one, the exchange reaction occurs between any two aggregates Ak^m and Af^m of the same species with the rate kernels Km（k,j）= Kmkj （m = 1, 2,... ,n, n ≥ 2）, and aggregates of A^n species catalyze a monomer-birth of A^l species （l = 1, 2 , n - 1） with the catalysis rate kernel Jl（k,j） -Jlkj^v. The kinetic behaviors are investigated by means of the mean-field theory. We find that the evolution behavior of aggregate-size distribution ak^l（t） of A^l species depends crucially on the value of the catalysis rate parameter v： （i） ak^l（t） obeys the conventional scaling law in the case of v ≤ 0, （ii） ak^l（t） satisfies a modified scaling form in the case of v 〉 0. In the second model, the mechanism of monomer-birth of An-species catalyzed by A^l species is added on the basis of the first model, that is, the aggregates of A^l and A^n species catalyze each other to cause monomer-birth. The kinetic behaviors of A^l and A^n species are found to fall into two categories for the different v： （i） growth obeying conventional scaling form with v ≤ 0, （ii） gelling at finite time with v 〉 0.     

Aggregation Behaviors of a Two-Species System with Lose-Lose Interactions

We propose an aggregation evolution model of two-species （A- and B-species） aggregates to study the prevalent aggregation phenomena in social and economic systems. In this model, A- and B-species aggregates perform self-exchange-driven growths with the exchange rate kernels K（k, l） = Kkl and L（k, l） = Lkl, respectively, and the two species aggregates perform self-birth processes with the rate kernels J1（k） = J1 k and J2（ k ） = J2k, and meanwhile the interaction between the aggregates of different species A and B causes a lose-lose scheme with the rate kernel H（k,l） = Hkl. Based on the mean-field theory, we investigated the evolution behaviors of the two species aggregates to study the competitions among above three aggregate evolution schemes on the distinct initial monomer concentrations A0 and B0 of the two species. The results show that the evolution behaviors of A- and B-species are crucially dominated by the competition between the two self-birth processes, and the initial monomer concentrations Ao and Bo play important roles, while the lose-lose scheme play important roles in some special cases.     

Kinetics of an n-Species Aggregation Chain Model with Complete Annihilation

The kinetic behavior of an n-species (n ≥ 3) aggregation-annihilation chain reaction model is studied. In this model, an irreversible aggregation reaction occurs between any two clusters of the same species, and an irreversible complete annihilation reaction occurs only between two species with adjacent number. Based on the rnean-field theory, we investigate the rate equations of the process with constant reaction rates to obtain the asymptotic solutions of the clustermass distributions for the system. The results show that the kinetic behavior of the system not only depends crucially on the ratio of the aggregation rate I to the annihilation rate J, but also has relation with the initial concentration of each species and the species number's odevity. We find that the cluster-mass distribution of each species obeys always a scaling law. The scaling exponents may strongly depend on the reaction rates for most cases, however, for the case in which the ratio of the aggregation rate to the annihilation rate is equal to a certain value, the scaling exponents are only dependent on the initial concentrations of the reactants.     

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 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.     

Novel Biased Aggregation-Annihilation Model

We propose a novel two-species aggregation-annihilation model,in which irreversible aggregation reactions occur between any two aggregates of the same species and biased annihilations occur simultaneously between two different species.The kinetic scaling behavior of the model is then analytically investigated by means of the mean-field rate equation.For the system without the self-aggregation of the un-annihilated species,the aggregate size distribution of the annihilated species always approaches a modified scaling form and vanishes finally; while for the system with the self-aggregation of the un-annihilated species,its scaling behavior depends crucially on t,he details of the rate kernels.Moreover,the results also exhibit that both species are conserved together in some cases,while only the un-annihilated species survives finally in other cases.     

Competition Between Self-birth and Catalyzed Death in Aggregation Growth with Catalysis Injection

We propose two irreversible aggregation growth models of aggregates of two distinct species （.4 and B） to study the interactions between virus aggregates and medicine efficacy aggregates in the virus-medicine cooperative evolution system. The A-species aggregates evolve driven by self monomer birth and B-species aggregate-catalyzed monomer death in model I and by self birth, catalyzed death, and self monomer exchange reactions in model II, while the catalyst B-species aggregates are assumed to be injected into the system sustainedly or at a periodic time-dependent rate. The kinetic behaviors of the A-species aggregates are investigated by the rate equation approach based on the mean-field theory with the self birth rate kernel IA（k） = Ik, catalyzed death rate kernel JAB（k） = Jk and self exchange rate kernel KA （k, l） = Kkl. The kinetic behaviors of the A-species aggregates are mainly dominated by the competition between the two effects of the self birth （with the effective rate I） and the catalyzed death （with the effective rate JB0）, while the effects of the self exchanges of the A-species aggregates which appear in an effective rate KAo play important roles in the cases of I 〉 JBo and I = JBo. The evolution behaviors of the total mass M1^A（t） and the total aggregate number MA（t） are obtained, and the aggregate size distribution ak（t） of species A is found to approach a generalized scaling form in the case of I ≥ JBo and a special modified scaling form in the case of I 〈 JB0. The periodical evolution of the B-monomers concentration plays an exponential form of the periodic modulation.