Scaling in Rate-Changeable Birth and Death Processes with Random Removals

We propose a monomer birth-death model with random removals, in which an aggregate of size k can produce a new monomer at a time-dependent rate I（t）k or lose one monomer at a rate J（t）k, and with a probability P（t） an aggregate of any size is randomly removed. We then anedytically investigate the kinetic evolution of the model by means of the rate equation. The results show that the scaling behavior of the aggregate size distribution is dependent crucially on the net birth rate I（t） - J（t） as well as the birth rate I（t）. The aggregate size distribution can approach a standard or modified scaling form in some cases, but it may take a scale-free form in other cases. Moreover, the species can survive finally only if either I（t） - J（t） ≥ P（t） or [J（t） ＋ P（t） - I（t）]t ≈ 0 at t ≥ 1; otherwise, it will become extinct.     

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.     

Kinetics of catalytically activated duplication in aggregation growth

We propose a catalytically activated duplication model to mimic the coagulation and duplication of the DNA polymer system under the catalysis of the primer RNA. In the model, two aggregates of the same species can coagulate themselves and a DNA aggregate of any size can yield a new monomer or double itself with the help of RNA aggregates. By employing the mean-field rate equation approach we analytically investigate the evolution behaviour of the system. For the system with catalysis-driven monomer duplications, the aggregate size distribution of DNA polymers a_k(t) always follows a power law in size in the long-time limit, and it decreases with time or approaches a time-independent steady-state form in the case of the duplication rate independent of the size of the mother aggregates, while it increases with time increasing in the case of the duplication rate proportional to the size of the mother aggregates. For the system with complete catalysis-driven duplications, the aggregate size distribution a_k(t) approaches a generalized or modified scaling form.     

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.     

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 (A 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 andB-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 I_A(K)=Ik, catalyzed death rate kernel J_AB(k)=Jk and self exchange rate kernel K_A(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 effectiverate KA₀ play important roles in the cases of I>JB₀ and I=JB₀. The evolution behaviors of the total mass M^A(t)₁ and the total aggregate number M^A(t)₀ are obtained, and the aggregate size distribution a_k(t) of species A is found toapproach a generalized scaling form in the case of I ≧ JB₀ and a special modified scaling form in the case of I0. The periodical evolution of the B-monomers concentration plays an exponential form of the periodic modulation.     

Migration-driven aggregate growth on scale-free networks

We study the kinetics of migration-driven aggregate growth on completely connected scale-free networks. A reversible migration system is considered with the size-dependent rate kernel K(k; l/i;j) approximately k(u)i(v)(lj)(v), at which an i-mer aggregate located on the node with j links gains one monomer from a k-mer aggregate on the node with l links. The results show that the evolution behavior of the aggregate size distribution is drastically different from that for the corresponding same system in normal space. This model can be used to mimic some phenomena such as the distribution of city populations. Moreover, we verify our analytic results in good agreement with the data of the population distributions of all U.S. counties.     

Competition between aggregation and migration processes of a multi-species system

We propose a solvable multi-species aggregation--migration model, in which irreversible aggregations occur between any two aggregates of the same species and reversible migrations occur between any two different species. The kinetic behaviour of an aggregation--migration system is then studied by means of the mean-field rate equation. The results show that the kinetics of the system depends crucially on the details of reaction events such as initial concentration distributions and ratios of aggregation rates to migration rate. In general, the aggregate mass distribution of each species always obeys a conventional or a generalized scaling law, and for most cases at least one species is scaled according to a conventional form with universal constants. Moreover, there is at least one species that can survive finally.     

A Solvable Symbiosis-Driven Growth Model

We introduce a two-species symbiosis-driven growth model, in which two species can mutually benefit for their monomer birth and the self-death of each species simultaneously occurs. By means of the generalized rate equation, we investigate the dynamic evolution of the system under the monodisperse initial condition. It is found that the kinetic behaviour of the system depends crucially on the details of the rate kernels as well as the initial concentration distributions. The cluster size distribution of either species cannot be scaled in most cases; while in some special cases, they both consistently take the universal scaling form. Moreover, in some cases the system may undergo a gelation transition and the pre-gelation behaviour of the cluster size distributions satisfies the scaling form in the vicinity of the gelation point. On the other hand, the two species always live and die together.     

Evolution behavior of catalytically activated replication-decline in a coagulation process

We propose a catalytically activated replication-decline model of three species, in which two aggregates of the same species can coagulate themselves, an A aggregate of any size can replicate itself with the help of B aggregates, and the decline of A aggregate occurs under the catalysis of C aggregates. By means of mean-field rate equations, we derive the asymptotic solutions of the aggregate size distribution ak（t） of species A, which is found to depend strongly on the competition among three mechanisms： the self-coagulation of species A, the replication of species A catalyzed by species B, and the decline of species A catalyzed by species C. When the self-coagulation of species A dominates the system, the aggregate size distribution a~（t） satisfies the conventional scaling form. When the catalyzed replication process dominates the system, ak（t） takes the generalized scaling form. When the catalyzed decline process dominates the system, ak（t） approaches the modified scaling form.     

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

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 1through a monomer-birth catalyzed by a population aggregate Aj of size j is J(k,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 assets,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 ofη =λ= 0,and they obey the modified scalingform 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.     

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.     

Kinetics of a Migration-Driven Aggregation-Fragmentation Process

We propose a reversible model of the migration-driven aggregation-fragmentation process with the sym-metric migration rate kernels K(k;j) = K‘(k;j) = λkjv and the constant aggregation rates I1, I2 and fragmentationrates J1, J2. Based on the mean-field theory, we investigate the evolution behavior of the aggregate size distributions inseveral cases with different values of index v. We find that the fragmentation reaction plays a more important role in the kinetic behaviors of the system than the aggregation and migration. When J1 = 0 and J2 = 0, the aggregate sizedistributions ak(t) and bk(t) obey the conventional scaling law, while when J1 ＞ 0 and J2 ＞ 0, they obey the modifiedscaling law with an exponential scaling function. The total mass of either species remains conserved.     

Kinetic Behaviors of a Competitive Population and Fitness System in Exchange-Driven Growth

We proposed an aggregation model of two species aggregates of fitness and population to study the interaction between the two species in their exchange-driven processes of the same species by introducing the monomer birth of fitness catalyzed by the population, where the fitness aggregates perform self-death process and the population aggregates perform self-birth process. The kinetic behaviors of the aggregate size distributions of the fitness and population were analyzed by the rate equation approach with their exchange rate kernel K₁(k,j)=K₁kj and K₂(k,j)=K₂kj, the fitness aggregate's self-death rate kernel J₁(k)=J₁k, population aggregate's self-birth rate kernel J₂(k)=J₂k and population-catalyzed fitness birth rate kernel I(k,j)=Ikj^u. The kinetic behavior of the fitness was found depending crucially on the parameter u, which reflects the dependence of the population-catalyzed fitness birth rate on the size of the catalyst (population) aggregate. (i) In the u ≤0 case, the effect of catalyzed-birth of fitness is rather weak and the exchange-driven aggregation and self-death of the fitness dominate the process, and the fitness aggregate size distribution a_k(t) does not have scale form. (ii) When u>0, the effect of the population-catalyzed birth of fitness gets strong enough, and the catalyzed-birth and self-death of the fitness aggregates, together with the self-birth of the population aggregates dominate the evolution process of the fitness aggregates. The aggregate size distribution a_k(t) approaches a generalized scaling form.     

Kinetics of a Migration－Driven Aggregation－Fragmentation Process

We propose a reversible model of the migration-driven aggregation-fragmentation process with the symmetric migration rate kernels K(k;j)=K'(k;j)=λkj^υ and the constant aggregation rates I₁, I₂ and fragmentation rates J₁, J₂. Based on the mean-field theory, we investigate the evolution behavior of the aggregate size distributions in several cases with different values of index υ. We find that the fragmentation reaction plays a more important role in the kinetic behaviors of the system than the aggregation and migration. When J₁=0 and J₂ =0, the aggregate size distributions a_k(t) and b_k(t) obey the conventional scaling law, while when J₁>0 and J₂>0, they obey the modified scaling law with an exponential scaling function. The total mass of either species remains conserved.     

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.     

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.     

Solvable Aggregation-Migration-Annihilation Processes of a Multispecies System

An aggregation-migration-annihilation model is proposed for a two-species-group system. In the system,aggregation reactions occur between any two aggregates of the same species and migration reactions between two different species in the same group and joint annihilation reactions between two species from different groups. The kinetics of the system is then investigated in the framework of the mean-field theory. It is found that the scaling solutions of the aggregate size distributions depend crucially on the ratios of the equivalent aggregation rates of species groups to the annihilation rates. Each species always scales according to a conventional or modified scaling form; moreover, the governing scaling exponents are nonuniversal and dependent on the reaction details for most cases.     

Solvable Aggregation-Migration-Annihilation Processes of a Multispecies System

An aggregation-migration-annihilation model is proposed for a two-species-group system. In the system, aggregation reactions occur between any two aggregates of the same species and migration reactions between two different species in the same group and joint annihilation reactions between two species from different groups. The kinetics of the system is then investigated in the framework of the mean-field theory. It is found that the scaling solutions of the aggregate size distributions depend crucially on the ratios of the equivalent aggregation rates of species groups to the annihilation rates. Each species always scales according to a conventional or modified scaling form; moreover, the governing scaling exponents are nonuniversal and dependent on the reaction details for most cases.