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 ） = J1 k, population aggregate＇s self-birth rate kernel J2（ k ） = J2k and population-catalyzed fitness birth rate kernel I（k,j） = Ikj＇. The kinetic behavior of the fitness was found depending crucially on the parameter v, which reflects the dependence of the population-catalyzed fitness birth rate on the size of the catalyst （population） aggregate. （i） In the v ≤ 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 αk（t） does not have scale form. （ii） When v ≥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 αk （t） approaches a generalized scaling form.