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 monoiner 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 K-1(k, j) = K(1)kj and K-2(k, j) = K(2)kj, the fitness aggregate's self-death rate kernel J(1)(k) = J(1)k, population aggregate's self-birth rate kernel J(2)(k) = J(2)k and population-catalyzed fitness birth rate kernel I(k,j) = Ikj(upsilon). 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 upsilon <= 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 riot have scale form. (ii) When upsilon > 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.