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) = Ikjυ, and the birth and death rates are proportional to the aggregate's size. The long time asymptotic behavior of the aggregate size distribution ak(t) is found to obey a much unusual scaling law with an exponentially growing scaling function Φ(x) = exp (x).