Evolution equation of population genetics: relation to the density-matrix theory of quasiparticles with general dispersion laws.

The Waxman-Peck theory of population genetics is discussed in regard of soil bacteria. Each bacterium is understood as a carrier of a phenotypic parameter p. The central objective is the calculation of the probability density with respect to p, Phi(p,t;p(0)), of the carriers living at time t>0, provided that initially at t(0)=0, all bacteria carried the phenotypic parameter p(0)=0. The theory involves two small parameters: the mutation probability mu and a parameter gamma involved in a function w(p) defining the fitness of the bacteria to survive the generation time tau and give birth to an offspring. The mutation from a state p to a state q is defined by a Gaussian with a dispersion sigma(2)(m). The author focuses our attention on a function phi(p,t) which determines uniquely the function Phi(p,t;p(0)) and satisfies a linear equation (Waxman's equation). The Green function of this equation is mathematically identical with the one-particle Bloch density matrix, where mu characterizes the order of magnitude of the potential energy. (In the x representation, the potential energy is proportional to the inverted Gaussian with the dispersion sigma(2)(m)). The author solves Waxman's equation in the standard style of a perturbation theory and discusses how the solution depends on the choice of the fitness function w(p). In a sense, the function c(p)=1-w(p)/w(0) is analogous to the dispersion function E(p) of fictitious quasiparticles. In contrast to Waxman's approximation, where c(p) was taken as a quadratic function, c(p) approximately gammap(2), the author exemplifies the problem with another function, c(p)=gamma[1-exp(-ap(2))], where gamma is small but a may be large. The author shows that the use of this function in the theory of the population genetics is the same as the use of a nonparabolic dispersion law E=E(p) in the density-matrix theory. With a general function c(p), the distribution function Phi(p,t;0) is composed of a delta-function component, N(t)delta(p), and a blurred component. When discussing the limiting transition for t--> infinity, the author shows that his function c(p) implies that N(t)-->N( infinity ) not equal 0 in contrast with the asymptotics N(t)-->0 resulting from the use of Waxman's function c(p) approximately p(2).