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

Abstract

The Waxman-Peck theory of the population genetics is discussed in regard of soil bacteria. Each bacterium is understood as a carrier of a phenotypic parameter p. The central aim is the calculation of the probability density with respect to p of the carriers living at time t>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 offspring. The mutation from a state p to a state q is defined by a Gaussian. The author focuses attention on an equation generalizing Waxman's equation. The author solves this 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. With a general function c(p), the distribution function ${\mathit\Phi}(p,t;0)$ is composed of a delta-function component, $N(t)\delta(p)$, and a blurred component. The author shows that asymptotically N(t) may tend to a positive value, in contrast with zero resulting from Waxman's approximation where $c(p)\sim p^2$.