  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 $c(p)\sim p^2$.
