In the first part of the paper, the scalar Kolmogorov equation
is considered under the assumptions that is continuous and uniformly almost periodic in for , that the mean value of is positive and the mean value of is not positive for some , and that for , where and are continuous and satisfy for , for , moreover there are two positive constants and such that
Under these assumptions, equation has a unique positive almost periodic solution. Using this result, the author formulates similar conditions for the system
to have at least one positive almost periodic solution. Finally, he applies the result to Lotka-Volterra systems.