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The almost periodic Kolmogorov competitive systems. (English) Zbl 1135.34319

In the first part of the paper, the scalar Kolmogorov equation

du dt=ug(t,u)(*)

is considered under the assumptions that g:× + is continuous and uniformly almost periodic in t for u + , that the mean value of g(t,0) is positive and the mean value of g(t,k 0 ) is not positive for some k 0 >0, and that g u (t,u)-q(t)p(u) for (t,x)× + , where q and p are continuous and satisfy q(t)0 for t, p(u)>0 for u>0, moreover there are two positive constants λ and α such that

t t+λ q(s)ds>αforallt·

Under these assumptions, equation (*) has a unique positive almost periodic solution. Using this result, the author formulates similar conditions for the system

du i dt=u i f i (t,u 1 ,,u n ),1in

to have at least one positive almost periodic solution. Finally, he applies the result to Lotka-Volterra systems.

MSC:
34C27Almost and pseudo-almost periodic solutions of ODE
92D25Population dynamics (general)