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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 $${du\over dt}= ug(t,u)\tag{$*$}$$ is considered under the assumptions that $g: \bbfR\times \bbfR^+\to \bbfR$ is continuous and uniformly almost periodic in $t$ for $u\in\bbfR^+$, 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 $g_u(t, u)\le -q(t) p(u)$ for $(t,x)\in\bbfR\times \bbfR^+$, where $q$ and $p$ are continuous and satisfy $q(t)\ge 0$ for $t\in\bbfR$, $p(u)> 0$ for $u> 0$, moreover there are two positive constants $\lambda$ and $\alpha$ such that $$\int^{t+\lambda}_t q(s)\,ds>\alpha\qquad\text{for all }t\in\bbfR.$$ Under these assumptions, equation $(*)$ has a unique positive almost periodic solution. Using this result, the author formulates similar conditions for the system $${du_i\over dt}= u_i f_i(t, u_1,\dots, u_n),\qquad 1\le i\le n$$ to have at least one positive almost periodic solution. Finally, he applies the result to Lotka-Volterra systems.

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