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

MSC:

34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
92D25 Population dynamics (general)
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References:

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