On the positive almost periodic solutions of a class of Lotka-Volterra type systems with delays. (English) Zbl 0967.34064

The following almost-periodic Lotka-Volterra type systems with delays are investigated \[ {dx_i(t)\over dt}= x_i(t)(a_i(t)- b_i(t) x_i(t)- f_i(t, x_t)),\quad i= 1,2,\dots, n,\tag{1} \] with \(t\in\mathbb{R}\), \(x(t)= (x_1(t),x_2(t),\dots, x_n(t))\in \mathbb{R}^n\), \(x_t= (x_{1t}, x_{2t},\dots, x_{nt})\in C^n[-\tau, 0]\), and \(x_{it}(s)= x_i(t+ s)\), \(i= 1,2,\dots, n\), for all \(s\in [-\tau,0]\); \(a_i(t)\) and \(b_i(t)\) are continuous almost-periodic functions with respect to \(t\in \mathbb{R}\) and \(b_i(t)\geq 0\) for all \(t\in\mathbb{R}\).
The author establishes a general criterion for the existence of positive almost-periodic solutions to system (1). The conditions required in the criterion are quite weak, and can be easily checked and reduced to some well-known results.


34K14 Almost and pseudo-almost periodic solutions to functional-differential equations
92D25 Population dynamics (general)
34K20 Stability theory of functional-differential equations
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