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Global stability in delay differential systems without dominating instantaneous negative feedbacks. (English) Zbl 0828.34066
The systems \(x_i'(t)= x_i(t) f_i(t, x_t(\cdot))\), \(i= 1,\dots, n\) and \(x_i'(t)= x_i(t) f_i(t, x_t(\cdot), y_t(\cdot))\), \(i= 1,\dots, n\), \(y_j'(t)= g_j(t, x_t(\cdot), y_t(\cdot))\), \(j= 1,\dots, m\) are considered with \(f_i\) and \(g_j\) continuous linear operators.
It is shown that if the systems have globally asymptotically stable nonnegative equilibria in the absence of the delays and if the systems are dissipative if delays are present (solutions are eventually uniformly bounded), then the equilibria remain globally asymptotically stable as long as the time lags involved in some part of the negative feedbacks are small enough.

MSC:
34K20 Stability theory of functional-differential equations
92D25 Population dynamics (general)
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