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For autonomous systems of ordinary differential equations with fractional-rational right-hand sides for which the nonnegative orthant is positively invariant, sufficient conditions for the existence of nontrivial localizing sets corresponding to coordinate functions are found. Conditions for the existence of an attractor for such systems are established, and its position is investigated. For systems modeling population dynamics, the relationship of the results obtained here with the conditions for the extinction or survival of populations is indicated. [ABSTRACT FROM AUTHOR]