Global attractor in competitive Lotka-Volterra systems with retardation (Q1871712)

The author considers an autonomous system of delayed differential equations which generalizes the classical Lotka-Volterra system, namely, \[ x_i'(t)=b_i x_i(t)\Biggl(1-\sum_{j=1}^{n} L_{ij}((x_j)_t)\Biggr), \quad i=1,2,\dots , N, \] where \(b_i>0\), each \(L_{ij}\) is a bounded positive linear operator defined in \(C([-r,0])\), \(r>0\), and, as usual, \((x_j)_t(s)=x_j(t+s)\), \(s\in [-r,0]\). The main results are devoted to find sufficient conditions for the existence of a global attractor for this system. Comparisons with previous results are also established; in particular, some results given in section 5.7 of the book by \textit{H. L. Smith} [Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, Mathematical Surveys and Monographs. 41. Providence, RI: American Mathematical Society (AMS) (1995; Zbl 0821.34003)] are improved.