On a delay population model with quadratic nonlinearity

A nonlinear delay differential equation with quadratic nonlinearity <(X)over dot>(t) = r(t) [Sigma(m)(k=1)alpha X-k(h(k)(t) - beta X-2(t)], t >= 0, is considered, where alpha(k) and beta are positive constants, hk : [0, infinity). R are continuous functions such that t - tau <= h(k)( t) <= t, tau = const, tau > 0, for any t > 0 the inequality h(k)( t) < t holds for at least one k, and r : [0, infinity).( 0,8) is a continuous function satisfying the inequality r(t) >= r(0) = const for an r(0) > 0. It is proved that the positive equilibrium is globally asymptotically stable without any further limitations on the parameters of this equation.