On a delay population model with a quadratic nonlinearity without positive steady state

A population model described by a nonlinear delay differential equation with a quadratic nonlinearity (x) over dot(t) = Sigma(m)(k=1)alpha(k)(t)x(h(k)(t)) - beta(t)x(2)(t), t >= 0 is considered where m >= 1 is an integer, functions alpha(k), beta : [0, infinity) -> (0, infinity) are continuous, functions h(k) : [0, infinity) -> ER are continuous such that t - tau <= h(k)(t) <= t, tau = const, tau > 0, and, for any t >= 0, the inequality h(j)(t) < t holds for at least one index j is an element of {1, ..., m}. Although this equation does not have a positive steady state, a new method not based on the existence of a positive steady state is developed and used to investigate the permanence, global attractivity conditions and nonoscillation properties. (C) 2013 Elsevier Inc. All rights reserved.