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 ? ( t ) = ? k = 1 m α k ( t ) x ( h k ( t ) ) - β ( t ) x 2 ( t ) , t ? 0 is considered where m ? 1 is an integer, functions α k , β : 0 , ∞ ) ? ( 0 , ∞ ) are continuous, functions h k : 0 , ∞ ) ? R are continuous such that t - ? ≤ h k ( t ) ≤ t , ? = const , ? > 0 , and, for any t ? 0 , the inequality h j ( t ) < t holds for at least one index j ? { 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.