On a delay population model with quadratic nonlinearity

Abstract

Abstract A nonlinear delay differential equation with quadratic nonlinearity, x ˙ ( t ) = r ( t ) [ ∑ k = 1 m α k x ( h k ( t ) ) − β x 2 ( t ) ] , t ≥ 0 , is considered, where α k and β are positive constants, h k : [ 0 , ∞ ) → R are continuous functions such that t − τ ≤ h k ( t ) ≤ t , τ = const , τ > 0 , for any t > 0 the inequality h k ( t ) < t holds for at least one k, and r : [ 0 , ∞ ) → ( 0 , ∞ ) 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.