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

Abstract

A population model described by a nonlinear delay differential equation with a quadratic nonlinearityẋ(t)=∑k=1mαk(t)x(hk(t))-β(t)x2(t),t⩾0is considered where m⩾1 is an integer, functions αk,β:[0,∞)→(0,∞) are continuous, functions hk:[0,∞)→R are continuous such that t-τ⩽hk(t)⩽t,τ=const,τ>0, and, for any t⩾0, the inequality hj(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.