Bifurcation and periodic solutions to population models with two dependent delays

Abstract

We investigate the scalar autonomous equation with two discrete delays $$ \dot{x}(t)=f(x(t),x(t-r),x(t-\sigma)), $$ where $f:\mathbb{R}^3\rightarrow \mathbb{R}$ is a continuously differentiable non-linear function such that $f(0,0,0)=0$. It is shown that if the difference between the delays is constant, then one of the delays becomes a Hopf-bifurcation parameter and, in addition, the absolute stability of the trivial solution can be established. Moreover, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by using normal form theory. The main results are applied to guarantee the existence of positive periodic solutions to Nicholson's blowflies and Mackey-Glass models, both with a delayed harvesting term. The conclusions are illustrated by numerical simulations.