On positive solutions and the Omega limit set for a class of delay differential equations

TL;DR

This paper investigates positive solutions of delay differential equations modeling cell populations, providing optimal initial conditions and analyzing their long-term behavior through dynamical systems and omega limit sets.

Contribution

It introduces an optimal initial condition criterion for positivity and characterizes the omega limit set for the long-term dynamics of solutions.

Findings

Established a sufficient condition for positive solutions for all time.

Characterized the omega limit set of the dynamical system.

Provided insights into the long-term behavior of solutions.

Abstract

This paper studies the positive solutions of a class of delay differential equations with two delays. These equations originate from the modeling of hematopoietic cell populations. We give a sufficient condition on the initial function for $t\leq 0$ such that the solution is positive for all time $t>0$. The condition is "optimal". We also discuss the long time behavior of these positive solutions through a dynamical system on the space of continuous functions. We give a characteristic description of the $\omega$ limit set of this dynamical system, which can provide informations about the long time behavior of positive solutions of the delay differential equation.