On positive solutions and the omega limit set for a class of delay differential equations (Q380055)

This paper considers delay differential equations of the form NEWLINE\[NEWLINE\begin{aligned} \frac{dy}{dt}&= -\gamma u+ f(u(t-\tau_0))- f(u(t-\tau))e^{-\gamma(\tau-\tau_0)}, \quad t >0,\\ u(t)&=\phi(t),\quad \tau \leq t\leq 0,\end{aligned}NEWLINE\]NEWLINE where \(\tau > \tau_0 > 0\), \(f\in C(\mathbb {R}^+,\mathbb {R}^+)\) and \(\phi (t) > 0\). These equations are originated from mathematical model of hematopoietic dynamics. The authors give an optimal formulation of initial conditions for \(t \leq 0\) such that the solutions are positive for \(t >0\). Long time behaviors of these positive solutions are also discussed for a dynamical system defined in the space of continuous functions. A characteristic description of the \(\omega\)-limit set of this dynamical system is obtained. This provides information on the long time behavior of positive solutions of delay differential equations.